\providecommand\noopsort

[1]

Support theories for non-Noetherian
tensor triangulated categories

Changhan Zou Changhan Zou, Mathematics Department, UC Santa Cruz, 95064 CA, USA [email protected] https://people.ucsc.edu/

\sim

czou3

(Date: December 13, 2023)

Abstract.

We extend the support theory of Benson–Iyengar–Krause to the non-Noetherian setting by introducing a new notion of small support for modules. This enables us to prove that the stable module category of a finite group is canonically stratified by the action of the Tate cohomology ring, despite the fact that this ring is rarely Noetherian. In the tensor triangular context, we compare the support theory proposed by W. Sanders (which extends the Balmer–Favi support theory beyond the weakly Noetherian setting) with our generalized BIK support theory. When the Balmer spectrum is homeomorphic to the Zariski spectrum of the endomorphism ring of the unit, the two support theories coincide as do their associated theories of stratification. We also prove a negative result which states that the Balmer–Favi–Sanders support theory can only stratify categories whose spectra are weakly Noetherian. This provides additional justification for the weakly Noetherian hypothesis in the work of Barthel, Heard and B. Sanders. On the other hand, the detection property and the local-to-global principle remain interesting in the general setting.

1. Introduction

The fundamental theorem of tensor triangular geometry [Bal05] unifies major classification theorems in algebraic geometry, modular representation theory, and stable homotopy theory [Tho97, BCR97, HS98]. The theorem states that the radical thick ideals of an essentially small tensor triangulated category are classified by the Thomason subsets of its Balmer spectrum via the universal theory of support. However, such categories often arise as the subcategory of compact objects inside a bigger rigidly-compactly generated tensor triangulated category. Understanding these big tt-categories leads to the problem of classifying their localizing ideals via some theory of support for big (non-compact) objects. The first such classification theorem was obtained by Neeman [Nee92], who proved that for a commutative Noetherian ring $A$ , the usual cohomological support (defined by tensoring with the residue fields) induces a bijection

\{\text{localizing subcategories of }\operatorname{D}(A)\}\xrightarrow{\sim}\{\text{subsets of }\operatorname{Spec}(A)\}.

Neeman [Nee00] also showed that such a classification can fail if $A$ is not Noetherian. He exhibited a truncated polynomial ring $A$ in infinitely many variables with the property that $\operatorname{D}(A)$ has lots of localizing subcategories while $\operatorname{Spec}(A)$ consists of a single point. In fact, Dwyer and Palmieri [DP08] constructed examples of nontrivial tensor-nilpotent objects in $\operatorname{D}(A)$ . This demonstrates that the cohomological support need not even detect vanishing of objects if the ring is non-Noetherian.

Nevertheless, various authors have constructed support theories for big categories under certain Noetherian hypotheses and used them to prove analogous classification theorems in different subjects. The current paper aims to study these notions of support without making any Noetherian assumptions.

\ast\ast\ast

In a series of papers [BIK08, BIK11a, BIK11b] Benson, Iyengar, and Krause developed a theory of support for objects in a compactly generated triangulated category $\mathscr{T}$ equipped with an action of a graded-commutative Noetherian graded ring $R$ . If $\mathscr{T}$ is a rigidly-compactly generated tensor triangulated category then their support function induces a map

\{\text{localizing ideals of }\mathscr{T}\}\xrightarrow{\operatorname{Supp}_{\textup{BIK}}}\{\text{subsets of }\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})\}

where $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})\subseteq\operatorname{Spec^{h}}(R)$ is the space of supports. The category $\mathscr{T}$ is said to be BIK-stratified by $R$ if this map is a bijection. A major application of this machinery is that the stable module category $\operatorname{StMod}(kG)$ of a finite group $G$ (or more generally, a finite group scheme) over a field $k$ of positive characteristic is BIK-stratified by the group cohomology ring $H^{*}(G,k)$ (see [BIK11a, BIKP18]).

