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Title: a tour of support theory for triangulated categories through tensor triangular geometry.

Abstract: These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during the Spring 2015 program on 'Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM in Barcelona.

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A Tour of Support Theory for Triangulated Categories Through Tensor Triangular Geometry

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Greg Stevenson 9

Part of the book series: Advanced Courses in Mathematics - CRM Barcelona ((ACMBIRK))

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These notes give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer’s construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during the Spring 2015 program on ‘Interactions between Representation Theory, Algebraic Topology and Commutative Algebra’ (IRTATCA) at the Centre de Recerca Matemàtica CRM-Barcelona.

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Greg Stevenson

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Matemàtiques, Eix Central, Universitat Autònoma de Barcelona Matemàtiques, Eix Central, Bellaterra, Barcelona, Spain

Dolors Herbera

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Wolfgang Pitsch

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Santiago Zarzuela

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Stevenson, G. (2018). A Tour of Support Theory for Triangulated Categories Through Tensor Triangular Geometry. In: Herbera, D., Pitsch, W., Zarzuela, S. (eds) Building Bridges Between Algebra and Topology. Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70157-8_2

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Tensor-triangular fields: ruminations

The frame of smashing tensor-ideals, adelic models of tensor-triangulated categories, stratified noncommutative geometry, on the surjectivity of the map of spectra associated to a tensor-triangulated functor, triangulated categories in the representation of finite dimensional algebras, prime ideal structure in commutative rings, maximal cohen-macaulay modules and tate-cohomology over gorenstein rings, homotopy limits in triangulated categories, nilpotence and stable homotopy theory ii, related papers (5), the spectrum of prime ideals in tensor triangulated categories, support theory via actions of tensor triangulated categories, generalized tensor idempotents and the telescope conjecture, smashing subcategories and the telescope conjecture - an algebraic approach, triangulated categories: cluster algebras, quiver representations and triangulated categories.

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These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during the Spring 2015 program on 'Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM in Barcelona.

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Stevenson, G. (2018) A tour of support theory for triangulated categories through Tensor triangular geometry. In: Herbera, D., Pitsch, W. and Zarzuela, S. (eds.) Building Bridges Between Algebra and Topology. Series: Advanced courses in mathematics - CRM Barcelona. Birkhäuser: Cham, Switzerland, pp. 63-101. ISBN 9783319701561.

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Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk A TOUR OF SUPPORT THEORY FOR TRIANGULATED CATEGORIES THROUGH TENSOR TRIANGULAR GEOMETRY

GREG STEVENSON

Abstract. These notes attempt to give a short survey of the approach to sup- port theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the ma- terial necessary to state and prove the local-to-global principle. In particular, we discuss Balmer’s construction of the spectrum, generalised Rickard idempo- tents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during the Spring 2015 program on ‘Interactions between Representation Theory, Algebraic Topology and Commutative Algebra’ (IRTATCA) at the CRM in Barcelona.

Contents Introduction 2 1. The Balmer spectrum 2 1.1. Rigid tensor triangulated categories 3 1.2. Ideals and the spectrum 4 1.3. Supports and the Zariski topology 5 1.4. The classiﬁcation theorem 8 1.5. An explicit example 11 2. Generalised Rickard idempotents and supports 12 2.1. Rigidly-compactly generated tensor triangulated categories 12 2.2. Localising sequences and smashing localisations 13 2.3. Generalised Rickard idempotents and supports 16 3. Tensor actions and the local-to-global principle 20 3.1. Actions and submodules 20 3.2. Supports 23 3.3. The local-to-global principle and parametrising submodules 23 3.4. An example: commutative noetherian rings 28 4. Further applications and examples 30 4.1. Singularity categories of hypersurfaces 30 4.2. Representations of categories over commutative noetherian rings 34 References 36

I am grateful to the CRM for both the very pleasant and productive environment provided and the opportunity to give the course which led to these notes during the IRTATCA program. Thanks are also due to the participants in the course for helpful feedback and pointing out several typographical misfortunes in the original version of these notes. 1 2 GREG STEVENSON

Introduction The aim of these notes is to give both an introduction to, and an overview of, certain aspects of the developing ﬁeld of tensor triangular geometry and abstract support varieties. We seek to understand the coarse structure, i.e. lattices of suitable subcategories, of triangulated categories. Put another way, given a triangulated category T and objects X,Y ∈ T, we study the question of when one can obtain Y from X by taking cones, suspensions, and (possibly inﬁnite) coproducts. The notes are structured as follows. In the ﬁrst section we review work of Paul Balmer in the case of essentially small (rigid) tensor triangulated categories. Here one can introduce a certain topological space , the spectrum, which solves the classiﬁcation problem for thick tensor ideals. The spectrum is the universal example of a so-called “support variety” and provides a conceptual framework which unites earlier results in various examples such as perfect complexes over schemes, stable categories of modular representations, and the ﬁnite stable homotopy category. In this section we ﬁrst meet the notion of supports and lay the foundations for deﬁning supports in more general settings. In the second section we turn to the inﬁnite case, i.e. compactly generated tensor triangulated categories. Following work of Balmer and Favi we use the compact objects and the Balmer spectrum to deﬁne a notion of support for objects of such categories. Along the way we discuss smashing localisations and the associated generalised Rickard idempotents which are the key to the deﬁnition of the support given in [BF11]. The third section serves as an introduction to actions of tensor triangulated categories. This allows us to deﬁne a relative version of the supports introduced in the second section; in this way we can, at least somewhat, escape the tyranny of monoidal structures. After introducing actions, the associated support theory, and outlining some of the fundamental lemmas concerning supports and actions, we come to the main abstract result of this course - the local-to-global principle. This theorem, which already provides new insight in the situation of Section 2, reduces the study of lattices of localising subcategories to the computation of these lattices in smaller (and hopefully simpler) subcategories. Finally, the fourth section focusses on illustrating some applications of the ab- stract machinery from the preceding sections; there will be, of course, examples along the way, but in this section we concentrate on giving more details on some more recent examples where the machinery of actions has been successfully ap- plied. In particular, we discuss singularity categories of aﬃne hypersurfaces and the corresponding classiﬁcation problem for localising subcategories as well as ap- plications to studying derived categories of representations of quivers over arbitrary commutative noetherian rings.

1. The Balmer spectrum The main reference for this section is the paper [Bal05] by Paul Balmer. We follow his exposition fairly closely, albeit with two major diﬀerences: ﬁrstly, in order to simplify the discussion, and since we will not require it later on, we do not work in full generality, and secondly we omit many of the technical details. The interested reader should consult [Bal05], [Bal10] and the references within for further details. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 3

This section mainly consists of deﬁnitions, but we provide several examples on the way and ultimately get to Balmer’s classiﬁcation of thick tensor ideals in terms of the spectrum. 1.1. Rigid tensor triangulated categories. Throughout this section K will de- note an essentially small triangulated category. We use lowercase letters k, l, m for objects of K and denote its suspension functor by Σ. We begin by introducing the main player in this section. Deﬁnition 1.1. An essentially small tensor triangulated category is a triple (K, ⊗, 1), where K is an essentially small triangulated category and (⊗, 1) is a symmetric monoidal structure on K such that ⊗ is an exact functor in each variable. Slightly more explicitly, − ⊗ −: K × K −→ K is a symmetric monoidal structure on K with unit 1 and with the property that, for all k ∈ K, the endofunctors k ⊗ − and − ⊗ k are exact. Remark 1.2. Throughout we shall not generally make explicit the associativity, symmetry, and unit constraints for the symmetric monoidal structure on a tensor triangulated category. By standard coherence results for monoidal structures this will not get us into any trouble. Deﬁnition 1.3. Let K be an essentially small tensor triangulated category. Assume that K is closed symmetric monoidal, i.e. for each k ∈ K the functor k ⊗ − has a right adjoint which we denote hom(k, −). These functors can be assembled into a bifunctor hom(−, −) which we call the internal hom of K. By deﬁnition one has, for all k, l, m ∈ K, the tensor-hom adjunction K(k ⊗ l, m) =∼ K(l, hom(k, m)), with corresponding units and counits

ηk,l : l −→ hom(k, k ⊗ l) and k,l : hom(k, l) ⊗ k −→ l. The dual of k ∈ K is the object k∨ = hom(k, 1). Given k, l ∈ K there is a natural evaluation map k∨ ⊗ l −→ hom(k, l), which is deﬁned by following the identity map on l through the composite

∼ K(l⊗k,1,l) ∼ K(l, l) / K(l ⊗ 1, l) / K(k ⊗ k∨ ⊗ l, l) / K(k∨ ⊗ l, hom(k, l)). We say that K is rigid if for all k, l ∈ K this natural evaluation map is an isomor- phism k∨ ⊗ l −→∼ hom(k, l). Remark 1.4. If K is rigid then, given k ∈ K, there is a natural isomorphism (k∨)∨ =∼ k, and the functor k∨ ⊗ − is both a left and a right adjoint to k ⊗ −, the functor given by tensoring with k. Example 1.5. Let us provide some standard examples of essentially small rigid tensor triangulated categories: 4 GREG STEVENSON

(1) Given a commutative ring R the category Dperf (R) of perfect complexes i.e., those complexes in the derived category which are quasi-isomorphic to a bounded complex of ﬁnitely generated projectives, is symmetric monoidal L via the left derived tensor product ⊗R (which we will usually denote just as ⊗R or ⊗ if the ring is clear) with unit the stalk complex R sitting in degree 0. The category Dperf (R) is easily checked to be rigid. (2) Let G be a ﬁnite group and k be a ﬁeld whose characteristic divides the order of G (this is not necessary but rules out trivial cases). We write mod kG for the category of ﬁnite dimensional kG-modules. This is a Frobenius category, so its stable category mod kG, which is obtained by factoring out maps factoring through projectives, is triangulated (see [Hap88] for instance for more details). Moreover, it is a rigid tensor triangulated category via the usual tensor product ⊗k with the diagonal action and unit object the trivial representation k. (3) The ﬁnite stable homotopy category SHﬁn together with the smash product of spectra is a rigid tensor triangulated category with unit object the sphere spectrum S0. 1.2. Ideals and the spectrum. From this point onward K denotes an essentially small rigid tensor triangulated category with tensor product ⊗ and unit 1. For much of what follows the rigidity assumption is overkill, but it simpliﬁes the discussion in several places and will be a necessary hypotheses in the sections to come. We begin by recalling the subcategories of K with which we will be concerned. All subcategories of K are assumed to be full and replete (i.e. closed under isomor- phisms). The simple, but beautiful, idea is to view K as a very strange sort of ring. One takes thick subcategories to be the analogue of additive subgroups and then deﬁnes ideals and prime ideals in the naive way. Deﬁnition 1.6. A triangulated subcategory I of K is thick if it is closed under taking direct summands, i.e. if k ⊕ k0 ∈ I then both k and k0 lie in I. To be very explicit: a full subcategory I of K is thick if it is closed under suspensions, cones, and direct summands. A thick subcategory I of K is a (thick) tensor-ideal if given any k ∈ K and l ∈ I the tensor product k ⊗ l lies in I. Put another way we require the functor K × I −→⊗ K to factor through I. Finally, a proper thick tensor-ideal P of K is prime if given k, l ∈ K such that k ⊗ l ∈ P then at least one of k or l lies in P. We will often just call prime tensor-ideals in K prime ideals. Remark 1.7. Since we have assumed K is essentially small the collections of thick subcategories, thick tensor-ideals, and prime tensor-ideals each form a set. Remark 1.8. Rigidity of K provides the following simpliﬁcation when dealing with tensor-ideals. Given k ∈ K the adjunctions between k ⊗ − and k∨ ⊗ − imply that k is a summand of k ⊗ k ⊗ k∨ and that k∨ is a summand of k ⊗ k∨ ⊗ k∨. The former implies that every tensor-ideal is radical, i.e. if I is a tensor-ideal and k⊗n ∈ I then k ∈ I. The latter implies that every tensor-ideal is closed under taking duals. An important special case of the above discussion is the following: if k ∈ K is nilpotent, i.e. k⊗n =∼ 0, then k =∼ 0. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 5

Given a collection of objects S ⊆ K we denote by thick(S) (resp. thick⊗(S)) the smallest thick subcategory (resp. thick tensor-ideal) containing S. We use Thick(K) and Thick⊗(K) respectively to denote the sets of thick subcategories and thick tensor-ideals of K. Both of these sets of subcategories are naturally ordered by inclusion and form complete lattices whose meet is given by intersection. Lemma 1.9. Suppose that K is generated by the tensor unit, i.e. thick(1) = K. Then every thick subcategory is a tensor-ideal: Thick(K) = Thick⊗(K).

Proof. Exercise. Example 1.10. The lemma applies to both Dperf (R) and SHﬁn which are generated by their respective tensor units R and S0. It is not necessarily the case that mod kG is generated by its tensor unit k. However, if G is a p-group then the trivial module k is the unique simple kG-module and thus generates mod kG. The prime ideals, somewhat unsurprisingly, receive special attention (and nota- tion). Deﬁnition 1.11. The spectrum of K is the set Spc K = {P ⊆ K | P is prime} of prime ideals of K. We next record some elementary but crucial facts about prime ideals and Spc K. Proposition 1.12 ([Bal05, Proposition 2.3]). Let K be as above. (a) Let S be a set of objects of K containing 1 and such that if k, l ∈ S then k ⊗ l ∈ S. If S does not contain 0 then there exists a P ∈ Spc K such that P ∩ S = ∅. (b) For any proper thick tensor-ideal I ( K there exists a maximal proper thick tensor-ideal M with I ⊆ M. (c) Maximal proper thick tensor-ideals are prime. (d) The spectrum is not empty: Spc K 6= ∅. Remark 1.13. It is worth noting that, as in the common proof of the analogous statement in commutative algebra, the proof of this proposition appeals to Zorn’s lemma. 1.3. Supports and the Zariski topology. We now deﬁne, for each object k ∈ K, a subset of prime ideals at which k is “non-zero”. This is the central construction of the section, allowing us to both put a topology on Spc K and to understand Thick⊗(K) in terms of the resulting topological space. Deﬁnition 1.14. Let k be an object of K. The support of k is the subset supp k = {P ∈ Spc K | k∈ / P}. We denote by U(k) = {P ∈ Spc K | k ∈ P} the complement of supp k. 6 GREG STEVENSON

Let us give some initial intuition for the way in which one can think of k as being supported at P ∈ supp k. We can form the Verdier quotient K/P of K by P, which comes with a canonical projection π : K −→ K/P. Since P is thick the kernel of π is precisely P and so, as k∈ / P, the object π(k) is non-zero in the quotient; this is the sense in which k is supported at P. The reason this may, at ﬁrst, look at odds with the analogous deﬁnition in commutative algebra is that one inverts morphisms in a triangulated category by taking such quotients. One can, at least in many cases, make a precise connection between the support deﬁned above and the usual support of modules over a (graded) commutative ring. We now list the main properties of the support. Lemma 1.15 ([Bal05, Lemma 2.6]). The assignment k 7→ supp k given by the support satisﬁes the following properties: (a) supp 1 = Spc K and supp 0 = ∅; (b) supp(k ⊕ l) = supp(k) ∪ supp(l); (c) supp(Σk) = supp k; (d) for any distinguished triangle k −→ l −→ m −→ Σk in K there is a containment supp l ⊆ (supp k ∪ supp m); (e) supp(k ⊗ l) = supp(k) ∩ supp(l). Proof. We refer to Balmer’s paper for the details, where the corresponding results are proved for the subsets U(k). These properties, as stated above, appear in [Bal05, Deﬁnition 3.1]. However, we suggest proving these statements as doing so provides an instructive exercise. Another very important property of the support is that it can detect whether or not an object is zero.

Lemma 1.16. Given k ∈ K we have supp k = ∅ if and only if k =∼ 0. Proof. By [Bal05, Corollary 2.4] the support of k is empty if and only if k is tensor- nilpotent i.e., k⊗n =∼ 0. As K is rigid there are no non-zero tensor-nilpotent objects by Remark 1.8. A fairly immediate consequence of Lemma 1.15 is that the family of subsets {supp k | k ∈ K} form a basis of closed subsets for a topology on Spc K. Deﬁnition 1.17. The Zariski topology on Spc K is the topology deﬁned by the basis of closed subsets {supp k | k ∈ K} The closed subsets of Spc K are of the form Z(S) = {P ∈ Spc K | S ∩ P = ∅} \ = supp(k) k∈S where S ⊆ K is an arbitrary family of objects. Before continuing to the universal property of Spc K and the classiﬁcation theo- rem we discuss some topological properties of Spc K. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 7

Proposition 1.18 ([Bal05, Proposition 2.9]). Let P ∈ Spc K. The closure of P is {P} = {Q ∈ Spc K | Q ⊆ P}.

In particular, Spc K is T0 i.e., for P1, P2 ∈ Spc K

{P1} = {P2} ⇒ P1 = P2.

Proof. Let S0 = K \ P denote the complement of P. It is immediate that P ∈ Z(S0) and one easily checks that if P ∈ Z(S) then S ⊆ S0. Thus for any such S we have Z(S0) ⊆ Z(S), i.e. Z(S0) is the smallest closed subset containing P. This shows

{P} = Z(S0) = {Q ∈ Spc K | Q ⊆ P} as claimed. The assertion that Spc K is T0 follows immediately. Remark 1.19. This proposition is the ﬁrst indication of the mental gymnastics that occur when dealing with Spc K versus Spec R for a commutative ring R.A wealth of further information on the relationship between prime ideals in R and prime tensor-ideals in Spc Dperf (R) can be found in [Bal10].

In fact Spc K is much more than just a T0 space. Deﬁnition 1.20. A topological space X is a spectral space if it veriﬁes the following properties:

(1) X is T0; (2) X is quasi-compact; (3) the quasi-compact open subsets of X are closed under ﬁnite intersections and form an open basis of X; (4) every non-empty irreducible closed subset of X has a generic point. Given spectral spaces X and Y , a spectral map f : X −→ Y is a continuous map such that for any quasi-compact open U ⊆ Y the preimage f −1(U) is quasi-compact. Typical examples of spectral spaces are given by Spec R where R is a commu- tative ring. In fact Hochster has shown in [Hoc69] that any spectral topological space is of this form. Further examples include the topological space underlying any quasi-compact and quasi-separated scheme. Another class of examples is provided by the following result. Theorem 1.21 ([BKS07]). Let K be as above. The space Spc K is spectral. Thus the spaces we produce by taking spectra of tensor triangulated categories are particularly nice and enjoy many desirable properties. Aside 1.22. There is a deep connection between spectral spaces and the theory of particularly nice lattices (coherent frames to be precise). One consequence of this is that the topology of a spectral space is determined by the specialisation ordering on points. This ordering is given by deﬁning, for a spectral space X, a point y ∈ X to be a specialisation of x ∈ X if y is in the closure of x. Viewed through the lens of lattice theory, the key fact which makes Balmer’s theory work so well is that the lattice of thick tensor-ideals is distributive. More details on this point of view can be found in [KP13]. As a complement to Theorem 1.21 let us record a few related facts which appear in Balmer’s work. 8 GREG STEVENSON

Proposition 1.23. For any rigid tensor triangulated category K the following as- sertions hold. (a) Any non-empty closed subset of Spc K contains at least one closed point. (b) For any k ∈ K the open subset U(k) = Spc K \ supp(k) is quasi-compact. (c) Any quasi-compact open subset of Spc K is of the form U(k) for some k ∈ K. Proof. Both results can be found in [Bal05]: the ﬁrst is Corollary 2.12, and the second two are the content of Proposition 2.14. In particular, it follows from the above proposition and Lemma 1.15 that the U(k) for k ∈ K give a basis of quasi-compact open subsets for Spc K that is closed under ﬁnite intersections as is required in the deﬁnition of a spectral space. Remark 1.24. Recall, or prove as an exercise, that a space is noetherian (i.e. satisﬁes the descending chain condition for closed subsets) if and only if every open subset is quasi-compact. Combining this with (c) above we see that if Spc K is noetherian then every open subset is of the form U(k) for some k ∈ K and hence every closed subset is the support of some object. 1.4. The classiﬁcation theorem. We now come to the ﬁrst main result of these notes, namely the abstract classiﬁcation of thick tensor-ideals of K in terms of Spc K. The starting point to this story is an abstract axiomatisation of the properties of the support, based on Lemma 1.15. Deﬁnition 1.25. A support data on (K, ⊗, 1) is a pair (X, σ) where X is a topolog- ical space and σ is an assignment associating to each object k of K a closed subset σ(k) of X such that: (a) σ(1) = X and σ(0) = ∅; (b) σ(k ⊕ l) = σ(k) ∪ σ(l); (c) σ(Σk) = σ(k); (d) for any distinguished triangle k −→ l −→ m −→ Σk in K there is a containment σ(l) ⊆ (σ(k) ∪ σ(m)); (e) σ(k ⊗ l) = σ(k) ∩ σ(l). A morphism of support data f :(X, σ) −→ (Y, τ) on K is a continuous map f : X −→ Y such that for every k ∈ K the equality σ(k) = f −1(τ(k)) is satisﬁed. An isomorphism of support data is a morphism f of support data where f is a homeomorphism . It should, given the deﬁnition of the support, come as no surprise that the pair (Spc K, supp) is a universal support data for K. Theorem 1.26 ([Bal05, Theorem 3.2]). Let K be as throughout. The pair (Spc K, supp) is the terminal support data on K. Explicitly, given any support data (X, σ) on K there is a unique morphism of support data f :(X, σ) −→ (Spc K, supp), i.e. a unique continuous map f such that σ(k) = f −1 supp(k) SUPPORT THEORY FOR TRIANGULATED CATEGORIES 9 for all k ∈ K. This unique morphism is given by sending x ∈ X to f(x) = {k ∈ K | x∈ / σ(k)}. Sketch of proof. We give a brief sketch of the argument. We have seen in Lemma 1.15 that the pair (Spc K, supp) is in fact a support data for K. By the same lemma it is clear that for x ∈ X the full subcategory f(x), deﬁned as in the statement, is a proper thick tensor-ideal: it is proper by (a), closed under summands by (b), closed under Σ by (c), closed under extensions by (d), and a tensor-ideal by (e). To see that it is prime suppose k ⊗ l lies in f(x). By axiom (e) for a support data we have x∈ / σ(k ⊗ l) = σ(k) ∩ σ(l). and hence x must fail to lie in at least one of σ(k), σ(l) implying that one of k or l lies in f(x). The map f is a map of support data as f(x) ∈ supp k if and only if k∈ / f(x) if and only if x ∈ σ(k), i.e. f −1 supp(k) = {x ∈ X | f(x) ∈ supp k} = {x ∈ X | x ∈ σ(k)} = σ(k). This also proves continuity of f by deﬁnition of the Zariski topology on Spc K. We leave the unicity of f to the reader (or it can be found as [Bal05, Lemma 3.3]). Before stating the promised classiﬁcation theorem we need one more deﬁnition (we also provide two bonus deﬁnitions which will be useful both here and later). Deﬁnition 1.27. Let X be a spectral space and let W ⊆ X be a subset of X. We say W is specialisation closed if for any w ∈ W and w0 ∈ {w} we have w0 ∈ W . That is, W is specialisation closed if it is the union of the closures of its elements. Given w, w0 as above we call w0 a specialisation of w. Dually, we say W is generisation closed if given any w0 ∈ W and a w ∈ X such that w0 ∈ {w} then we have w ∈ W . In this situation we call w a generisation of w0. A Thomason subset of X is a subset of the form [ Vλ λ∈Λ where each Vλ is a closed subset of X with quasi-compact complement. Note that any Thomason subset is specialisation closed. We denote by Thom(X) the collection of Thomason subsets of X. It is a poset with respect to inclusion and in fact also carries the structure of a complete lattice where the join is given by taking unions. Remark 1.28. As observed in Remark 1.24 if X is noetherian then every open subset is quasi-compact. Thus the Thomason subsets of X are precisely the spe- cialisation closed subsets. Remark 1.29. By Proposition 1.23 for any k ∈ K the subset supp(k) is a Thomason subset of Spc K, as is any union of supports of objects. Part (c) of the same proposition implies the converse, namely that any Thomason subset V can be written as a union of supports of objects of k. This observation explains the role of Thomason subsets in the following theorem. Theorem 1.30 ([Bal05, Theorem 4.10]). The assignments [ σ : Thick⊗(K) −→ Thom(Spc K), σ(I) = supp(k) k∈I 10 GREG STEVENSON and

τ : Thom(Spc K) −→ Thick⊗(K), τ(V ) = {k ∈ K | supp(k) ⊆ V }, for I ∈ Thick⊗(K) and V ∈ Thom(K) give an isomorphism of lattices

Thick⊗(K) =∼ Thom(K). Sketch of proof. By the previous remark and the properties of the support given in Lemma 1.15 both assignments are well deﬁned, i.e. σ(I) is a Thomason subset and τ(V ) is a thick tensor-ideal. It is clear that both morphisms are inclusion preserving, so we just need to check that they are inverse to one another. Consider the thick tensor-ideal τσ(I). It is clear that I ⊆ τσ(I). The equality is proved by showing that \ I = P = τσ(I). P∈Spc K I⊆P We refer to [Bal05] for the details. On the other hand consider the Thomason subset στ(V ). In this case it is clear that στ(V ) ⊆ V . By the remark preceding the theorem we know V can be written as a union of supports of objects and so this containment is in fact an equality.

Aside 1.31. By Stone duality the lattice Thom(Spc K) actually determines Spc K. So by the theorem if we know Thick⊗(K) we can recover Spc K. In fact this gives another way of describing the universality of Spc K.

To conclude we brieﬂy explain what the spectrum is in each of our current running examples and then give a slightly more detailed treatment of a particular case.

Example 1.32. Let R be a commutative ring. Then

Spc Dperf (R) =∼ Spec R. This computation is due to Neeman [Nee92] (and Hopkins) in the case that R is noetherian and Thomason [Tho97] in general. Applying the above theorem, in combination with Lemma 1.9, tells us that thick subcategories of Dperf (R) are in bijection with Thomason subsets of Spec R.

Example 1.33. As previously let SHﬁn denote the ﬁnite stable homotopy category. The description of the thick subcategories of SHﬁn by Devinatz, Hopkins, and Smith [DHS88] allows one to compute Spc SHﬁn. A nice picture of this space can be found in [Bal10, Corollary 9.5].

Example 1.34. Let G be a ﬁnite group and k a ﬁeld whose characteristic divides the order of G. By a result of Benson, Carlson, and Rickard [BCR97] we have

Spc mod kG =∼ Proj H•(G; k) i.e. the spectrum of the stable category is the space underlying the projective scheme associated to the group cohomology ring. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 11

1.5. An explicit example. We now describe, in some detail but mostly without proofs, the explicit example of Db(Z), the bounded derived category of ﬁnitely generated abelian groups. Our aim is to highlight some of the phenomena involved in studying the spectrum rather than to give a proof of the classiﬁcation in this case. As Z is a hereditary ring, i.e. has global dimension 1, every complex E ∈ Db(Z) is formal. Explicitly, this means that for every E there is an isomorphism M E =∼ Σ−iHi(E). i∈Z Thus to understand tensor-ideals in Db(Z) it is suﬃcient to understand what hap- pens to ﬁnitely generated abelian groups. Recall from Lemma 1.9 that since Z generates Db(Z) every thick subcategory is a tensor-ideal. Our ﬁrst observation is that if p 6= q are distinct primes in Z then /p ⊗L /q =∼ 0. Z Z Z Z Z If P is a prime tensor-ideal then it certainly contains 0. This observation then shows that P must contain Z/pZ for all but possibly one prime p ∈ Z. In fact, this missing prime (or lack thereof) completely determines the prime tensor-ideal: the homeomorphism b φ: Spec Z −→ Spc D (Z) can be described explicitly as

φ((p)) = thick(Z/qZ | q 6= p). In particular, φ sends the 0 ideal to the thick subcategory consisting of all complexes with torsion cohomology, which is the maximal proper thick subcategory of Db(Z). As this maximal proper thick subcategory certainly contains φ((p)) for all p 6= 0 it does indeed give the generic point of Spc Db(Z) under the Zariski topology. On the other hand φ−1 sends a prime tensor-ideal P to

−1 n φ (P) = {n ∈ Z | cone(Z −→ Z) ∈/ P}, which one can check is really a prime ideal. This gives a concrete illustration of the reversal of inclusions (noted in Re- mark 1.19) upon passing from prime ideals in Z to prime tensor-ideals in Db(Z). To localise at a prime ideal (p) ∈ Z we simply invert those elements in Z \ (p). On the other hand, in Db(Z), we invert these elements by killing their cones, i.e. taking the quotient by φ((p)). The analogue is true for general commutative rings—localising at a smaller prime ideal inverts more elements and thus the corresponding thick subcategory in the perfect complexes is larger. One can also use this example to gain further intuition for the support and to somewhat explain the motivation behind deﬁning it in terms of failure to lie in prime tensor-ideals. There is an equivalence, up to idempotent completion, for each prime p ∈ Z b b D (Z)/φ((p))−→D (Z(p)) and one can identify, under this functor, the projection

b π b D (Z) −→ D (Z)/φ((p)) 12 GREG STEVENSON with localisation at (p). Thus, for E ∈ Db(Z), we have φ((p)) ∈ supp E if and only if π(E) 6= 0 if and only if E(p) 6= 0. This identiﬁes the abstract support with the usual homological support of complexes.

2. Generalised Rickard idempotents and supports In this section we use the Balmer spectrum to construct a theory of supports for triangulated categories admitting arbitrary set-indexed coproducts (so being interesting and essentially small become mutually exclusive). Our main reference is [BF11] and again, we follow it relatively closely. 2.1. Rigidly-compactly generated tensor triangulated categories. We start by introducing the main players of this section, rigidly-compactly generated tensor triangulated categories. In order to deﬁne such a creature it makes sense to start with the notion of a compactly generated triangulated category. Deﬁnition 2.1. Let T be a triangulated category admitting all set-indexed coprod- ucts. We say an object t ∈ T is compact if T(t, −) preserves arbitrary coproducts, i.e. if for any family {Xλ | λ ∈ Λ} of objects in T the natural morphism M a T(t, Xλ) −→ T(t, Xλ) λ∈Λ λ∈Λ is an isomorphism. We say T is compactly generated if there is a set G of compact objects of T such that an object X ∈ T is zero if and only if i T(g, Σ X) = 0 for every g ∈ G and i ∈ Z. We denote by Tc the full subcategory of compact objects of T and note that it is an essentially small thick subcategory of T. Example 2.2. Let us give some examples of compactly generated triangulated categories which will be used as illustrations throughout. (1) Let R be a ring and denote by D(R) the unbounded derived category of R. Then D(R) is compactly generated and R is a compact generator for D(R). (2) Let SH denote the stable homotopy category. The sphere spectrum S0 is a compact generator for SH and hence it is compactly generated. (3) Let G be a ﬁnite group and let k be a ﬁeld whose characteristic divides the order of G. Then Mod kG, the stable category of aribtrary kG-modules is a compactly generated triangulated category. Deﬁnition 2.3. A compactly generated tensor triangulated category is a triple (T, ⊗, 1) where T is a compactly generated triangulated category, and (⊗, 1) is a symmetric monoidal structure on T such that the tensor product ⊗ is a coproduct preserving exact functor in each variable and the compact objects Tc form a tensor subcategory. In particular, we require that the unit 1 is compact. Remark 2.4. We will frequently supress the tensor product and unit and just refer to T as a compactly generated tensor triangulated category. Remark 2.5. By Brown representability [Nee96, Theorem 3.1], and the assumption that the tensor product is coproduct preserving, it is automatic that T is closed symmetric monoidal—for each X ∈ T the functor X ⊗ − has a right adjoint which we denote by hom(X, −). SUPPORT THEORY FOR TRIANGULATED CATEGORIES 13

Finally, we come to the combination of hypotheses that we can get the most mileage out of. Deﬁnition 2.6. A rigidly-compactly generated tensor triangulated category T is a compactly generated tensor triangulated category such that the full subcategory Tc of compact objects is rigid. Explicitly, not only is Tc a tensor subcategory of T, but it is closed under the internal hom (which exists by the previous remark) and is rigid in the sense of Deﬁnition 1.3. Example 2.7. Each of the triangulated categories covered in Example 2.2 (with the proviso in (1) that the ring is commutative) is a rigidly-compactly generated tensor triangulated category via the left derived tensor product, smash product, and tensor product over the ground ﬁeld with the diagonal action respectively. 2.2. Localising sequences and smashing localisations. We now review some deﬁnitions concerning the particular subcategories of a rigidly-compactly generated tensor triangulated category that we will wish to consider. Of course some of the deﬁnitions we make are valid more generally, but we will restrict ourselves to what we need. Throughout we ﬁx a rigidly-compactly generated tensor triangulated category T. Deﬁnition 2.8. Let L be a triangulated subcategory of T. The subcategory L is localising if it is closed under arbitrary coproducts in T. Dually, it is colocalising if it is closed under arbitrary products in T. Put another way, a full subcategory of T is localising if it is closed under suspensions, cones, and coproducts and colocalising if it is closed under suspensions, cones, and products. Let I be a localising subcategory of T. We call I a localising tensor-ideal if for any X ∈ T and Y ∈ I the object X ⊗ Y lies in I. Given a family of objects S ⊆ T we denote by loc(S) (resp. loc⊗(S)) the small- est localising subcategory (resp. localising tensor-ideal) of T containing S and by Loc(T) (resp. Loc⊗(T)) the collection of all localising subcategories (resp. localising tensor-ideals) of T. Remark 2.9. There is, a priori, no reason that either Loc(T) or Loc⊗(T) should be sets. In fact, whether or not these collections can form proper classes in the given situation is a longstanding open problem. Remark 2.10. By [Nee01, Proposition 1.6.8] every localising subcategory and every colocalising subcategory of T is automatically closed under direct summands, i.e. (co)localising implies thick. Remark 2.11. One can show, see for instance [Nee96, Lemma 3.2], that a set G of compact objects in T is a set of compact generators for T if and only if loc(G) = T. This last remark is very important in practice. For instance, it is a key point in proving the following elementary but useful lemma. Perhaps more importantly, the proof illustrates an argument which appears frequently in this subject. Lemma 2.12. Let T be a rigidly-compactly generated tensor triangulated category. Then, for t ∈ Tc the isomorphism of endofunctors of Tc t∨ ⊗ (−) −→α hom(t, −) extends to an isomorphism of the corresponding endofunctors of T. That is, for any X ∈ T we have t∨ ⊗ X =∼ hom(t, X). In particular, hom(t, −) preserves coproducts and t ⊗ (−) is left and right adjoint to t∨ ⊗ (−) when viewed as endofunctors of T. 14 GREG STEVENSON

Proof. As in Deﬁnition 1.3 we can deﬁne a natural transformation α: t∨ ⊗ (−) −→ hom(t, −) of functors T −→ T. Deﬁne a full subcategory of T as follows ∨ S = {X ∈ T | αX : t ⊗ X −→ hom(t, X) is an isomorphism}. As both functors involved preserve suspensions and coproducts, and α is also com- patible with suspensions and coproducts, we see that S is closed under coproducts and suspensions. Given a triangle X −→ Y −→ Z −→ ΣX with X,Y ∈ S the naturality of α guarantees the commutativity of the following diagram t∨ ⊗ X / t∨ ⊗ Y / t∨ ⊗ Z / Σ(t∨ ⊗ X)

αX αY αZ αΣX hom(t, X) / hom(t, Y ) / hom(t, Z) / Σ hom(t, X)

As both functors are exact the rows are triangles and so since αX and αY are isomorphisms it follows that αZ must also be an isomorphism and hence Z ∈ S. Thus S is a localising subcategory and, by rigidity of Tc, we have Tc ⊆ S. It then follows from [Nee96, Lemma 3.2] that S = T which proves the ﬁrst assertion of the Lemma; the rest of the statements are immediate. Another illustration is given by the following analogue of Lemma 1.9. Lemma 2.13. Let T be a rigidly-compactly generated tensor triangulated category. If 1 is a compact generator for T, i.e. if T = loc(1), then every localising subcategory of T is a localising tensor-ideal.

Proof. Exercise. Localising subcategories are not imaginatively named—they are precisely the kernels of localisation functors. We will only give a brief outline of the formalism that we need and do not go into the details of forming Verdier quotients. There are many excellent sources for further information on this topic such as [Nee01], [BN93], and [Kra10]. Deﬁnition 2.14. A localisation sequence is a diagram

∗ i∗ j L o / T o / C i! j∗ ! ∗ where i is right adjoint to i∗ and j∗ is right adjoint to j , both i∗ and j∗ are fully faithful and hence embed L and C as a localising and a colocalising subcategory respectively, and we have equalities ⊥ ⊥ (i∗L) = j∗C and (j∗C) = i∗L, where ⊥ (i∗L) = {Y ∈ T | T(i∗L, Y ) = 0 for all L ∈ L} and ⊥ (j∗C) = {Y ∈ T | T(Y, j∗C) = 0 for all C ∈ C}. ! ∗ We refer to the composite i∗i as the acyclisation functor and j∗j as the localisation functor corresponding to this localisation sequence. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 15

We will frequently abuse the notation in such situations and identify L and C with their images under the fully faithful functors i∗ and j∗. The existence of a localisation sequence provides a great deal of information and has many consequences. The following proposition lists a few of the ones we will need. We suggest perusing the references given before the above deﬁnition for more details as well as proofs of the statements made below (there are, in addition, relevant references in [BF11, Theorem 2.6] where these statements also appear). Proposition 2.15. If we have a localisation sequence as in the deﬁnition then the following statements hold. ∗ ! ∗ (a) The composites j i∗ and i j∗ are zero. Moreover, the kernel of j is precisely L. (b) The composite j C −→∗ T −→ T/L is an equivalence. In particular, the Verdier quotient T/L is locally small and the canonical projection T −→ T/L has a right adjoint. (c) For every X ∈ T there is a distinguished triangle ! ∗ ! i∗i X −→ X −→ j∗j X −→ Σi∗i X. These triangles are functorial and unique in the sense that given any dis- tinguished triangle X0 −→ X −→ X00 −→ ΣX with X0 ∈ L and X00 ∈ C 0 ∼ ! 00 ∼ ∗ there are unique isomorphisms X = i∗i X and X = j∗j X. (d) A localisation sequence is completely determined by either of the pairs of ! ∗ adjoint functors (i∗, i ) or (j , j∗). In nature localising sequences are actually rather easy to come by as a conse- quence of the following form of Brown representability. Theorem 2.16. Let T be a compactly generated triangulated category, L ⊆ T a set of objects, and set L = loc(L). Then the inclusion i∗ : L −→ T admits a right adjoint. In particular, L ﬁts into a localisation sequence

∗ i∗ j L o / T o / T/L i! j∗ Proof. This follows from the results in Section 7.2 of [Kra10] combined with [Nee01, Theorem 8.4.4]. ∗ By deﬁnition, given any localisation sequence as above, the functors i∗ and j preserve coproducts by virtue of being left adjoints. The condition that the right ! adjoints i and j∗ also preserve coproducts turns out to be very strong and very useful. Deﬁnition 2.17. A localisation sequence as in Deﬁnition 2.14 is smashing if i! (or equivalently j∗) preserves coproducts. In this case we call L a smashing subcategory of T. If L is, moreover, a tensor-ideal we call it a smashing tensor-ideal.

In a smashing localisation sequence C is also, under the embedding j∗, a localising subcategory of T. The standard source of smashing localisation sequences is the following well known lemma. 16 GREG STEVENSON

Lemma 2.18. Let T be a compactly generated triangulated category and S ⊆ Tc a set of compact objects. Then S = loc(S) is a smashing subcategory, i.e. the inclusion ! i∗ : S −→ T admits a coproduct preserving right adjoint i . ! Proof. We know from Theorem 2.16 that i∗ admits a right adjoint i . It follows from Thomason’s localisation theorem [Nee96, Theorem 2.1] together with [Nee96, ! Theorem 5.1] that i preserves coproducts. 2.3. Generalised Rickard idempotents and supports. Throughout this sec- tion we will assume T is a rigidly-compactly generated tensor triangulated category such that Spc Tc, the spectrum of the compact objects, is a noetherian topological space. Recall from Remark 1.28 that in this case the Thomason subsets of Spc Tc are precisely the specialisation closed subsets. Aside 2.19. The assumption that Spc Tc is noetherian is not necessary, but the situation is signiﬁcantly more complicated without this assumption. If the spectrum is not noetherian then one can not necessarily deﬁne a tensor-idempotent, as in Deﬁnition 2.22, for every point. However, the main ideas, as introduced in [BF11] and discussed below, do work to some extent without the noetherian hypotheses and as shown in [Ste14] one always has enough information to recover the supports of compact objects via the support of Deﬁnition 2.25. We now introduce the objects which allow us to extend the support deﬁned in the ﬁrst section for objects of Tc to arbitrary objects of T. The construction was motivated by Rickard’s work in modular representation theory [Ric97] and the work of Hovey, Palmieri, and Strickland [HPS97] on the correct axiomatic framework in which to perform such constructions. We produce the objects we desire, named Rickard idempotents, via the following key construction based on certain smashing localisations. This result is now some- what standard, but we include some details. Further information and references can be found in [BF11]. Theorem 2.20. Let T be a rigidly-compactly generated tensor triangulated cate- gory, S ⊆ Tc a set of compact objects, and set S = loc⊗(S). Consider the corre- sponding smashing localisation sequence

i j∗ ∗ ⊥ S o / T o / S i! j∗ Then: (a) S⊥ is a localising tensor-ideal; (b) there are isomorphisms of functors ! ∼ ! ∗ ∼ ∗ i∗i 1 ⊗ (−) = i∗i and j∗j 1 ⊗ (−) = j∗j ; ! ∗ (c) the objects i∗i 1 and j∗j 1 satisfy ! ! ∼ ! ∗ ∗ ∼ ∗ ! ∗ ∼ i∗i 1 ⊗ i∗i 1 = i∗i 1, j∗j 1 ⊗ j∗j 1 = j∗j 1, and i∗i 1 ⊗ j∗j 1 = 0, i.e. they are tensor idempotent and tensor to 0. Proof. The claimed smashing localisation sequence exists by Lemma 2.18. The only hitch is that one needs to know S is in fact generated by objects of Tc. In fact loc⊗(S) = loc(thick⊗(S)) and this can be proved directly or deduced from [Ste13, Lemma 3.8] and is left as an exercise. We begin by proving (a); this is SUPPORT THEORY FOR TRIANGULATED CATEGORIES 17 the main point and the other statements follow in a straightforward manner from abstract properties of localisations. Consider the following full subcategory of T M = {X ∈ T | X ⊗ S⊥ ⊆ S⊥}. As ⊗ is exact and coproduct preserving in each variable and S⊥ is localising it is straightforward to verify that M is a localising subcategory. Let t be a compact object of T and Y ∈ S⊥. We have isomorphisms for any Z ∈ S T(Z, t ⊗ Y ) =∼ T(Z, hom(t∨,Y )) =∼ T(Z ⊗ t∨,Y ) = 0 where the ﬁrst isomorphism is via Lemma 2.12, the second is by adjunction, and the ﬁnal equality holds as S is a tensor ideal, so Z ⊗ t∨ ∈ S, together with the fact that Y ∈ S⊥. As Z ∈ S was arbitrary this shows t ⊗ Y ∈ S⊥, i.e. Tc ⊆ M. It follows from Remark 2.11 that M = T which says precisely that S⊥ is a tensor-ideal. Now we show (b). Consider the localisation triangle ! ∗ i∗i 1 −→ 1 −→ j∗j 1 −→ from Proposition 2.15 (c). Given X ∈ T we can tensor this triangle with X to obtain the distinguished triangle ! ∗ i∗i 1 ⊗ X −→ X −→ j∗j 1 ⊗ X −→ As both S and S⊥ are tensor-ideals the leftmost and rightmost terms of this latter triangle lie in S and S⊥ respectively. Uniqueness and functoriality of localisation triangles, also observed in (c) of the aforementioned proposition, then guarantees unique isomorphisms ! ∼ ! ∗ ∼ ∗ i∗i X = i∗i 1 ⊗ X and j∗j X = j∗j 1 ⊗ X which can be assembled to the desired isomorphisms of functors. We leave (c) as an exercise so the interested reader can familiarise themselves with the properties ∗ ! of the localisation and acyclisation functors (j∗j and i∗i respectively) associated to localisation sequences. We now apply this proposition to certain subcategories arising from Theorem 1.30. Let V be a specialisation closed subset of Spc Tc. Recall that τ(V) denotes the as- sociated thick tensor-ideal τ(V) = {t ∈ Tc | supp t ⊆ V}. We set ΓV T = loc(τ(V)). By Lemma 2.18 there is an associated smashing localisation sequence

ΓV T o / T o / LV T and we denote the corresponding acyclisation and localisation functors by ΓV and LV respectively. By [BF11, Theorem 4.1] ΓV T is not only a smashing subcategory but a smashing tensor-ideal. Thus applying the preceding theorem yields tensor idempotents ΓV 1 and LV 1 which give rise to the acyclisation and localisation func- tors by tensoring. It is these idempotents that are used to deﬁne the support; the 18 GREG STEVENSON intuition is that, for X ∈ T, ΓV 1 ⊗ X is the “piece of X supported on V” and LV 1 ⊗ X is the “piece of X supported on the complement of V”. Let us make a quick comment on notation before continuing. From this point forward we will generally denote points of Spc Tc by x, y, . . . rather than in a no- tation suggestive of prime tensor-ideals as in the last section. This is due to the fact that we will work with Spc Tc as an abstract topological space which, together with supp, forms a support data rather than explicitly with prime tensor-ideals. Deﬁnition 2.21. For every x ∈ Spc Tc we deﬁne subsets of the spectrum V(x) = {x} and Z(x) = {y ∈ Spc Tc | x∈ / V(y)}. Both of these subsets are specialization closed and hence Thomason as we have assumed Spc Tc is noetherian. We note that V(x) \ (Z(x) ∩ V(x)) = {x} i.e. these two Thomason subsets let us pick out the point x. Deﬁnition 2.22. Let x be a point of Spc Tc. We deﬁne a tensor-idempotent

Γx1 = (ΓV(x)1 ⊗ LZ(x)1).

Following the intuition above, for an object X ∈ T, the object Γx1 ⊗ X is supposed to be the “piece of X which lives only over the point x ∈ Spc Tc”. Remark 2.23. Given any Thomason subsets V and W of Spc Tc such that V\ (V ∩ W) = {x} we can deﬁne a similar object by forming the tensor product ΓV 1 ⊗ LW 1. By [BF11, Corollary 7.5] any such object is uniquely isomorphic to Γx1. Example 2.24. As a brief respite from the abstraction let us provide a detailed example describing what these idempotents look like in D(R), the derived category of a commutative noetherian ring. Given an element f ∈ R we deﬁne the stable Koszul complex K∞(f) to be the complex concentrated in degrees 0 and 1

· · · −→ 0 −→ R −→ Rf −→ 0 −→ · · · where the only non-zero morphism is the canonical map to the localisation. Given a sequence of elements f = {f1, . . . , fn} of R we set

K∞(f) = K∞(f1) ⊗ · · · ⊗ K∞(fn).

There is a canonical morphism K∞(f) −→ R which is clearly a degreewise split epimorphism. We deﬁne the Cechˇ complex of f to be the suspension of the kernel of this map. As the stable Koszul complex and the desuspension of the Cechˆ complex ﬁt into a degreewise split short exact sequence there is a triangle

In particular, the objects ΓpR = ΓV(p)R ⊗ LZ(p)R, which will give rise to supports on D(R), are naturally isomorphic to K∞(p) ⊗ Rp. Statements (1) and (2) are already essentially present in [Har67]. In the form stated here they are special cases of [Gre01, Lemma 5.8]. The third statement can be proved by noting that the full subcategory of complexes with homological support in U(p) = Spec R \Z(p) is the essential image of the inclusion of D(Rp) and then appealing to the fact that, up to direct summands, perf ∼ perf D (R)/τ(Z(p)) = D (Rp). We now come to the main deﬁnition of this section. Deﬁnition 2.25. For X ∈ T we deﬁne the support of X to be c supp X = {x ∈ Spc T | Γx1 ⊗ X 6= 0}. We do not introduce notation to distinguish this notion of support, applied to compact objects, from the one for the ﬁrst section as they agree. The following proposition records this fact as well as several other important properties of the support and the tensor-idempotents we have deﬁned. Proposition 2.26 ([BF11, 7.17, 7.18]). The support deﬁned above satisﬁes the following properties. (a) The two notions of support coincide for any compact object of T. (b) For any set-indexed family of objects {Xλ | λ ∈ Λ} we have a [ supp( Xλ) = supp Xλ. λ∈Λ λ∈Λ (c) For every X ∈ T supp(ΣX) = supp(X). (d) Given a distinguished triangle X −→ Y −→ Z −→ in T we have supp(Y ) ⊆ supp(X) ∪ supp(Z). (e) For any X,Y ∈ T we have supp(X ⊗ Y ) ⊆ supp(X) ∩ supp(Y ). (f) For any X ∈ T and Thomason subset V ⊆ Spc Tc we have

supp(ΓV 1 ⊗ X) = (supp X) ∩ V c supp(LV 1 ⊗ X) = (supp X) ∩ (Spc T \V). 20 GREG STEVENSON

Remark 2.27. Note that (e) is weaker than the corresponding statement given in Lemma 1.15. It is also worth noting that we have not mentioned that an object X ∈ T is 0 if and only if supp X = ∅. In fact this latter statement is much more subtle for rigidly-compactly generated tensor triangulated categories. We will show, with an additional hypothesis, that it does indeed hold though when we discuss the local-to-global principle in the next section.

We defer any detailed examples to the end of the next section. One can, using just the deﬁnition of the support and these properties, already make interesting statements but life is much easier once we have the local-to-global principle.

3. Tensor actions and the local-to-global principle In the last section we saw how to deﬁne a notion of support for rigidly-compactly generated tensor triangulated categories using the spectrum of the compacts. In this section we will explain a relative version, where one deﬁnes supports in terms of the action of a rigidly-compactly generated tensor triangulated category. What we have seen is then the special case of such a category acting on itself in the obvious way. The main reference for this section is [Ste13]. Continuing our analogy from the ﬁrst section: if a tensor triangulated category is viewed as a strange sort of ring, we will now study the basics of the corresponding module theory.

3.1. Actions and submodules. We deﬁne here the notion of left action and express a sinistral bias by only considering left actions. We will omit the “left” and refer to them simply as actions. One can, of course, deﬁne right actions in the analogous way, although since we deal with symmetric monoidal categories this is mostly cosmetic. We ﬁx throughout this section a rigidly-compactly generated tensor triangulated category T and a compactly generated triangulated category K. We will continue to assume throughout that Spc Tc is noetherian. Again a lot of the theory can be developed without this hypothesis, but we avoid the related technicalities for the sake of simplicity.

Deﬁnition 3.1. A (left) action of T on K is a functor

∗: T × K −→ K which is exact in each variable, i.e. for all X ∈ T and A ∈ K the functors X ∗ (−) and (−) ∗ A are exact, together with natural isomorphisms

∼ aX,Y,A :(X ⊗ Y ) ∗ A −→ X ∗ (Y ∗ A) and ∼ lA : 1 ∗ A −→ A for all X,Y ∈ T, A ∈ K, compatible with the exactness hypotheses on (−) ∗ (−) and satisfying the following compatibility conditions: SUPPORT THEORY FOR TRIANGULATED CATEGORIES 21

(1) The associator a makes the following diagram commute for all X,Y,Z in T and A in K X ∗ (Y ∗ (Z ∗ A)) 5 i X∗aY,Z,A aX,Y,Z∗A

X ∗ ((Y ⊗ Z) ∗ A) (X ⊗ Y ) ∗ (Z ∗ A) O O aX,Y ⊗Z,A aX⊗Y,Z,A

(X ⊗ (Y ⊗ Z)) ∗ A o ((X ⊗ Y ) ⊗ Z) ∗ A

where the bottom arrow is the associator of (T, ⊗, 1). (2) The unitor l makes the following squares commute for every X in T and A in K

X∗l l X ∗ (1 ∗ A) A / X ∗ A 1 ∗ (X ∗ A) X∗A / X ∗ A O O

aX,1,A 1X∗A a1,X,A 1X∗A (X ⊗ 1) ∗ A / X ∗ A (1 ⊗ X) ∗ A / X ∗ A

where the bottom arrows are the right and left unitors of (T, ⊗, 1). (3) For every A in K and r, s ∈ Z the diagram ∼ Σr1 ∗ ΣsA / Σr+sA

r s r+ s Σ (1 ∗ Σ A) ∼ / Σ A is commutative, where the left vertical map comes from exactness in the ﬁrst variable of the action, the bottom horizontal map is the unitor, and the top map is given by the composite

Σr1 ∗ ΣsA −→ Σs(Σr1 ∗ A) −→ Σr+s(1 ∗ A) −→l Σr+sA whose ﬁrst two maps use exactness in both variables of the action. (4) The functor ∗ distributes over coproducts. Explicitly, for families of objects {Xi}i∈I in T and {Aj}j∈J in K, and X in T, A in K the canonical maps a ∼ a (Xi ∗ A) −→ ( Xi) ∗ A i i and a ∼ a (X ∗ Aj) −→ X ∗ ( Aj) j j are isomorphisms. Remark 3.2. Given composable morphisms f, f 0 in T and g, g0 in K one has (f 0 ∗ g0)(f ∗ g) = (f 0f ∗ g0g) by functoriality of T × K −→∗ K. 22 GREG STEVENSON

We also note it follows easily from the deﬁnition that both 0T ∗ (−) and (−) ∗ 0K are isomorphic to the zero functor. Indeed if X ∈ T then one has a triangle

1 X X / X / 0 / ΣX and applying − ∗ A for A ∈ K we get a triangle

1 X ∗ A X∗A / X ∗ A / 0 ∗ A / ΣX ∗ A. ∼ Thus 0 ∗ A = 0 for every A in K and the argument for (−) ∗ 0K is similar. We view K as a module over T and from now on we will use the terms module and action interchangeably. There are of course, depending on the context, natural notions of T-submodule. Deﬁnition 3.3. Let L ⊆ K be a localising (thick) subcategory of K. We say L is a localising (thick) T-submodule of K if the functor T × L −→∗ K factors via L, i.e. L is closed under the action of T. In the case that K = T acts on itself by ⊗ this recovers the notion of a localising (thick) tensor-ideal of T. We mean the obvious things by a smashing or compactly generated (by compact objects in the ambient category) submodule. Notation 3.4. For a collection of objects A in K we denote, as before, by loc(A) the smallest localising subcategory containing A and we will use loc∗(A) to denote the smallest localising T-submodule of K containing A. Extending our earlier notation ∗ we let LocT(K) denote the lattice of localising submodules of K with respect to the action of T. Usually the action in question is clear and we omit the T from the notation. Given in addition a class X of objects of T we denote by X ∗ A = loc∗(X ∗ A | X ∈ X ,A ∈ A) the localising submodule generated by products, with respect to the action, of the objects from X and A. We have the following useful technical result for working with tensor-ideals and submodules generated by prescribed sets of objects. Lemma 3.5 ([Ste13, Lemma 3.12]). Formation of localising T-submodules com- mutes with the action, i.e. given a collection of objects X of T and a collection of objects A of K we have loc⊗(X ) ∗ loc(A) = loc(X ) ∗ loc(A) = X ∗ A = loc(Z ∗ (X ∗ A) | Z ∈ T,X ∈ X ,A ∈ A). Another useful lemma in examples, which frequently removes the need to keep track of submodules versus localising subcategories, is the following analogue of Lemmas 1.9 and 2.13. Lemma 3.6 ([Ste13, Lemma 3.13]). If T is generated as a localising subcategory by the tensor unit 1 then every localising subcategory of K is a T-submodule. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 23

3.2. Supports. We now introduce supports for K relative to the action of T. This is done in the naive way, extending what we have seen in the last section. Recall from Deﬁnition 2.22 that for each x ∈ Spc Tc we have objects

Γx1 = ΓV(x)1 ⊗ LZ(x)1. c Notation 3.7. We use ΓxK, for x ∈ Spc T , to denote the essential image of Γx1 ∗ (−). It is a T-submodule as for any X ∈ T and A ∈ ΓxK ∼ 0 ∼ 0 X ∗ A = X ∗ (Γx1 ∗ A ) = Γx1 ∗ (X ∗ A ) 0 for some A ∈ K as, by virtue of being in the essential image of Γx1 ⊗ (−), we must have such an A0 and an isomorphism ∼ 0 A = Γx1 ⊗ A .

Similarly we will often write ΓxA as shorthand for Γx1 ∗ A. Deﬁnition 3.8. Given A in K we deﬁne the support of A to be the set c supp(T,∗) A = {x ∈ Spc T | ΓxA 6= 0}. When the action in question is clear we will omit the subscript from the notation. We have analogous properties for this notion of support as for the support when T acts on itself as in Proposition 2.26.

Proposition 3.9. The support assignment supp(T,∗) satisﬁes the following proper- ties: (a) given a triangle A −→ B −→ C −→ ΣA in K we have supp B ⊆ supp A ∪ supp C; (b) for any A in K and i ∈ Z supp A = supp ΣiA;

(c) given a set-indexed family {Aλ}λ∈Λ of objects of K there is an equality a [ supp Aλ = supp Aλ; λ λ (d) the support satisﬁes the separation axiom, i.e. for every specialization closed subset V ⊆ Spc Tc and every object A of K

supp ΓV 1 ∗ A = (supp A) ∩ V c supp LV 1 ∗ A = (supp A) ∩ (Spc T \V). 3.3. The local-to-global principle and parametrising submodules. As we saw an action allows us to deﬁne a sublattice Loc∗(K) of the collection of all local- ising subcategories of K. The hope is twofold: on one hand we hope that the lattice Loc∗(K) is more tractable than Loc(K) and on the other hand we hope the action gives us the necessary tools to analyse Loc∗(K). In this section we will deﬁne as- signments relating supports to localising submodules. These assignments are a step toward achieving our two hopes. We then introduce the local-to-global principle, a version of which was ﬁrst formalised in [BIK11] but is already implicit in the work of Neeman [Nee92], and explain some of its consequences for these assignments. We will prove a general result showing that the local-to-global principle holds in many cases of interest (see also Aside 3.20 for a brief discussion of an even more 24 GREG STEVENSON general approach). This powerful technical result allows us to reduce classiﬁcation problems into smaller (but usually still diﬃcult) pieces. Along the way we will ﬁnally show that the support, in the sense we have deﬁned above, is suﬃciently reﬁned to determine whether or not an object is 0. Deﬁnition 3.10. We deﬁne order preserving assignments τ subsets of Spc Tc / Loc∗(K) o σ where both collections are ordered by inclusion. For a localising submodule L we set c σ(L) = supp L = {x ∈ Spc T | ΓxL 6= 0} and for a subset W ⊆ Spc Tc we set τ(W ) = {A ∈ K | supp A ⊆ W }. Both of these are well deﬁned; this is clear for σ and for τ it follows from Proposition 3.9. Deﬁnition 3.11. We say T × K −→∗ K satisﬁes the local-to-global principle if for each A in K ∗ ∗ c loc (A) = loc (ΓxA | x ∈ Spc T ). The local-to-global principle has the following rather pleasing consequences for the assignments σ and τ of Deﬁnition 3.10. Lemma 3.12. Suppose the local-to-global principle holds for the action of T on K and let W be a subset of Spc Tc. Then ∗ τ(W ) = loc (ΓxK | x ∈ W ∩ σK). Proof. By the local-to-global principle we have for every object A of K an equality ∗ ∗ c loc (A) = loc (ΓxA | x ∈ Spc T ). Thus τ(W ) = loc∗(A | supp A ⊆ W ) ∗ = loc (ΓxA | A ∈ K, x ∈ W ) ∗ = loc (ΓxA | A ∈ K, x ∈ W ∩ σK) ∗ = loc (ΓxK | x ∈ W ∩ σK). Proposition 3.13. Suppose the local-to-global principle holds for the action of T on K and let W be a subset of Spc Tc. Then there is an equality of subsets στ(W ) = W ∩ σK. In particular, τ is injective when restricted to subsets of σK. Proof. With W ⊆ Spc Tc as in the statement we have στ(W ) = supp τ(W ) ∗ = supp loc (ΓxK | x ∈ W ∩ σK), the ﬁrst equality by deﬁnition and the second by the last lemma. Thus στ(W ) = W ∩ σK as claimed: by the properties of the support (Proposition 3.9) we have SUPPORT THEORY FOR TRIANGULATED CATEGORIES 25

στ(W ) ⊆ W ∩ σK and it must in fact be all of W ∩ σK as x ∈ σK if and only if ΓxK contains a non-zero object. Now we go about showing that we have access to these results in signiﬁcant generality. Deﬁnition 3.14. We will say the tensor triangulated category T has a model if it occurs as the homotopy category of a monoidal model category. Our main interest in such categories is that the existence of a monoidal model provides a good theory of homotopy colimits compatible with the tensor product. Remark 3.15. Of course instead of requiring that T arose from a monoidal model category we could, for instance, ask that T was the underlying category of a stable monoidal derivator. In fact we will only use directed homotopy colimits so one could use a weaker notion of a stable monoidal “derivator” only having homotopy left and right Kan extensions for certain diagrams; to be slightly more precise one could just ask for homotopy left and right Kan extensions along the smallest full 2-subcategory of the category of small categories satisfying certain natural closure conditions and containing the ordinals (one can see the discussion before [Gro13] Deﬁnition 4.21 for further details). We begin by showing that, when T has a model, taking the union of a chain of specialisation closed subsets is compatible with taking the homotopy colimit of the associated idempotents.

Lemma 3.16. Suppose T has a model. Then for any chain {Vi}i∈I of specialisation closed subsets of Spc Tc with union V there is an isomorphism ∼ ΓV 1 = hocolim ΓVi 1 where the structure maps are the canonical ones.

Proof. Each Vi is contained in V so there are corresponding inclusions for i < j

ΓVi T ⊆ ΓVj T ⊆ ΓV T. These inclusions induce canonical morphisms between the corresponding tensor- idempotents which ﬁt into a commutative triangle

ΓV 1 / ΓV 1 i <

We remind the reader that ΓVi T denotes the localising tensor-ideal generated by those compact objects with support in Vi and ΓVi 1 is the associated idempotent for the acyclisation (see the discussion before Deﬁnition 2.21). Commutativity of these triangles gives us an induced morphism from the homo- topy colimit of the ΓVi 1 to ΓV 1 which we complete to a triangle

hocolimI ΓVi 1 −→ ΓV 1 −→ Z −→ Σ hocolimI ΓVi 1. In order to prove the lemma it is suﬃcient to show that Z is isomorphic to the zero object in T. If this is the case then the ﬁrst map in the above triangle is the desired isomorphism. 26 GREG STEVENSON

The argument given in [Bou83] can be extended to prove that localising subcat- egories are closed under homotopy colimits and so this triangle consists of objects of ΓV T. By deﬁnition ΓV T is the full subcategory of T generated by those objects of Tc whose support is contained in V. Thus Z =∼ 0 if and only if for each compact object t with supp t ⊆ V we have Hom(t, Z) = 0. We note there is no ambiguity here as by Proposition 2.26 the two notions of support, that of [Bal05] and [BF11], agree for compact objects. In particular, the support of any compact object is closed. Recall from Theorem 1.21 that Spc Tc is spectral (see Deﬁnition 1.20), and we have assumed it is also noetherian. Thus the closed subset supp t can be written as a ﬁnite union of irreducible closed subsets. We can certainly ﬁnd a j ∈ I so that Vj contains the generic points of these ﬁnitely many irreducible components which implies supp t ⊆ Vj by specialisation closure. Therefore, by adjunction, it is enough to show ∼ Hom(t, Z) = Hom(ΓVj t, Z) ∼ = Hom(t, ΓVj Z) is zero, as this implies Z =∼ 0 and we get the claimed isomorphism.

In order to show the claimed hom-set vanishes let us demonstrate that ΓVj Z is zero. Observe that tensoring the structure morphisms Γ 1 −→ Γ 1 for i ≥ Vi1 Vi2 2 i1 ≥ j with ΓVj 1 yields canonical isomorphisms ∼ Γ 1 =∼ Γ 1 ⊗ Γ 1 −→ Γ 1 ⊗ Γ 1 =∼ Γ 1. Vj Vj Vi1 Vj Vi2 Vj

Thus applying ΓVj to the sequence {ΓVi 1}i∈I gives a diagram whose homotopy colimit is ΓVj 1. From this we deduce that the ﬁrst morphism in the resulting triangle

ΓVj 1 ⊗ hocolimI ΓVi 1 −→ ΓVj 1 −→ ΓVj Z is an isomorphism; since T is the homotopy category of a monoidal model cate- ∼ gory the tensor product commutes with homotopy colimits. This forces ΓVj Z = 0 completing the proof.

Lemma 3.17. Let P ⊆ Spc Tc be given and suppose A is an object of K such that ∼ c ΓxA = 0 for all x ∈ (Spc T \ P ). If T has a model then A is an object of the localising subcategory

L = loc(ΓyK | y ∈ P ). Proof. Denote by P(Spc Tc) the power set of Spc Tc and let Λ ⊆ P(Spc Tc) be the set of specialisation closed subsets W such that ΓW A is in L = loc(ΓyK | y ∈ P ). We ﬁrst note that Λ is not empty. Indeed, as Tc is rigid the only compact objects with empty support are the zero objects by [Bal07, Corollary 2.5] so

c Γ∅T = loc(t ∈ T | supp(T,⊗) t = ∅) = loc(0) giving Γ∅A = 0 and hence ∅ ∈ Λ. Since L is localising, Lemma 3.16 shows the set Λ is closed under taking increasing unions: as mentioned above the argument in [Bou83] extends to show that localising subcategories are closed under directed homotopy colimits in our situation. Thus Λ contains a maximal element Y by Zorn’s lemma. We will show that Y = Spc Tc. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 27

Suppose Y 6= Spc Tc. Since Spc Tc is noetherian the subspace Spc Tc \Y contains an element z maximal with respect to specialisation. We have ∼ LY 1 ⊗ ΓY ∪{z}1 = Γz1 as Y ∪ {z} is specialisation closed by maximality of z and Remark 2.23 tells us that ∼ we can use any suitable pair of Thomason subsets to deﬁne Γz1. So LY ΓY ∪{z}A = ΓzA and by our hypothesis on vanishing either ΓzK ⊆ L or ΓzA = 0. Consider the triangle

ΓY ΓY ∪{z}A / ΓY ∪{z}A / LY ΓY ∪{z}A

o o ΓY A ΓzA

We see that in either case, since ΓY A is in L and L is localising, the object ΓY ∪{z}A is in L. Thus Y ∪{z} ∈ Λ which contradicts the maximality of Y . Hence Y = Spc Tc and so A is in L. Proposition 3.18. Suppose T has a model. Then the local-to-global principle holds for the action of T on K. Explicitly, for any A in K there is an equality of T- submodules ∗ ∗ loc (A) = loc (ΓxA | x ∈ supp A). Proof. By Lemma 3.17 applied to the action T × T −→⊗ T c ⊗ we see T = loc(ΓxT | x ∈ Spc T ). Since ΓxT = loc (Γx1) it follows that the set of c objects {Γx1 | x ∈ Spc T } generates T as a localising tensor-ideal. By Lemma 3.5 given an object A ∈ K we get a generating set for T ∗ loc(A): ⊗ ∗ T ∗ loc(A) = loc (Γx1 | x ∈ Spc T) ∗ loc(A) = loc (ΓxA | x ∈ supp A). But it is also clear that T = loc⊗(1) so, by Lemma 3.5 again, T ∗ loc(A) = loc⊗(1) ∗ loc(A) = loc∗(A) and combining this with the other string of equalities gives ∗ ∗ loc (A) = T ∗ loc(A) = loc (ΓxA | x ∈ supp A) which completes the proof. We thus have the following theorem concerning the local-to-global principle for actions of rigidly-compactly generated tensor triangulated categories. Theorem 3.19. Suppose T is a rigidly-compactly generated tensor triangulated category with a model and such that Spc Tc is noetherian. Then for any compactly generated triangulated category K, for instance K = T, we have: (i) The local-to-global principle holds for any action of T on K; (ii) The associated support theory detects vanishing of objects, i.e. A ∈ K is zero if and only if supp X = ∅; c (iii) For any chain {Vi}i∈I of specialisation closed subsets of Spc T with union V there is an isomorphism ∼ ΓV 1 = hocolim ΓVi 1 where the structure maps are the canonical ones. 28 GREG STEVENSON

Proof. That (iii) always holds is the content of Lemma 3.16 and we have proved in Proposition 3.18 that (i) holds. To see (i) implies (ii), for an action T ∗ K −→ K, observe that if supp X = ∅ for an object X of K then the local-to-global principle yields ∗ ∗ c ∗ loc (X) = loc (ΓxX | x ∈ Spc T ) = loc (0) ∼ so X = 0. Aside 3.20. There are other situations in which the local-to-global principle is known to hold—one is not forced to only consider T whose compacts have noether- ian spectrum. A general criterion for verifying the local-to-global principle is given in [Ste16] and it is shown that there are other settings in which one has access to this tool. However, the local-to-global principle does not come for free. In [Ste14] examples are given where the local-to-global principle fails. 3.4. An example: commutative noetherian rings. As an example we sketch a proof of Neeman’s classiﬁcation, as proved in [Nee92], of the localising subcategories of D(R) for a commutative noetherian ring R. Our goal is to illustrate how one applies the machinery thusfar developed in a concrete setting. The details of the argument are not so diﬀerent from those in Amnon’s paper, although the abstract machinery we have developed takes care of many of the technical points. Let us begin by recalling the theorem. Theorem 3.21 ([Nee92, Theorem 2.8]). Let R be a commutative noetherian ring. There is an isomorphism of lattices τ subsets of Spec R / Loc D(R) o σ As we have claimed in Example 1.32 there is a homeomorphism Spc Dperf (R) =∼ Spec R and we regard this as an identiﬁcation. Let us now recall what the various relevant tensor-idempotents are in this situation. We introduced, in Example 2.24, the stable Koszul complex

K∞(f) = · · · −→ 0 −→ R −→ Rf −→ 0 −→ · · · and the analogue for any ideal I in R. In this example we also noted that there are isomorphisms in D(R) for an ideal I and a prime ideal p ∼ (1) ΓV(I)R = K∞(I); ∼ ˇ (2) LV(I)R = C(I); ∼ (3) LZ(p)R = Rp. ∼ In particular, ΓpR = K∞(p)p for each prime ideal p ∈ Spec R. The unbounded derived category D(R) is certainly rigidly-compactly generated and we have assumed R is noetherian so Theorem 3.19 guarantees for us that the local-to-global principle holds. Thus, considering the assignments τ subsets of Spec R / Loc D(R) o σ given by σ(L) = supp L = {p ∈ Spec R | ΓpL 6= 0} for a localising subcategory L, and τ(W ) = {A ∈ D(R) | supp A ⊆ W } SUPPORT THEORY FOR TRIANGULATED CATEGORIES 29 for a subset W of Spec R, we know by Proposition 3.13 that τ is a split monomor- phism when restricted to σD(R). One sees easily, using the explicit description, that none of the ΓpR are zero and so, in fact, τ is just injective. In order to show τ is an isomorphism it is enough to check that each ΓpD(R) is minimal in the sense that it has no proper non-trivial localising subcategories. Minimality of ΓpD(R) is equivalent to the statement that for any non-zero A ∈ ΓpD(R) we have loc(A) = ΓpD(R). Suppose this is the case. Then for L ∈ Loc D(R) we clearly have a containment L ⊆ τσL. On the other hand, if A ∈ τσL then we know supp A ⊆ σL and so by the local-to- global principle A ∈ loc(A) = loc(ΓpA | p ∈ σL).

If p ∈ σL then ΓpL 6= 0 and so, by our minimality assumption, ΓpD(R) ⊆ L. This shows A ∈ L proving L = τσL. We now sketch the proof that the ΓpD(R) are minimal. For each prime ideal p we will denote by k(p) = Rp/pRp the associated residue ﬁeld. Lemma 3.22. There are equalities of localising subcategories

ΓpD(R) = loc(ΓpR) = loc(k(p)). Proof. The ﬁrst equality follows from the fact that, by Lemma 2.13, every localising subcategory of D(R) is a tensor-ideal. In order to prove the second equality, ﬁrst note that ∼ ΓpR ⊗ k(p) = k(p) and so k(p) ∈ ΓpD(R). To prove the reverse containment, we begin by noting that i i K(p )p, the usual Koszul complex for p localised at p is contained in thick(k(p)). i This is a straightforward consequence of the fact that over Rp the complex K(p )p has ﬁnite length homology. One can write the localised stable Koszul complex i K∞(p)p as a homotopy colimit of the K(p )p and so K∞(p)p ∈ loc(k(p)). This completes the proof of the second equality. Remark 3.23. Using this lemma one can check that the support we have deﬁned for D(R) agrees with the support of Neeman which is deﬁned in terms of the residue ﬁelds.

Lemma 3.24. The localising subcategory ΓpD(R) is minimal. Proof. If A ∈ D(R) then one can check, by factoring the functor k(p) ⊗ − through D(k(p)), that M k(p) ⊗ A =∼ Σik(p)(λi) i∈Z (λi) for some set of cardinals λi, where k(p) denotes the coproduct of λi copies of k(p). One then proceeds by arguing that if A 6= 0 lies in ΓpD(R) we must have k(p) ⊗ A 6= 0 and so loc(A) contains k(p). Hence

loc(k(p)) ⊆ loc(A) ⊆ ΓpD(R), which shows loc(A) = ΓpD(R). 30 GREG STEVENSON

4. Further applications and examples In this ﬁnal section we concentrate on giving some further examples to illustrate how one can use the local-to-global principle in practice. 4.1. Singularity categories of hypersurfaces. Here we outline the solution to the classiﬁcation problem for localising subcategories of singularity categories of hypersurface rings. The intention is to indicate the manner in which these compu- tations usually proceed. In particular, we further highlight the manner in which the local-to-global principle is used. We begin by deﬁning the main players and explaining the action we will use. Further details, including the analogous results in the non-aﬃne case, can be found in [Ste14]. We can, for the time being, work quite generally. Suppose R is an arbitrary commutative noetherian ring. Then one deﬁnes a category b perf DSg(R) := D (mod R)/D (R), where Db(mod R) is the bounded derived category of ﬁnitely generated R-modules and Dperf (R) is the full subcategory of complexes locally isomorphic to bounded complexes of ﬁnitely generated projectives. This category measures the singularities of R. In particular, DSg(R) vanishes if and only if R is regular, it is related to other measures of the singularities of R for example maximal Cohen-Macaulay modules (see [Buc87]), and its properties reﬂect the severity of the singularities of R. The particular category which will concern us is the stable derived category of Krause [Kra05], namely S(R) := Kac(Inj R) the homotopy category of acyclic complexes of injective R-modules. We slightly abuse standard terminology by calling S(R) the singularity category of R. We will also need to consider the following categories D(R) := D(Mod R), and K(R) := K(Mod R) the unbounded derived category of R and the homotopy category of R-modules. As indicated above we denote by Inj R the category of injective R-modules and we write Flat R for the category of ﬂat R-modules. The main facts about S(R) that we will need are summarised in the following theorem of Krause. Theorem 4.1 ([Kra05] Theorem 1.1). Let R be a commutative noetherian ring. (1) There is a recollement

Iλ Qλ u I t Q S(R) / K(Inj R) / D(R) i j Iρ Qρ where each functor is right adjoint to the one above it. (2) The triangulated category K(Inj R) is compactly generated, and Q induces an equivalence K(Inj R)c −→ Db(mod R). (3) The sequence

Qλ I D(R) / K(Inj R) λ / S(R) SUPPORT THEORY FOR TRIANGULATED CATEGORIES 31

is a localisation sequence. Therefore S(R) is compactly generated, and the composite Iλ ◦ Qρ induces (up to direct factors) an equivalence c DSg(R) −→ S(R) . Now let us outline the proof that there is an action D(R) × S(R) −→ S(R) in the sense of Deﬁnition 3.1. This comes down to showing one can take K-ﬂat resolutions (see Deﬁnition 4.4) of objects of D(R) in a way which, although not necessarily initially functorial, becomes functorial after tensoring with an acyclic complex of injectives. Our starting point is work of Neeman [Nee08] and Murfet [Mur07] which provides a category that naturally acts on K(Inj R). As we work here in the aﬃne case Neeman’s results are suﬃcient for our purposes. However, we provide references to the work of Murfet as this makes the results necessary to generalise to the non-aﬃne case apparent. There is an obvious action of the category K(Flat R) on K(Inj R). It is given by simply taking the tensor product of complexes. This does in fact deﬁne an action as, since R is noetherian, the tensor product of a complex of ﬂats with a complex of injectives is again a complex of injectives. However, K(Flat R) may contain many objects which act trivially. Thus we will now pass to a more manageable quotient. In order to do so we need the notion of pure acyclicity. In [MS11] a complex F in K(Flat R) is deﬁned to be pure acyclic if it is exact and has ﬂat syzygies. Such complexes form a triangulated subcategory of K(Flat R) which we denote by Kpac(Flat R) and we say that a morphism with pure acyclic mapping cone is a pure quasi-isomorphism. The point is that tensoring a pure acyclic complex of ﬂats with a complex of injectives yields a contractible complex by [Nee08, Corollary 9.7]. Following Neeman and Murfet we consider N(Flat R), which is deﬁned to be the quotient K(Flat R)/Kpac(Flat R). What we have observed above shows the action of K(Flat R) on K(Inj R) factors via the projection K(Flat R) −→ N(Flat R). Next we describe a suitable subcategory of N(Flat R) which will act on S(R). Recall from above that S(R) is a compactly generated triangulated category. ` Consider E = λ Eλ where Eλ runs through a set of representatives for the iso- morphism classes of compact objects in S(R). We deﬁne a homological functor H : K(Flat R) −→ Ab by setting, for F an object of K(Flat R), 0 H(F ) = H (F ⊗R E) where the tensor product is taken in K(R). This is a coproduct preserving homolog- ical functor since we are merely composing the exact coproduct preserving functor 0 (−) ⊗R E with the coproduct preserving homological functor H . In particular every pure acyclic complex lies in the kernel of H.

Deﬁnition 4.2. With notation as above we denote by A⊗(Inj R) the quotient ker(H)/Kpac(Flat R), where i ker(H) = {F ∈ K(Flat R) | H(Σ F ) = 0 ∀i ∈ Z}. The next lemma shows this is the desired subcategory of N(Flat R). Lemma 4.3 ([Ste14, Lemma 4.3]). An object F of K(Flat R) lies in ker(H) if and only if the exact functor

F ⊗R (−): K(Inj R) −→ K(Inj R) 32 GREG STEVENSON restricts to F ⊗R (−): S(R) −→ S(R).

In particular, A⊗(Inj X) consists of the pure quasi-isomorphism classes of objects which act on S(X). Before continuing we recall the notion of K-ﬂatness. By taking K-ﬂat resolutions we get an action of D(R) via the above category. Deﬁnition 4.4. We say that a complex of ﬂat R-modules F is K-ﬂat provided F ⊗R(−) sends quasi-isomorphisms to quasi-isomorphisms (or equivalently if F ⊗RE is an exact complex for any exact complex of R-modules E). Lemma 4.5. There is a fully faithful, exact, coproduct preserving functor

D(R) −→ A⊗(Inj R). Proof. There is, by the proof of Theorem 5.5 of [Mur07], a fully faithful, exact, coproduct preserving functor D(R) −→ N(Flat R) given by taking K-ﬂat resolutions and inducing an equivalence ∼ ⊥ D(R) = Nac(Flat R).

This functor given by taking resolutions factors via A⊗(Inj R) since K-ﬂat complexes send acyclics to acyclics under the tensor product. Taking K-ﬂat resolutions and then tensoring gives the desired action (−) (−): D(R) × S(R) −→ S(R) by an easy argument: K-ﬂat resolutions are well behaved with respect to the tensor product so the necessary compatibilities follow from those of the tensor product of complexes. Remark 4.6. Recall that every complex in K−(Flat R), the homotopy category of bounded above complexes of ﬂat R-modules, is K-ﬂat. Thus when acting by the − subcategory K (Flat R) there is an equality = ⊗R. We can now apply all of the machinery we have introduced for actions of rigidly- compactly generated triangulated categories. Recall from Example 1.32 that Spc D(R)c = Spc Dperf (R) =∼ Spec R Henceforth we will identify these spaces. As in Deﬁnition 3.8 we get a notion of support on S(R) with values in Spec R; we will denote the support of an object A of S(R) simply by supp A. As the category D(R) has a model the local-to-global principle (Theorem 3.19) holds. In particular, not only do we have assignments as in Deﬁnition 3.10 τ (4.1) subsets of Spec R / localising subcategories of S(R) o σ where for a localising subcategory L we set

σ(L) = supp L = {p ∈ Spec R | ΓpL 6= 0} and for a subset W ⊆ Spec R we set τ(W ) = {A ∈ S(R) | supp A ⊆ W }. we have the following additional information. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 33

Proposition 4.7. Given a subset W ⊆ Spec R there is an equality of subcategories

τ(W ) = loc(ΓpS(R) | p ∈ W ) Proof. This is just a restatement of Lemma 3.12 combined with Lemma 2.13. We now describe σS(R), the support of the singularity category. In order to describe it we need to recall the following deﬁnition. Deﬁnition 4.8. The singular locus of R is the set

Sing(R) = {p ∈ Spec R | Rp is not regular}, i.e. the collection of those prime ideals p such that Rp has inﬁnite global dimension. Recall that this is equivalent to the residue ﬁeld k(p) having inﬁnite projective dimension over Rp.

Proposition 4.9 ([Ste14, Proposition 5.7]). For any p ∈ Sing R the object ΓpIλQρk(p) is non-zero in S(R). Thus ΓpS(R) is non-trivial for all such p yielding the equality σS(R) = Sing R. We now restrict ourselves to hypersurfaces, which we deﬁne below, so we can state the classiﬁcation theorem, and provide a rough sketch of the proof. Deﬁnition 4.10. Let (R, m, k) be a noetherian local ring. We say R is a hyper- surface if its m-adic completion Rˆ can be written as the quotient of a regular ring by a regular element (i.e. a non-zero divisor). Given a not necessarily local commutative noetherian ring R if, when localized at each prime ideal p in Spec R, Rp is a hypersurface we say that R is locally a hypersurface. Theorem 4.11 ([Ste14, Theorem 6.13]). If R is a noetherian ring which is locally a hypersurface then there is an isomorphism of lattices τ subsets of Sing R / localising subcategories of S(R) o σ given by the assignments of (4.1). This restricts to the equivalent lattice isomor- phisms τ specialization closed / localising subcategories of S(R) c subsets of Sing R o σ generated by objects of S(R) and specialization closed / thick subcategories of D (R) . subsets of Sing R o Sg Sketch of proof. We only sketch the proof of the ﬁrst bijection, the others follow from it but the proof is somewhat more involved. By Proposition 3.13, combined with the computation of σS(R) given in Propo- sition 4.9, we know that τ is injective with left inverse σ. Suppose then that L is a localising subcategory and consider

τσL = loc(ΓpS(R) | ΓpL 6= 0) where we have this equality by Proposition 4.7. By [Ste14, Theorem 6.12] (together with a short reduction to the local case) each of the localising subcategories ΓpS(R) is minimal, i.e. it has no proper non-zero localising subcategories. Thus, if ΓpL is 34 GREG STEVENSON non-zero it must, by this minimality, be equal to ΓpS(R). By the local-to-global principle we know

L = loc(ΓpL | p ∈ Sing R) and so putting all this together yields

L = loc(ΓpL | p ∈ Sing R) = loc(ΓpS(R) | ΓpL 6= 0) = τσL. This proves we have the claimed bijection. Remark 4.12. One should compare the above approach with that used in Sec- tion 3.4 (and note the similarity). The strategy evident in these proof sketches is really the thing to remember. Given the local-to-global principle for an action of T on K it is suﬃcient to check that each ΓxK is either zero or has no non-trivial, proper localising submodules; if this is the case one has an isomorphism of lattices Loc∗(K) =∼ {subsets of σK} giving a solution to the classiﬁcation problem for submodules. 4.2. Representations of categories over commutative noetherian rings. As another application of the local-to-global principle we sketch here some ideas coming from recent work of the author with Benjamin Antieau [AS15]. We are quite light on details; the idea is just to give another taste of the kind of reductions one can do with the local-to-global principle. The idea is to reduce the study of localising subcategories of derived categories of representations of a small category in R-modules, for a commutative noetherian ring R, to the study of the same problem over ﬁelds. This allows one, for instance, to give a classiﬁcation of localising subcategories in the derived category of the R-linear path algebra of a Dynkin quiver. We begin with some setup. Again we do not work in the maximum possible generality, but restrict the discussion to make it less technical. Fix a commutative noetherian ring R and let C be a small category. For instance C could be the path category of a quiver, a groupoid, or a poset. The category of (right) R-modules is the category op ModR C = [C , Mod R] of contravariant functors from C to R-modules. By standard nonsense ModR C is a Grothendieck abelian category with enough projectives and is equivalent to the category of R-linear functors op [RC , Mod R]R where RC denotes the free R-linear category on C. There is an action

⊗R Mod R × ModR C −→ ModR C deﬁned by pointwise tensoring, i.e. given an R-module M and a C-module F we have for c ∈ C

(M ⊗R F )(c) = M ⊗R (F (c)). Given a homomorphism of rings φ: R −→ S we have the obvious base change and restriction functors giving rise to an adjunction

φ∗ ModR C o / ModS C φ∗ SUPPORT THEORY FOR TRIANGULATED CATEGORIES 35

∗ and we can write φ in terms of the action as S ⊗R − together with the obvious S-action. We can now derive all of this to obtain an action D(R) × D(RC) −→ D(RC) where D(RC) is our shorthand for the unbounded derived category D(ModR C) of C-modules. As the projectives in ModR C are pointwise R-ﬂat we can resolve in D(RC) to compute this left derived action and so given φ: R −→ S as above L ∗ S ⊗R − realises Lφ . As a particular instance of this we can take, for p ∈ Spec R, the morphism R −→ k(p). We thus get an adjunction

L k(p)⊗R− D(RC) o / D(k(p)C) relating representations over R to representations over k(p) and arising from the D(R) action. We can use these adjunctions to split up the problem of understanding the lattice of localising subcategories of D(RC) into pieces. Deﬁne F −→s Spec R by declaring that s−1(p) = Loc(D(k(p)C)), i.e. we have a F = Loc(D(k(p)C)) p∈Spec R and s is the obvious “collapsing map”. We get poset morphisms f / sections l of Loc D(RC) s o g F −→ Spec R deﬁned on a localising subcategory L and a section l by f(L): p 7→ loc(k(p) ⊗ L) ⊆ D(k(p)C) and g(l) = {X ∈ D(RC) | k(p) ⊗ X ∈ l(p) ∀ p ∈ Spec R}. One of the main theorems of [AS15] is: Theorem 4.13. The morphisms f and g are inverse to one another. This fact is a direct application of the local-to-global principle for the action of D(R) on D(RC). Let us close by giving an outline of the proof. By Theorem 3.19 applied to the D(R) action we can reduce to checking that there is a lattice isomorphism ∼ Loc ΓpD(RC) = Loc D(k(p)C) for each p ∈ Spec R. However, one can proceed more directly. The main inputs for this direct approach to proving the theorem are the following two technical results. The ﬁrst is essentially a consequence of the observation we have made in Lemma 3.22 Proposition 4.14. Given X ∈ D(RC) there is an equality loc(X) = loc(k(p) ⊗ X | p ∈ Spec R). In particular, k(p) ⊗ X =∼ 0 for all p if and only if X =∼ 0. 36 GREG STEVENSON

Proposition 4.15. Given a section l of F −→s Spec R there is an equality g(l) = {X | k(p) ⊗ X ∈ l(p) ∀ p ∈ Spec R} = loc(l(p) | p ∈ Spec R) where l(p) is viewed as a subcategory of D(RC) via restricting along R −→ k(p). Fix a localising subcategory L of D(RC) and a section l of s. Given these two results one proves the theorem via the computations gf(L) = loc(f(L)(p) | p ∈ Spec R) = loc(k(p) ⊗ L | p ∈ Spec R) = L where the ﬁnal equality holds by the ﬁrst proposition, and fg(l)(p) = loc(k(p) ⊗ g(l)) = loc(k(p) ⊗ l(p)) = l(p). It remains an open and challenging problem to better understand the smashing subcategories and telescope conjecture for D(RC). Furthermore, very little is known about the lattices Loc D(kC) for a ﬁeld k except in certain very special cases.

References [AS15] Benjamin Antieau and Greg Stevenson, Derived categories of representations of small categories over commutative noetherian rings, Paciﬁc J. Math (2015). to appear. [Bal05] Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168. [Bal07] , Supports and ﬁltrations in algebraic geometry and modular representation the- ory, Amer. J. Math. 129 (2007), no. 5, 1227–1250. [Bal10] , Spectra, spectra, spectra - Tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebraic and Geometric Topology 10 (2010), no. 3, 1521–1563. [BCR97] Dave Benson, Jon F. Carlson, and Jeremy Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80. [BF11] Paul Balmer and Giordano Favi, Generalized tensor idempotents and the telescope con- jecture, Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1161–1185. [BIK11] Dave Benson, Srikanth B. Iyengar, and Henning Krause, Stratifying triangulated cate- gories, J. Topol. 4 (2011), no. 3, 641–666. [BKS07] Aslak Bakke Buan, Henning Krause, and Øyvind Solberg, Support varieties: an ideal approach, Homology, Homotopy Appl. 9 (2007), no. 1, 45–74. [BN93] Marcel B¨okstedtand Amnon Neeman, Homotopy limits in triangulated categories, Com- positio Math. 86 (1993), no. 2, 209–234. [Bou83] A. K. Bousﬁeld, Correction to: “The Boolean algebra of spectra” [Comment. Math. Helv. 54 (1979), no. 3, 368–377; ], Comment. Math. Helv. 58 (1983), no. 4, 599–600. [Buc87] Ragnar-Olaf Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished (1987). [DHS88] Ethan S. Devinatz, Michael J. Hopkins, and Jeﬀrey H. Smith, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), no. 2, 207–241. [Gre01] J. P. C. Greenlees, Tate cohomology in axiomatic stable homotopy theory, Cohomological methods in homotopy theory (Bellaterra, 1998), 2001, pp. 149–176. [Gro13] Moritz Groth, Derivators, pointed derivators, and stable derivators, Algebr. Geom. Topol. 13 (2013), 313–374. [Hap88] Dieter Happel, Triangulated categories in the representation theory of ﬁnite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge Uni- versity Press, Cambridge, 1988. [Har67] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961, Springer-Verlag, Berlin, 1967. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 37

[Hoc69] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60. [HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy the- ory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. [KP13] Joachim Kock and Wolfgang Pitsch, Hochster duality in derived categories and point-free reconstruction of schemes, arXiv:1305.1503 (2013). [Kra05] Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. [Kra10] , Localization theory for triangulated categories, Triangulated categories, 2010, pp. 161–235. [MS11] Daniel Murfet and Shokrollah Salarian, Totally acyclic complexes over Noetherian schemes, Adv. Math. 226 (2011), no. 2, 1096–1133. [Mur07] Daniel Murfet, The mock homotopy category of projectives and Grothendieck duality, Ph.D. thesis (2007). Available at: http://www.therisingsea.org/thesis.pdf. [Nee01] Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR1812507 (2001k:18010) [Nee08] , The homotopy category of ﬂat modules, and Grothendieck duality, Invent. Math. 174 (2008), no. 2, 255–308. [Nee92] , The chromatic tower for D(R), Topology 31 (1992), no. 3, 519–532. With an appendix by Marcel B¨okstedt. [Nee96] , The Grothendieck duality theorem via Bousﬁeld’s techniques and Brown repre- sentability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. [Ric97] Jeremy Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), no. 1, 149–170. [Ste13] Greg Stevenson, Support theory via actions of tensor triangulated categories, J. Reine Angew. Math. 681 (2013), 219–254. [Ste14] , Derived categories of absolutely ﬂat rings, Homology Homotopy Appl. 16 (2014), no. 2, 45–64. [Ste16] , The local-to-global principle for triangulated categories via dimension functions, arXiv:1601.01205 (2016). [Ste14] , Subcategories of singularity categories via tensor actions, Compos. Math. 150 (2014), no. 2, 229–272. [Tho97] R. W. Thomason, The classiﬁcation of triangulated subcategories, Compositio Math. 105 (1997), no. 1, 1–27.

Greg Stevenson, Universitat¨ Bielefeld, Fakultat¨ fur¨ Mathematik, BIREP Gruppe, Postfach 10 01 31, 33501 Bielefeld, Germany. E-mail address: [email protected]

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In This Article Expand or collapse the "in this article" section Social Support

Introduction, overview and history.

Theoretical Perspectives
Social Support and Physical Health
Social Support and Mental Health
Mechanisms and Pathways
Moderating and Mediating Factors in Social Support
Receiving and Providing Support: Costs and Benefits
Interventions
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Social Support by Robert G. Kent de Grey , Bert N. Uchino , Ryan Trettevik , Sierra Cronan , Jasara Hogan LAST REVIEWED: 11 January 2018 LAST MODIFIED: 11 January 2018 DOI: 10.1093/obo/9780199828340-0204

Social support is a broad construct comprising both the social structure of an individual’s life and the specific functions served by various interpersonal relationships. Structural aspects of support are often measured by assessing social integration, indicating the extent to which an individual is a part of social networks. Researchers usually divide functional support into two domains: perceived support, or people’s subjective construal of the support they believe to be available to them, and received (or enacted) support, which is aid actually rendered by other persons. Perceived and received support take a number of forms. Informational support involves the provision of recommendations, advice, and other helpful information. Tangible (or instrumental) support is the furnishing of financial, material, or physical assistance, such as the provision of money or labor. Emotional (or appraisal) support involves the expression of affection, empathy, caring, and so on. Belonging (or companionship) support creates a sense of belonging and can involve the presence or availability of others for social engagement. The different facets of social support often have distinct implications for psychological well-being and physical health. For example, higher perceived support is usually associated with preferable mental and physical health outcomes, including lower risk of cardiovascular disease (an association for which the evidence is particularly robust) and all-cause mortality. The proposed mechanisms by which social support influences health typically fall into one of two theoretical frameworks: the buffering model, which contends that support protects against the negative impacts of stressors, and the direct effects model, which holds that social support can also be beneficial in the absence of stressors. Not all support is beneficial; links between received support and health are fraught with complex moderators, such that received support that is beneficial in one circumstance may be ineffective or even detrimental under other conditions. More recent research has examined the factors that determine the direction and magnitude of the effects of received social support. Some evidence suggests support may be most beneficial when it is unobtrusive and matches the receiver’s needs. In contrast, unhelpful or unsolicited received support may feel controlling, frustrate receivers, or lower self-esteem and self-efficacy. Research has also started to consider the costs and benefits of providing social support, especially in the context of serving as a caregiver for a family member with chronic disease. Intervention attempts, especially for caregivers and individuals with unmet needs, increasingly recognize the importance of social support. A more recently emerging literature explores the ways in which computer-mediated communication and online social networks relate to social support.

Thoits 1995 provides a classic overview of the study of social support and the direction of the field at that time. A thorough and more current presentation of the development of social support and its implications for individuals is given by Taylor 2011 . Ditzen and Heinrichs 2014 provides a concise, more recent introduction. Cohen and Wills 1985 presents the two main competing hypotheses on how social support may influence health outcomes, and House, et al. 1988 uses relevant literature to establish a causal link between social support and health. Gottlieb 1985 expands the construct of social support to include the transactional process of giving and receiving support that occurs in interpersonal relationships. This bidirectional process is further discussed in Sarason and Sarason 2009 , with particular emphasis on the fit of an individual to his or her social support environment. Explorations of the potential pathways through which social support affects physiology and health are found in Uchino 2006 .

Cohen, S., and T. A. Wills. 1985. Stress, social support, and the buffering hypothesis. Psychological Bulletin 98.2: 310–357.

DOI: 10.1037/0033-2909.98.2.310

A seminal work examining the potential mechanisms through which social support influences physical health. Authors provide a thorough review of the literature on how social support is conceptualized and measured, as well as support for and against both the main effect and buffering hypotheses of social support.

Ditzen, B., and M. Heinrichs. 2014. Psychobiology of social support: The social dimension of stress buffering. Restorative Neurology and Neuroscience 32.1: 149–162.

DOI: 10.3233/rnn-139008

A brief but broad introduction to various subtopics in social support research. The structured organization renders this paper accessible to nonexperts, though the emphasis on health may make it more suitable for advanced readers. Includes citations of many classic papers in the field.

Gottlieb, B. H. 1985. Social support and the study of personal relationships. Journal of Social and Personal Relationships 2.3: 351–375.

DOI: 10.1177/0265407585023007

An exploration of the form and function of social support in both initiating and maintaining relationships. The author posits an expansion of the idea of social support from an intrapersonal, psychological construct to an interpersonal interaction, with an emphasis on the ways in which social support influences primary and secondary appraisals and coping processes. Additionally, the author provides a unique discussion of unintentional and indirect forms of social support.

House, J. S., K. R. Landis, and D. Umberson. 1988. Social relationships and health. Science 241.4865: 540–545.

DOI: 10.1126/science.3399889

A causal analysis of the association between low levels of social relationships and negative health outcomes. The authors argue that low-quality social support is a risk factor for, rather than a consequence of, poor health.

Sarason, I. G., and B. R. Sarason. 2009. Social support: Mapping the construct. Journal of Social and Personal Relationships 26.1: 113–120.

DOI: 10.1177/0265407509105526

This article provides a discussion on how social support is and should be both conceptualized and operationalized. Particular attention is paid to the bidirectional perspective of social support wherein an individual’s level of social support is the result of an interaction between the individual and his or her environment. The relative importance of fit in supportive relationships is discussed, and suggestions for future directions are given.

Taylor, S. E. 2011. Social support: A review. In The Oxford handbook of health psychology . Edited by H. S. Friedman, 189–214. Oxford: Oxford Univ. Press.

Comprehensive overview of social support. The author details the forms of social support and its benefits, as well as potential negative responses to social support efforts. Particular attention is paid to pathways by which social support influences health. Key factors that influence who receives social support and how it is received are discussed, as are implications for social support interventions.

Thoits, P. A. 1995. Stress, coping, and social support processes: Where are we? What next? In Special issue: Forty years of medical sociology: The state of the art and directions for the future . Edited by Mary L. Fennell. Journal of Health and Social Behavior 35 (extra issue): 53–79.

DOI: 10.2307/2626957

Comprehensive overview of previous literature examining social support in the context of coping with stress, for which a general introduction is also given. Summarizes then-current innovations in the field and suggests future directions.

Uchino, B. N. 2006. Social support and health: A review of physiological processes potentially underlying links to disease outcomes. Journal of Behavioral Medicine 29.4: 377–387.

DOI: 10.1007/s10865-006-9056-5

Thorough review of relevant literature examining which physiological mechanisms may bridge the relationship between social support and health outcomes. The author discusses differential functioning as a result of social support across the cardiovascular, neuroendocrine, and immune systems. Literature supports better functioning in all three systems in individuals with higher levels of social support.

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Challenge and Support Theory

The Challenge and Support Theory comes from Nevitt Sanford.

Description of this Theory

Nevitt Sanford theorized that student development depends on both challenge and support. Finding the sweet space f between challenge and support will lead to growth.

Challenges occur in situations where an individual does not have knowledge, skills, or attitudes to cope. This often occurs when entering a college campus.

The practitioner responsibility is to find balance between challenge and assist the students with developing and strengthening their coping mechanisms.

Coping can mean striking a balance between attitudes of differentiation (i.e., the student understands themselves as unique and differentiated), and i ntegration(i.e., students understand themselves as members of groups).

This Prezi on Suppor t & Challenge is interactive and incredibly useful. It was created by Peggy Sharp:

(According to the UNHCR) (According to France)

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Listen and share this podcast to learn more about supporting Afghan refugees: Podcast: Global Connections #10: [Special Episode] Supporting the Afghan Community

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Favorite Theory: Mattering & Marginality

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Astronomers' simulations support dark matter theory

The tests addressed the elusive matter's existence despite it never having been observed.

Computer simulations by astronomers support the idea that dark matter -- matter that no one has yet directly detected but which many physicists think must be there to explain several aspects of the observable universe -- exists, according to the researchers, who include those at the University of California, Irvine.

The work addresses a fundamental debate in astrophysics -- does invisible dark matter need to exist to explain how the universe works the way it does, or can physicists explain how things work based solely on the matter we can directly observe? Currently, many physicists think something like dark matter must exist to explain the motions of stars and galaxies.

"Our paper shows how we can use real, observed relationships as a basis to test two different models to describe the universe," said Francisco Mercado, lead author and recent Ph.D. graduate from the UC Irvine Department of Physics & Astronomy who is now a postdoctoral scholar at Pomona College. "We put forth a powerful test to discriminate between the two models."

The test involved running computer simulations with both types of matter -- normal and dark -- to explain the presence of intriguing features measured in real galaxies. The team reported their results in Monthly Notices of the Royal Astronomy Society .

The features in galaxies the team found "are expected to appear in a universe with dark matter but would be difficult to explain in a universe without it," said Mercado. "We show that such features appear in observations of many real galaxies. If we take these data at face value, this reaffirms the position of the dark matter model as the one that best describes the universe we live in."

These features Mercado noted describe patterns in the motions of stars and gas in galaxies that seem to only be possible in a universe with dark matter.

"Observed galaxies seem to obey a tight relationship between the matter we see and the inferred dark matter we detect, so much so that some have suggested that what we call dark matter is really evidence that our theory of gravity is wrong," said co-author James Bullock, professor of physics at UCI and dean of the UCI School of Physical Sciences. "What we showed is that not only does dark matter predict the relationship, but for many galaxies it can explain what we see more naturally than modified gravity. I come away even more convinced that dark matter is the right model."

The features also appear in observations made by proponents of a dark matter-free universe. "The observations we examined -- the very observations where we found these features -- were conducted by adherents of dark matter-free theories," said co-author Jorge Moreno, associate professor of physics and astronomy at Pomona College. "Despite their obvious presence, little-to-no analysis was performed on these features by that community. It took folks like us, scientists working with both regular and dark matter, to start the conversation."

Moreno added that he expects debate within his research community to follow in the wake of the study, but that there may be room for common ground, as the team also found that such features only appear in their simulations when there is both dark matter and normal matter in the universe.

"As stars are born and die, they explode into supernovae, which can shape the centers of galaxies, naturally explaining the existence of these features," said Moreno. "Simply put, the features we examined in observations require both the existence of dark matter and the incorporation of normal-matter physics."

Now that the dark matter model of the universe appears to be the leading one, the next step, Mercado explained, is to see if it remains consistent across a dark matter universe.

"It would be interesting to see if we could use this same relationship to even distinguish between different dark matter models," said Mercado. "Understanding how this relationship changes under distinct dark matter models could help us constrain the properties of dark matter itself."

Dark Matter
Astrophysics
Quantum Physics
Nuclear Energy
Inorganic Chemistry
Ultimate fate of the universe
Dark matter
Shape of the Universe
Dark energy
Extraterrestrial life
Interstellar medium

Story Source:

Materials provided by University of California - Irvine . Note: Content may be edited for style and length.

Journal Reference :

Francisco J Mercado, James S Bullock, Jorge Moreno, Michael Boylan-Kolchin, Philip F Hopkins, Andrew Wetzel, Claude-André Faucher-Giguère, Jenna Samuel. Hooks & Bends in the radial acceleration relation: discriminatory tests for dark matter and MOND . Monthly Notices of the Royal Astronomical Society , 2024; 530 (2): 1349 DOI: 10.1093/mnras/stae819

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UC Irvine astronomers’ simulations support dark matter theory

The tests addressed the elusive matter’s existence despite it never having been observed

Irvine, Calif., April 29, 2024 — Computer simulations by astronomers support the idea that dark matter – matter that no one has yet directly detected but which many physicists think must be there to explain several aspects of the observable universe – exists, according to the researchers, who include those at the University of California, Irvine.

The work addresses a fundamental debate in astrophysics – does invisible dark matter need to exist to explain how the universe works the way it does, or can physicists explain how things work based solely on the matter we can directly observe? Currently, many physicists think something like dark matter must exist to explain the motions of stars and galaxies.

“Our paper shows how we can use real, observed relationships as a basis to test two different models to describe the universe,” said Francisco Mercado, lead author and recent Ph.D. graduate from the UC Irvine Department of Physics & Astronomy who is now a postdoctoral scholar at Pomona College. “We put forth a powerful test to discriminate between the two models.”

The test involved running computer simulations with both types of matter – normal and dark – to explain the presence of intriguing features measured in real galaxies. The team reported their results in Monthly Notices of the Royal Astronomical Society .

The features in galaxies the team found “are expected to appear in a universe with dark matter but would be difficult to explain in a universe without it,” said Mercado. “We show that such features appear in observations of many real galaxies. If we take these data at face value, this reaffirms the position of the dark matter model as the one that best describes the universe we live in.”

These features Mercado noted describe patterns in the motions of stars and gas in galaxies that seem to only be possible in a universe with dark matter.

“Observed galaxies seem to obey a tight relationship between the matter we see and the inferred dark matter we detect, so much so that some have suggested that what we call dark matter is really evidence that our theory of gravity is wrong,” said co-author James Bullock, professor of physics at UCI and dean of the UCI School of Physical Sciences. “What we showed is that not only does dark matter predict the relationship, but for many galaxies it can explain what we see more naturally than modified gravity. I come away even more convinced that dark matter is the right model.”

The features also appear in observations made by proponents of a dark matter-free universe. “The observations we examined – the very observations where we found these features – were conducted by adherents of dark matter-free theories,” said co-author Jorge Moreno , associate professor of physics and astronomy at Pomona College. “Despite their obvious presence, little-to-no analysis was performed on these features by that community. It took folks like us, scientists working with both regular and dark matter, to start the conversation.”

“As stars are born and die, they explode into supernovae, which can shape the centers of galaxies, naturally explaining the existence of these features,” said Moreno. “Simply put, the features we examined in observations require both the existence of dark matter and the incorporation of normal-matter physics.”

Now that the dark matter model of the universe appears to be the leading one, the next step, Mercado explained, is to see if it remains consistent across a dark matter universe.

“It would be interesting to see if we could use this same relationship to even distinguish between different dark matter models,” said Mercado. “Understanding how this relationship changes under distinct dark matter models could help us constrain the properties of dark matter itself.”

Funding came from a National Science Foundation MSP-Ascend Award AST-2316748 to Mercado. Mercado and Bullock were supported by NSF grant AST-1910965 and NASA grant 80NSSC22K0827. Moreno receives funding from the Hirsch Foundation. Collaborators include Michael Boylan-Kolchin (The University of Texas at Austin), Philip F. Hopkins (California Institute of Technology), Andrew Wetzel (University of California, Davis) and Claude-André Faucher-Giguère (Northwestern University) and Jenna Samuel (The University of Texas at Austin).

UCI’s Brilliant Future campaign: Publicly launched on Oct. 4, 2019, the Brilliant Future campaign aims to raise awareness and support for the university. By engaging 75,000 alumni and garnering $2 billion in philanthropic investment, UCI seeks to reach new heights of excellence in student success, health and wellness, research and more. The School of Physical Sciences plays a vital role in the success of the campaign. Learn more at https://brilliantfuture.uci.edu/uci-school-of-physical-sciences/ .

About the University of California, Irvine: Founded in 1965, UC Irvine is a member of the prestigious Association of American Universities and is ranked among the nation’s top 10 public universities by U.S. News & World Report . The campus has produced five Nobel laureates and is known for its academic achievement, premier research, innovation and anteater mascot. Led by Chancellor Howard Gillman, UC Irvine has more than 36,000 students and offers 224 degree programs. It’s located in one of the world’s safest and most economically vibrant communities and is Orange County’s second-largest employer, contributing $7 billion annually to the local economy and $8 billion statewide. For more on UC Irvine, visit www.uci.edu .

Media access: Radio programs/stations may, for a fee, use an on-campus ISDN line to interview UC Irvine faculty and experts, subject to availability and university approval. For more UC Irvine news, visit news.uci.edu . Additional resources for journalists may be found at https://news.uci.edu/media-resources/ .

Kamala Harris Leads Push to Shore Up Democratic Support From Black Voters

Speaking in Atlanta, the vice president began a national tour to highlight how the Biden administration is trying to help Black Americans economically.

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Kamala Harris walks onto a stage while waving to the crowd. Some people up front are holding phones to capture the moment.

By Maya King

Reporting from Atlanta

April 29, 2024

Vice President Kamala Harris made a new effort to energize Black voters in battleground states on Monday, visiting Atlanta for the kickoff of a national economic tour that will highlight how the Biden administration says its policies are helping a constituency that will be vital to Democrats’ success in November.

Speaking to a largely Black crowd of about 400 people, Ms. Harris laid out ways that she and President Biden have sought to improve Black Americans’ upward mobility and help them realize their business ambitions. A chief objective of the tour, she said, was to let Black business owners and entrepreneurs know about the resources available to them.

“I need the help of the leaders who are here to get the word out so people know what is available to them,” she said during a conversation at the Georgia International Convention Center with Rashad Bilal and Troy Millings of the financial literacy podcast “Earn Your Leisure,” which offers business advice to its more than two million listeners, a majority of whom are Black.

Explaining how government policies have widened the racial wealth gap over the years, Ms. Harris pointed to the Biden administration’s attempts to try to narrow it, including small-business grants and efforts to forgive student loans.

“We want to make sure people know about it — and then know where they can receive — the support to be ready to take on the work and then to grow their capacity,” she said.

Her remarks at the official White House-hosted event — drier and less political than her forceful campaign speeches on abortion recently — meandered at times.

“As much as anything, the spirit behind the push for access to capital, and in particular, on this tour, focusing on minority small businesses and Black-owned small businesses, and small businesses and entrepreneurs who are Black men, is to recognize the disparities that have existed around the access to the opportunity to achieve success,” she said at one point.

But at other moments Ms. Harris was more pointed, including when she defended diversity, equity and inclusion initiatives that conservative lawmakers have pushed to dismantle in state governments and at colleges and universities .

“In spite of those who in certain parts of our country want to attack D.E.I., we understand that you can’t truly invest in the strength of our nation if you don’t pay attention to diversity, equity and inclusion,” she said.

The vice president’s Atlanta visit, her 12th trip to Georgia since taking office, was the first stop in a tour of several battleground states in the coming weeks. Much of the tour will focus on Black small businesses and economic issues that are especially pressing for Black communities. She will visit Detroit next week, aides said.

Ms. Harris’s tour will also seek to engage Black men, whom Democrats are urgently courting as polls show them softening in their support for Mr. Biden. Much of the crowd at the vice president’s event consisted of Black male political and business leaders, as well as a contingent of students from Morehouse College, a historically Black men’s college in Atlanta where President Biden will deliver the commencement address next month.

Before her speech, Ms. Harris visited an entrepreneurship hub near downtown Atlanta where she spoke with Black small-business owners.

Flanked by Georgia’s Democratic senators, Raphael Warnock and Jon Ossoff, as well as Representative Nikema Williams, the state Democratic Party chair, Ms. Harris highlighted the Biden administration’s marquee legislation, like the CHIPS and Science Act, which offers funding for small-business research projects.

She also underlined the roughly $7 billion that the administration has poured into historically Black colleges and universities, the largest investment of any White House administration.

Atlanta is set to receive more than $150 million for a project called The Stitch that is meant to revitalize the city’s downtown areas and connect them to its growing midtown neighborhoods. Georgia is one of 40 states that will receive money from a White House program that aims to repair the decades-old societal damage from federal transportation projects that disproportionately displaced Black communities.

Maya King is a politics reporter covering the Southeast, based in Atlanta. She covers campaigns, elections and movements in the American South, as well as national trends relating to Black voters and young people. More about Maya King

Our Coverage of the 2024 Election

Presidential Race

Protests and arrests on college campuses exploded into the forefront of the presidential race recently, opening up a new line of attack for Republicans and forcing President Biden to directly address an issue that has divided the liberal wing of his party.

In a stop in Wilmington, N.C., a possible swing state in the fall election, the president announced $3 billion in new spending to upgrade water systems around the country.

Biden has been under pressure to prove he can be tough at the border. But at a campaign reception, he also tried to voice his commitment to America’s long history of immigration, calling Japan and India “xenophobic” in the process.

A Key to 2024: As Democrats confront a presidential race against a resurgent and resilient Trump as well as a brutally challenging Senate map, they believe they have an increasingly powerful political weapon: ballot measures to protect abortion rights.

Presidential Medal of Freedom: With the election looming, Biden assembled a list of 19 people to honor that was heavy with major Democratic Party figures and others he has worked with over the years.

The Ultimate D.C. Veteran: In 30 years of Senate bids, Joe Biden was such a formidable incumbent that he did not face a serious threat to his return to office. But his last re-election is shaping up to be a fight.

Awash in Federal Money: Across Milwaukee, there is evidence of federal money from laws passed under the Biden administration. The president hopes voters will notice .

Astronomers' simulations support dark matter theory

C omputer simulations by astronomers support the idea that dark matter—matter that no one has yet directly detected but which many physicists think must be there to explain several aspects of the observable universe—exists, according to the researchers, who include those at the University of California, Irvine.

The work addresses a fundamental debate in astrophysics—does invisible dark matter need to exist to explain how the universe works the way it does, or can physicists explain how things work based solely on the matter we can directly observe? Currently, many physicists think something like dark matter must exist to explain the motions of stars and galaxies.

The test involved running computer simulations with both types of matter—normal and dark—to explain the presence of intriguing features measured in real galaxies. The team reported their results in Monthly Notices of the Royal Astronomical Society .

These features Mercado noted describe patterns in the motions of stars and gas in galaxies that seem to only be possible in a universe with dark matter.

"What we showed is that not only does dark matter predict the relationship, but for many galaxies it can explain what we see more naturally than modified gravity. I come away even more convinced that dark matter is the right model."

The features also appear in observations made by proponents of a dark matter-free universe. "The observations we examined—the very observations where we found these features—were conducted by adherents of dark matter-free theories," said co-author Jorge Moreno, associate professor of physics and astronomy at Pomona College.

"Despite their obvious presence, little-to-no analysis was performed on these features by that community. It took folks like us, scientists working with both regular and dark matter, to start the conversation."

Now that the dark matter model of the universe appears to be the leading one, the next step, Mercado explained, is to see if it remains consistent across a dark matter universe.

More information: Francisco J Mercado et al, Hooks & Bends in the radial acceleration relation: discriminatory tests for dark matter and MOND, Monthly Notices of the Royal Astronomical Society (2024). DOI: 10.1093/mnras/stae819

Provided by University of California, Irvine

Australia concerts in 2024: here’s a list of shows and tours coming up

Artists touring Down Under in 2024 include Coldplay, Pearl Jam, Hozier, Laufey, aespa, IVE and more

Will Champion, Chris Martin and Guy Berryman of Coldplay performing in 2023, photo by Jordi Vidal/Redferns via Getty Images

Fresh out of a pandemic, Australia’s 2023 touring calendar was one of the busiest in recent memory, as artists of all stripes took centrestage everywhere from stadium shows to intimate local gigs. With a starry schedule that included the likes of Post Malone , Red Hot Chili Peppers , and Mötley Crüe , it’ll be tricky for 2024’s schedule to outdo its predecessor – and yet, thanks to the likes of Taylor Swift , Blink-182 and Pink , plus Coldplay, Pearl Jam and SZA, it’s poised to do just that.

From debut headline shows for breakout artists to bustling festival sideshow programs, there’s something for everyone to enjoy as a stellar batch of musicians make their way Down Under in 2024. Read on for NME’s roundup of all the biggest concerts and tours coming to Australia this year.

Here are the concerts and tours coming to Australia in 2024:

James Taylor: An Evening with James Taylor & His All-Star Band When: April 12 – April 28 Find tickets and more info here

If there was ever a festival made for James Taylor & His All-Star Band at this juncture of his 50-year career, it’s Day on the Green . The singer will perform at wineries in Queensland (Sirromet Wines in Mount Cotton), and New South Wales (Bimbadgen in the Hunter Valley, and Centennial Vineyards in Bowral) across April 2024. Taylor will be accompanied by Aussie pair Josh Pyke and Ella Hooper .

Taylor is also playing his own headline shows. After selling out dates in Perth, Melbourne and Sydney, new shows have been added in those cities. A Live Nation pre-sale begins February 9 while general on-sale starts February 12.

Chase & Status When: April 19 – April 28 Find more info and ticket waitlists here

After a new mixtape, charting singles and landmark Boiler Room set, drum’n’bass lifers Chase & Status celebrated a huge 2023. This year, the British duo will bring the party to Australia and New Zealand on a sold-out co-headlining tour with Australia’s very own Luude. They’re notably playing RAC Arena in Perth, on top of Melbourne, Sydney, Adelaide and Brisbane.

Initially slated to appear in Australia in November 2023, Beth Orton rescheduled her Australian tour to April 2024 and added an extra show in Tasmania. The folk musician will now embark on a four-date run with performances in Melbourne, Hobart, Sydney and Brisbane from April 24 to April 30, with tickets for the original cities still valid for the rescheduled dates. Tickets for the additional date at Hobart’s Odeon Theatre are accessible here.

Nick Cave (solo) When: April 25 – May 7 Find tickets and more info

The legendary Nick Cave embarks on a solo tour of Australia (read: without the Bad Seeds) this April, playing two shows in Melbourne and five in Sydney. Radiohead’s Colin Greenwood will be on bass duties for these shows at Plenary Melbourne and State Theatre Sydney, which are all sold out.

The Dandy Warhols When: April 25 – May 1 Find tickets and more info

The Dandy Warhols will make their return to Australia in April 2024 armed with their new album ‘Rockmaker’. So far, they’ve scheduled dates in Brisbane , Adelaide, Melbourne, Perth and Sydney. If you missed them when they toured down under with Hoodoo Gurus in 2022, this is your chance. Find tickets here.

Elephant Gym When: April 26 – April 28 Find more info and tickets here

Elephant Gym isn’t the result of a random band name generator; “elephant” refers to the Taiwanese math rock band’s bass-led grooves, while “Gym” denotes the “agility” of their rhythm. The trio will perform their technical and idiosyncratic instrumentals at three shows on the Australian east coast for the first time this April, supporting sleepmakeswaves.

6LACK: Since I Have A Lover Tour When: April 26 – May 1 Find tickets and more info

R&B favourite 6LACK (pronounced ‘black’) will return to Australia in April to tour his new album, ‘Since I Have A Lover’. The four-date jaunt will take him to Sydney, Brisbane, Melbourne and Perth.

Niall Horan: ‘The Show’ Live On Tour When: April 26 – May 4 Find tickets and more info

Niall Horan will take ‘The Show’ on the road down under in 2024. The ex- One Direction member’s world tour comes in support of his third solo album ‘The Show’ , out June 9. For the Australian leg, he’ll perform arena shows in Brisbane, Sydney and Melbourne.

Groovin The Moo sideshows When: April 29 – May 11 Find tickets and more info

Sadly, Groovin The Moo is no longer going ahead this year – but many of the artists who would have played the touring festival will still be forging ahead with their own Australian headline shows. Singer-songwriter Claire Rosinkranz, Wu-Tang Clan legend GZA (performing a set dubbed ‘Liquid Swords Live’) and Stephen Sanchez will all perform headline shows in Melbourne, Sydney, Brisbane and Perth.

Toronto indie rockers The Beaches will also perform their own dates, playing shows in Sydney, Brisbane and Melbourne in May. Find more information and tickets for all sideshows here .

Jessie Reyez When: April 30 – May 7 Find tickets and more info

Jessie Reyez will tour Australia for the first time in autumn, playing three shows along the east coast. The Canadian R&B singer will bring cuts from 2020 debut ‘Before Love Came to Kill Us’ and 2022 follow-up ‘Yessie’ to Sydney’s Enmore Theatre on April 30, before shows at the Forum in Melbourne and the Tivoli in Brisbane. Find tickets here .

Nothing But Thieves: Welcome To The DCC World Tour When: April 30 – May 7 Find tickets and more info

Nothing But Thieves cleaned up on a sold-out tour of Australia last year – and they’re back for more. The UK alt-rockers have announced the second Aussie leg of their Dead Club City World tour, which will kick off at Sydney’s Hordern Pavilion on April 30, continue on to Brisbane and Melbourne, and wrap up at Hindley Street Music Hall in Adelaide on May 7.

Mahalia: In Real Life When: April 30 – May 8 Find tickets and more info

UK R&B artist Mahalia has booked a five-date tour of Australia, her biggest yet and her first time back down under since 2020. Audiences in Perth, Adelaide, Melbourne, Sydney and Brisbane should get ready for smooth tunes off her 2023 album ‘IRL’ .

READ MORE: Mahalia on her emotional, empowering new record: “I didn’t think I would finish this album”

Jonas Brothers: Five Albums. One Night. Tour When: May 1 – 9 Find tickets and more info

Joe , Nick and Kevin Jonas – the Jonas Brothers – play Australia for the first time between May 1 and May 9 as part of their massive Five Albums. One Night. Tour. The tour will see the brothers perform hits from all five of their albums across one night, including fan favourites.

Tesseract When: May 2 – May 9 Find tickets and more info

UK prog metal outfit Tesseract are set to make the highly anticipated return to Australia in May 2024, marking their first shows Down Under since 2018. Between May 2 and May 9, the band will perform in Brisbane, Sydney, Melbourne, Adelaide and Perth.

With 2023 album ‘War of Being’ and songs from 2021’s ‘Portals’ not played in Australia yet, Tesseract are set to bring with them a fresh bounty of new music to Australia.

The Vaccines and Everything Everything Dates: May 4-11 Find tickets and more info here

UK indie rockers The Vaccines and Everything Everything are banding together for a co-headline tour of Australia, their first time back in the country since 2019 and 2018 respectively. Embrace the indie disco in May when they head to Perth, Adelaide, Melbourne, Brisbane and finally Sydney.

Sammy Virji When: May 10 – May 18 Find more info and tickets here

UK garage has been making a steady comeback, and DJ/producer Sammy Virji is one of its frontrunners. He’s set to come back to Australia for his biggest headline tour of the country yet. Virji will kick off his tour at Metro City in Perth on May 10, before heading through Melbourne, Adelaide, Brisbane and finishing at The Roundhouse in Sydney May 18.

Melissa Etheridge: I’m Not Broken Tour When: May 10 – May 17 Find tickets and more info here

Melissa Etheridge is bringing her trailblazing heartland rock to Australian shores for the first time in five years in May 2024. The ‘I’m Not Broken’ tour kicks off in Perth on May 10, before heading through Adelaide, Melbourne and finishing up in Sydney on May 17. Expect to hear a blend of the songwriter’s greatest hits as heard on her latest live album ‘Beautiful Day’.

Macklemore When: May 11 – May 20 Find tickets and more info

Fresh off the release of latest album ‘Ben’, Macklemore will embark on an Australian tour next May. The hip-hop artist will perform at Hordern Pavillion in Sydney on May 11-12, before taking to Melbourne’s John Cain Arena and Brisbane’s Riverstage on May 15 and May 17, respectively. Macklemore will perform at BASSINTHEGRASS in Darwin, and then conclude his Australian run at HBF Stadium in Perth on May 20. Find tickets here.

READ MORE: Does Rock ‘N’ Roll Kill Braincells?! – Macklemore

Sleater-Kinney When: May 17 – May 23 Find tickets and more info

In their first shows here since 2016, Sleater-Kinney are embarking on a theatre tour of Australia. Corin Tucker and Carrie Brownstein will play four headline shows down under in support of their latest album, ‘Little Rope’ , in what the latter has called a “homecoming” of sorts.

READ MORE: Sleater-Kinney talk new single ‘Untidy Creature’ and grief-driven new album: “This is not a somber record”

“For all intents and purposes, Sleater-Kinney got its start in Australia,” Brownstein said in a statement. “We recorded our first album and played our first ever shows there. Because of that, Australia feels like one of the band’s spiritual homes, and returning there always feels like a homecoming, a reunion.”

Jungle When: May 17 – May 22 Find more info

Jungle have unleashed more dancey goodness with their latest album, ‘Volcano’. Get ready to boogie in May when they tour Australia. The entire run, comprising the Hordern Pavilion in Sydney, Festival Hall in Melbourne, and Fortitude Music Hall in Brissy, is sold out.

Botch When: May 17 – 25 Find tickets and more info

Cult-favourite mathcore/hardcore/metal band Botch from Tacoma, Washington reunited last year more than two decades after their split – and now they’re embarking on their first-ever Aussie tour. The jaunt will include two dates in Melbourne and Sydney apiece as well as shows in Adelaide, Perth and Brisbane.

Guitarist David Knudson has promised Botch will go all out: “We don’t want to half-ass it…we want to be fucking tight as we ever were, if not tighter. We don’t want to disappoint a fan that’s never seen us live. So you know, it’s all in or nothing.”

Peter Hook & The Light When: May 21 – 28 Find tickets and more info

Peter Hook , the legendary bassist and co-founder of Joy Division and New Order is set to return to Australia in May 2024, just two years after his last tour of the country in 2022. For the upcoming tour, Hook will perform the New Order and Joy Division substance compilations live, giving fans a chance to once again revel in both bands’ music.

The Snuts When: May 23 – May 26 Find tickets and more info

Scottish indie rockers The Snuts will return for their second Australian tour in as many years, following their debut run of headline shows in 2023. With third studio album ‘Millennials’ in tow, the band will kick off their 2024 tour with a show at Melbourne’s Northcote Theatre. They’ll play Sydney’s Metro Theatre on May 25, before a show at the Triffid in Brisbane the following evening. Find tickets here .

Tom Grennan When: May 29 – May 30 Find tickets and more info

Tom Grennan will duck over to Australia for a pair of headline shows this year, following a debut visit in 2022. The English singer-songwriter will play Sydney’s Metro Theatre and 170 Russell in Melbourne on May 29 and 30 respectively.

Since his last trip, Grennan has released his third studio album, ‘What Ifs & Maybes’. Find tickets here .

Sky Ferreira When: June 2 – June 4 Find more info and tickets here

It’s been 12 years since Sky Ferreira released her debut ‘Night Time, My Time’, and her mystique has only grown in the interim: Ferriera has explored the silver screen, modelling, and long teased the release of a second album ‘Masochism’ . She’ll play two rare headline shows at bespoke Melbourne and Sydney festivals RISING and Vivid LIVE this June – a decade after her last Australian performance.

Boney M featuring Maizie Williams: The Farewell Tour When: June 3 – July 6 Find tickets and more info

Legendary disco group Boney M and vocalist Maizie Williams will bid goodbye to Australia with an extensive national tour in June and July 2024. Don’t wait to get your tickets – 15 of the 20 shows are sold out. More info here.

Bar Italia When: June 4 – June 8 Find more info and tickets here

Buzzy London trio Bar Italia are proteges of Dean Blunt, and have emerged from relative anonymity in the last few years with two albums of sinister post-punk. They’ll tour Australia for the first time in June, playing at Oxford Art Factory in Sydney and Brisbane’s Black Bear Lodge before a matinee show in Melbourne as part of RISING festival.

Deerhoof: Miracle Level Tour When: June 12 – June 16 Find tickets and more info

Indie vets Deerhoof return to Australia this June for the Miracle Level Tour, in support of their 19th (!) album of the same name. They’ll play five cities in five days, kicking off in Melbourne and then heading to Sydney for Vivid before Brisbane, Adelaide and finally Perth.

LANY When: June 19 – June 28 Find tickets and more info

LANY – the pop duo of Paul Klein and Jake Goss – will return to Australia in mid-2024. After playing small, intimate shows down under in August 2023, they’ll go bigger in this national tour at venues including the Hordern Pavilion and Margaret Court Arena. See info on dates and tickets here.

Lizzy McAlpine: The Older Tour When: July 11 – July 18 Find tickets and more info

A year after breaking out with TikTok hit ‘Ceilings’, Lizzy McAlpine has released her major-label debut, the album ‘Older’. To celebrate, she’ll stage her debut headline shows in Australia with four stops across Brisbane, Sydney and Melbourne.

Conan Gray When: July 11 – July 19 Find more info and tickets here

Conan Gray is the archetypal Gen Z popstar, honing his craft as a teenage YouTube vlogger before unleashing his multimodal talent as a singer. Gray will play songs from his upcoming third album ‘Found Heaven’ in some of the biggest rooms in the country this July, as well as a headline spot at Adelaide’s Spin Off Festival on July 19.

Girl In Red: Doing It Again: Asia/Oceania Tour When: July 11 – July 21 Find tickets and more info

She’s doing it again: Marie Ulven aka Girl In Red returns to Australia after selling out her headline tour here last year. The Norwegian alt-pop sensation will play Perth, Sydney, Melbourne and Brisbane this go round. Pre-sale begins Friday April 26, while general on-sale kicks off Monday April 29.

READ MORE: Girl In Red on her new album: “It’s ballsy, big dick energy”

Tenacious D When: July 13 – July 22 Find tickets and more info

It’s been more than a decade since Tenacious D toured Australia, but that all changes in July. Jack Black and Kyle Gass will play their first shows in the country since 2013, performing six arena dates.

The tour will kick off with two shows at the ICC Sydney Theatre on July 13, continuing on to Newcastle, Brisbane and Melbourne before wrapping up at Adelaide Entertament centre on July 22. Find tickets here .

FLETCHER When: July 16 – July 28 Find tickets and more info

After postponing her original tour, the rescheduled dates for FLETCHER ’s long-awaited Australian visit have been locked in for mid-2024. Audiences across the country will revel in FLETCHER’s latest album ‘Girl Of My Dreams’ towards the end of July, with venues including Perth’s Metro City (July 16), Fortitude Music Hall in Brisbane (July 18), Festival Hall in Melbourne (July 23) and Sydney’s Hordern Pavilion (July 28). Find tickets here.

The Last Dinner Party When: July 20 – July 23 Find tickets and more info

Nothing else will matter when The Last Dinner Party touch down in Australia this July. See the buzzy baroque indie pop band – and NME Cover stars – in Brisbane (a co-headline with TV Girl), Melbourne and Sydney (both supported by Tia Gostelow).

READ MORE: The Last Dinner Party: the newly-coronated monarchs of baroque-pop

IVE: 1ST WORLD TOUR ‘SHOW WHAT I HAVE’ When: July 25 – July 28 Find tickets and more info

As part of their broader debut world tour, IVE will bring their ‘Show What I Have’ set to Australian shores in July, kicking off the two-date run at Melbourne’s Rod Laver Arena on July 25. The K-pop group will conclude the Australian leg of their tour in support of 2023 album ‘I’ve IVE’ with a show at Sydney’s Qudos Bank Arena on July 28. Find tickets here.

READ MORE: IVE – ‘I’VE MINE’ review: K-pop’s queen bees grow beyond the archetype

HEALTH: Rat Based Warfare Tour Down Under When: July 31 – August 4 Find more info and tickets here

The electronic body music of HEALTH flirts with metal, noise, and synth wave, but remains uniquely their own. The band are back in Australia after their 2023 Dark Mofo appearance for a full tour in support of their seventh album ‘Rat Wars’. They’ll play cosy rooms in Perth, Adelaide, Melbourne, Sydney and Brisbane from late July this year.

Greta Van Fleet: Starcatcher World Tour When: August 21 – August 27 Find tickets and more info

Classic rock revivalists Greta Van Fleet return to Australia for the first time in five years this August, supporting their 2023 album ‘Starcatcher’. They’ll kick things off in Brisbane then head to Sydney and Melbourne.

aespa: SYNK: Parallel Line Tour When: August 31 – September 2 Find more info

K-pop girl group aespa will bring all the dra-ma-ma-ma down under for two stops of their SYNK: Parallel Line tour. The four-piece of Karina , Giselle , Winter and NingNing will perform at Sydney’s Qudos Bank Arena on August 31 and head to Melbourne for the Rod Laver Arena two days later. A Telstra pre-sale begins May 7 while a TEG Live + Ticketek pre-sale begins two days later. General on-sale starts May 10.

READ MORE: The 25 best K-pop songs of 2023

Iron Maiden: The Future Past Tour When: September 1 – September 13 Find tickets and more info

In what will be their first trip Down Under since 2017, Iron Maiden are poised to bring The Future Past tour to stadiums in Perth, Adelaide, Brisbane, Sydney and Melbourne next September.

The first three cities will host the metal legends between September 1 to September 10, while Melbourne and Sydney will each enjoy a two-night outing at Rod Laver Arena (September 6 and 7) and Qudos Bank Arena (September 12 and 13), respectively. Find tickets here.

J Balvin: Que Bueno Volver a Verte Tour When: September 4 – September 10 Find more info and tickets here

J Balvin’s career as the Prince of Reggaeton has dovetailed with an explosion in the popularity of Latin music worldwide. The juggernaut will bring his Que Bueno Volver a Verte Tour (It’s Good To See You Tour) to Australian arenas in September this year. Sofi Tukker will support him as he plays the biggest rooms in Melbourne, Sydney, Brisbane and Perth.

Laufey: Bewitched: The Goddess Tour When: September 7 – September 14 Find tickets and more info

Get ready to fall under Laufey’s spell this September. The young jazz-pop star’s first Aussie tour last year sold out in minutes, and the Bewitched: The Goddess Tour looks set to do the same, with pre-sales already fully exhausted. She’ll play two shows in Sydney and Melbourne apiece, and also perform in Brisbane – general on-sale begins Friday April 26.

READ MORE: Laufey on winning her first Grammy: “There is no way to prepare. It was really amazing”

SiM: Playdead World Tour When: September 12 – September 15 Find tickets and more info here

Like a kind of Japanese Gorillaz , SiM are a content universe unto their own – spanning anime , video games and records. Australian audiences can experience their inimitable reggae-metal-punk at The Zoo in Brisbane September 12, Sydney’s Crowbar on September 14 or Max Watts in Melbourne on September 15.

Thirty Seconds To Mars: Seasons World Tour When: September 12 – September 17 Find tickets and more info

To announce Thirty Seconds To Mars ’ Seasons World Tour, Jared Leto pulled a massive stunt: scaling the Empire State Building . The tour comes in support of their album ‘It’s The End Of The World But It’s A Beautiful Day’ and will hit Australian shores in mid-September. It marks the band’s first headline tour in over five years, and will see them grace the stage at Melbourne’s Rod Laver Arena (September 12), Qudos Bank Arena in Sydney (September 14) and Brisbane’s Riverstage (September 17). Find tickets here.

Ne-Yo: Champagne & Roses Tour When: September 26 – October 3 Find tickets and more info

Come closer – Ne-Yo is coming back to Australia for the Champagne & Roses Tour with special guest Lloyd. Expect smooth throwback R&B in Melbourne, Adelaide, Perth, Brisbane and Sydney.

READ MORE: Soundtrack Of My Life: Ne-Yo

The Reytons When: September 29 – October 5 Find more info and tickets here

Yorkshire four-piece The Reytons are revivalists of another revival – 2000s British indie. But this band is independent in the truer sense of the word, self-releasing their latest album ‘Ballad of a Bystander’. They will tour their raucous, ungenteel rock from late September to early October this year, packing rooms in Perth, Adelaide, Brisbane, Sydney, and Melbourne.

The Kid LAROI When: October 2024 Find tickets and more info

US-based homegrown rap phenom The Kid LAROI ‘s return to Australia was supposed to take place in February, but it was postponed in December , with the Gamilaroi star calling February “logistically impossible”. Instead, he’ll fill stadiums around the country in October with tracks from his freshly released debut album ‘The First Time’. With shows locked in for Melbourne, Perth, Adelaide, Sydney and the Gold Coast, these all-ages shows will be his biggest to date. Stay tuned for more info on specific dates for the rescheduled tour.

READ MORE: The Kid LAROI live in Sydney: A hometown hero returns a certified legend

Passenger When: October 17 – October 23 Find tickets and more info

Passenger’s first Australian tour in six years also serves as a celebration of his 2012 album ‘All The Little Lights’, which received an anniversary edition last year. This jaunt will feature Josh Pyke as supporting act and, notably, two shows at the Sydney Opera House (one of which is already sold out).

READ MORE: Soundtrack Of My Life: Passenger

Kim Wilde: The Greatest Hits Tour When: October 17 – October 26 Find more info and tickets here

If you look at the pop charts for any given week in the 1980s, chances are Kim Wilde was on it. The ‘Kids in America’ singer is still performing, on a break from her new life as a gardener, and will play a whirlwind Australian tour this October. Wilde will begin in Brisbane on October 17, before heading through Tweed Heads, Sydney, Wollongong, Perth, Adelaide and finishing in Melbourne October 26.

READ MORE: Soundtrack of My Life: Kim Wilde

Buzzcocks When: October 24 – November 2 Find more info and tickets here

The Buzzcocks , progenitors of “love punk” and pop punk before it even existed, are still going strong almost 50 years after their formation. With Steve Diggle on vocals, replacing the late Pete Shelley, the band will return to Australia for seven shows in late October to early November. The Buzzcocks will begin on the Gold Coast on October 24, before travelling through Brisbane, Adelaide, Fremantle, Newcastle, Sydney and finishing in Melbourne on November 2.

PinkPantheress: Capable of Love Tour When: October 29 – November 5 Find tickets and more info here

PinkPantheress was once an anonymous beatmaker going viral on TikTok – she’s anything but anonymous now, as she comes to Australia for the first time promoting her debut album ‘Heaven’. Between opening for Coldplay , the 22-year-old Brit will headline shows at Melbourne’s Festival Hall (October 29) and Horden Pavilion in Sydney on November 5.

Coldplay: Music of the Spheres Tour When: October 30 – November 9 Find tickets and more info here

Coldplay will bring their ‘Music of the Spheres’ tour to Australia’s east coast at the end of a more than two-year stretch. The pop juggernauts have already sold 9 million tickets – the most for any tour in history – and are set to play eight stadium shows in Melbourne and Sydney after playing in Perth in 2023.

Expect to experience the galactic sprawl of the band’s recent rock operas, while still belting out the anthems that took them to the top.

Take That and Sophie Ellis-Bextor When: October 30 – November 10 Find tickets and more info

It’ll be murder on the dancefloor when this tour comes to town. Sophie Ellis-Bextor , enjoying a Saltburn -fueled resurgence, will accompany Take That on a six-show tour of Australia . Three dates of the tour, which marks Take That’s first live shows in the region since 2017, will take place at wineries as part of A Day on the Green, where Ricki-Lee Coulter will also appear.

READ MORE: Sophie Ellis-Bextor on the return of ‘Murder On The Dancefloor’ – and watching Saltburn with her mum

Hozier: Unreal Unearth Tour When: November 6 – November 18 Find more info and tickets here

Irish singer-songwriter Hozier will tour Down Under for the first time since 2019 this November, supporting his 2023 album ‘Unreal Unearth’ with a nation-wide arena sojourn. Hozier will kick off in Perth on November 6, before heading through Adelaide, Melbourne, Sydney, and Brisbane. He will also play regional areas Torquay and the Hunter Valley as part of Summersalt festival.

READ MORE: Hozier: “There has always been a space in my work for my own conscience”

Tate McRae: Think Later World Tour When: November 8 – November 17 Find tickets and more info

In what marks her biggest headline Australian shows to date, Tate McRae’s Think Later World Tour will head Down Under in November, encompassing five shows in Perth (November 8), Brisbane (November 10), Sydney (November 12), Adelaide (November 15) and Melbourne (November 17). The tour comes in support of McRae’s sophomore album ‘Think Later’, which features her massive single ‘Greedy’. Tickets are available here.

Pearl Jam: Dark Matter World Tour 2024 When: November 13 – November 23 Find tickets and more info

Pearl Jam have announced their new album ‘Dark Matter’, which they’ll support with a massive tour around the world . They’ll make a stop Down Under with Pixies in support, playing stadium shows in the Gold Coast, Melbourne and Sydney. The grunge giants have added bonus gigs in Melbourne on November 18 and Sydney on November 23 in response to overwhelming demand. Find tickets here.

James Blunt: The Who We Used To Be Tour When: November 21 – November 28 Find tickets and more info here

Self-deprecating superstar James Blunt is returning to Australian stages for the first time in over six years in November 2024. The ‘You’re Beautiful’ singer will begin a five-date arena tour at Brisbane’s Riverstage on November 21, before heading through Sydney’s ICC Super Theatre (November 23), Melbourne’s Rod Laver Arena (November 24), Adelaide’s Entertainment Centre Theatre (November 25) and Perth’s Red Hill Auditorium (November 28).

Wallows: Model Tour 2024 When: December 5 – December 14 Find more info and tickets here

US alt-rockers Wallows played some of their favourite shows ever on their previous tour of Australia – so they’ll undoubtedly be looking to top the experience when they return in December. They’ll kick things off in Perth, before heading to Adelaide, Melbourne, Sydney and finally Brisbane.

Martin Gore (left) and Dave Gahan (right) performing live onstage with Depeche Mode at the Golden 1 Center arena in Sacramento, California on March 23, 2023

Depeche Mode

Aussie Depeche Mode fans are an incredibly patient bunch. The last time the British synth-pop titans toured the country was in 1994, meaning nearly three decades have passed since they last paid us a visit. But good things come to those who wait, and it seems there is a glimmer of hope for those hoping to catch the band Down Under.

READ MORE: Depeche Mode: every single album ranked and rated

In March 2023, Depeche Mode’s Dave Gahan told an Italian blog that the band were eyeing “as-yet-unannounced dates in Asia and Australia” as part of a touring run that would lead into 2024. The band have toured the world in support of their latest album, ‘Memento Mori’ , and a trip to Australia to cap off the jaunt seems more likely than ever.

Additional reporting by Ellie Robinson, Tom Disalvo and Josh Martin

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Everything Everything
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Iron Maiden
James Blunt
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Schuyler Dixon

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Taylor Pendrith, right, laughs with fellow competitor Davis Riley, left, at the second hole tee box during the third round of the Byron Nelson golf tournament in McKinney, Texas, Saturday, May 4, 2024. (AP Photo/LM Otero)

MCKINNEY, Texas – Taylor Pendrith has been here before — holding or sharing a lead after three rounds on the PGA Tour.

The Canadian was one of six among the top 10 on the leaderboard seeking a first tour victory at the CJ Cup Byron Nelson.

IMAGES

Challenge and Support Theory (Nevitt Sanford)
Understanding The Social Support Theory [Guide For 2023]
Challenge and Support Theory (Nevitt Sanford)
Social Support Theory by Aristeo Ruiz on Prezi
The Broaden-and-Build Theory of Positive Emotions (Source: Cohn
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COMMENTS

PDF A Tour of Support Theory for Triangulated Categories Through Tensor
A TOUR OF SUPPORT THEORY FOR TRIANGULATED CATEGORIES THROUGH TENSOR TRIANGULAR GEOMETRY GREG STEVENSON Abstract. These notes attempt to give a short survey of the approach to sup-port theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the ma-
[PDF] A tour of support theory for triangulated categories through
These notes give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for ...
[1601.03595v1] A tour of support theory for triangulated categories
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support ...
A Tour of Support Theory for Triangulated Categories Through Tensor
A Tour of Support Theory for Triangulated Categories Through Tensor Triangular Geometry. In: Herbera, D., Pitsch, W., Zarzuela, S. (eds) Building Bridges Between Algebra and Topology. Advanced Courses in Mathematics - CRM Barcelona.
(PDF) A tour of support theory for triangulated categories through
These notes give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for ...
A tour of support theory for triangulated categories through tensor
Our theory of stratification is based on the approach of Stevenson which uses the Balmer-Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian.
A Tour of Support Theory for Triangulated Categories Through Tensor
Special cases include abelian groups (the Hasse square), chromatic homotopy theory (a module theoretic chromatic fracture square), and rational torus-equivariant homotopy theory (first step to the ...
A tour of support theory for triangulated categories through tensor
Upload an image to customize your repository's social media preview. Images should be at least 640×320px (1280×640px for best display).
A tour of support theory for triangulated categories through tensor
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support ...
A Tour of Support Theory for Triangulated Categories Through Tensor
A Tour of Support Theory for Triangulated Categories Through Tensor Triangular Geometry Arxiv:1810.04306V2 [Math.CT] 20 Jan 2021 Arxiv:1805.12454V1 [Math.GN] 31 May 2018 Date: September 21, 2018
A tour of support theory for triangulated categories through tensor
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of... Skip to main content. We will keep fighting for all libraries - stand with us! A line drawing of the Internet Archive headquarters building façade. ...
New Look at Social Support: A Theoretical Perspective on Thriving
This perspective requires us to take a new look at social support and to re-conceptualize it in terms of the promotion of positive well-being instead of only buffering stress - and to view it as an interpersonal process that unfolds over time instead of an attitude or expectation (e.g., perceived available support).
Social Support Theory
Abstract. Social support theory emerged from publications by Don Drennon-Gala and Francis Cullen, both of whom drew on insights from several theoretical traditions. The theory is centered on the proposition that instrumental, informational, and emotional supports reduce the likelihood of delinquency and crime. The theory incorporates macro- and ...
Social Support Theoretical Perspective
Social Support Theoretical Perspective. Overweight and obesity have been noted as a major health problems among Hispanics, particularly Hispanic women. 1 Data indicates that approximately 39% of Hispanic women are overweight, with 47% having a body mass index (BMI) greater than 27. 1 Mexican Americans exhibit the highest rate of obesity and the ...
Social Support
Overview and History. Thoits 1995 provides a classic overview of the study of social support and the direction of the field at that time. A thorough and more current presentation of the development of social support and its implications for individuals is given by Taylor 2011. Ditzen and Heinrichs 2014 provides a concise, more recent introduction. Cohen and Wills 1985 presents the two main ...
(PDF) Social support: Theory, Measurement and Intervention
PDF | On Jan 1, 2000, B. Lakey and others published Social support: Theory, Measurement and Intervention | Find, read and cite all the research you need on ResearchGate
Social Support Theory
Social support theory emerged from publications by Don Drennon-Gala and Francis Cullen, both of whom drew on insights from several theoretical traditions. The theory is centered on the proposition that instrumental, informational, and emotional supports reduce the likelihood of delinquency and crime. The theory incorporates macro- and micro ...
Social support theory.
Social support is considered a middle range theory that focuses on relationships and the interactions within those relationships. The importance of social relationships in contributing to health and well-being has been the focus of research by scientists and practitioners across a large number of social, behavioral, medical, and nursing disciplines.
Perceived organizational support and well‐being of tour guides: The
Drawing on the organizational support theory and theoretical conceptualizations on well-being, the present study proposes a conceptual model linking perceived organizational support (POS) with quality of work life, life satisfaction, and burnout. ... To communicate support to tour guides, travel agencies and tour operators should embrace a ...
Tour vs Path in graph theroy
1. A path is a walk with no repeated vertices. A trail is a walk with no repeated edges. A tour is a walk that visits every vertex returning to its starting vertex. A tour could visit some vertices more than once. If you visit them exactly once, then the tour is a Hamiltonian cycle. A cycle is a walk in which the end vertex is the same as the ...
PDF http://eprints.gla.ac.uk/167512/
A TOUR OF SUPPORT THEORY FOR TRIANGULATED CATEGORIES THROUGH TENSOR TRIANGULAR GEOMETRY GREG STEVENSON Abstract. These notes attempt to give a short survey of the approach to sup-port theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the ma-
Challenge and Support Theory (Nevitt Sanford)
Description of this Theory. Nevitt Sanford theorized that student development depends on both challenge and support. Finding the sweet space f between challenge and support will lead to growth. Challenges occur in situations where an individual does not have knowledge, skills, or attitudes to cope. This often occurs when entering a college campus.
Astronomers' simulations support dark matter theory
University of California - Irvine. (2024, April 29). Astronomers' simulations support dark matter theory. ScienceDaily. Retrieved May 4, 2024 from www.sciencedaily.com / releases / 2024 / 04 ...
UC Irvine astronomers' simulations support dark matter theory
Irvine, Calif., April 29, 2024 — Computer simulations by astronomers support the idea that dark matter - matter that no one has yet directly detected but which many physicists think must be there to explain several aspects of the observable universe - exists, according to the researchers, who include those at the University of California, Irvine.
Kamala Harris Leads Push to Shore Up Democratic Support From Black
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A Tour of Support Theory for Triangulated Categories Through Tensor Triangular Geometry

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A tour of support theory for triangulated categories through tensor triangular geometry

Tensor-triangular fields: ruminations

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In This Article Expand or collapse the "in this article" section Social Support

Social Support by Robert G. Kent de Grey , Bert N. Uchino , Ryan Trettevik , Sierra Cronan , Jasara Hogan LAST REVIEWED: 11 January 2018 LAST MODIFIED: 11 January 2018 DOI: 10.1093/obo/9780199828340-0204

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