Note that this construction of support relies on an auxiliary action by a Noetherian ring. Thanks to the monoidal structure, the graded endomorphism ring $\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1})$ of the unit object canonically acts on $\mathscr{T}$ but it has no reason to be Noetherian in general. For instance, $\operatorname{End}_{\operatorname{StMod}(kG)}^{*}(\mathbb{1})$ is the Tate cohomology ring $\smash{\hat{H}^{*}(G,k)}$ which is usually non-Noetherian. This motivated the author to develop a BIK-style support theory without assuming that the ring acting on the category is Noetherian:

Theorem A (4.13).

Let $\mathscr{T}$ be a compactly generated triangulated category equipped with an action by a graded-commutative graded ring $R$ . There is an associated support theory which assigns a subset

\operatorname{Supp}_{\textup{BIK}}(t)\subseteq\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))

to every object $t\in\mathscr{T}$ . This support theory recovers the one in [BIK08] when $R$ is Noetherian.

A key ingredient of this theory is a new notion (3.7) of small support for modules over commutative rings that are not necessarily Noetherian. This notion makes use of weakly associated primes which behave well for non-Noetherian rings. For example, we show that the local modules and torsion modules can still be recognized from their supports; see 3.19.

\ast\ast\ast

In the tensor triangular world, Balmer and Favi [BF11] proposed a notion of support for rigidly-compactly generated tensor triangulated categories which takes values in the Balmer spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ of compact objects. The construction of the Balmer–Favi support requires $\operatorname{Spc}(\mathscr{T}^{c})$ to be weakly Noetherian, meaning that for every $\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})$ there exist Thomason subsets $U$ and $V$ with $\{\mathscr{P}\}=U\cap V^{c}$ . A point satisfying this condition is called weakly visible. Based on this notion of support, Barthel, Heard and B. Sanders [BHS23b] developed a theory of stratification which applies to any category $\mathscr{T}$ whose spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian. In particular, $\mathscr{T}$ is said to be stratified if the map induced by the Balmer–Favi support

\{\text{localizing ideals of }\mathscr{T}\}\xrightarrow{\operatorname{Supp}_{\text{BF}}}\{\text{subsets of }\operatorname{Spc}(\mathscr{T}^{c})\}

is a bijection. Moreover, the Balmer–Favi support theory is the “universal” one for stratification in the weakly Noetherian context; see [BHS23b, Theorem 7.6] for a precise statement.

However, there are tensor triangulated categories whose spectra are not weakly Noetherian. A prominent example is the stable homotopy category. Another example is the derived category of a polynomial ring in infinitely many variables. Therefore, a more general notion of support is needed.

W. Sanders [San17] has proposed a generalization of the Balmer–Favi support theory which does not require the spectrum to be weakly Noetherian. We will call this Balmer–Favi–Sanders support the tensor triangular support. A crucial feature of this theory is that the support of an object $t\in\mathscr{T}$ is a closed subset with respect to certain topology on the Balmer spectrum called the localizing topology. This topology is generated by subsets of the form $U\cap V^{c}$ , where $U$ and $V$ are Thomason subsets. Hence the Balmer spectrum is weakly Noetherian precisely when its localizing topology is discrete. Therefore, the following definition recovers stratification in the sense of [BHS23b] when $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian:

Definition.

A rigidly-compactly generated tensor triangulated category $\mathscr{T}$ is stratified if the map induced by the tensor triangular support

(1.1)

\{\text{localizing ideals of }\mathscr{T}\}\xrightarrow{\operatorname{Supp}}\{\text{localizing closed subsets of }\operatorname{Spc}(\mathscr{T}^{c})\}

is a bijection.

At this point, one may wonder if there is any stratified category with non-weakly Noetherian spectrum. Perhaps surprisingly, the answer is no:

Theorem B (8.13).

If a rigidly-compactly generated tensor triangulated category $\mathscr{T}$ is stratified then the Balmer spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian.

However, without the weakly Noetherian assumption, we can still define a notion of local-to-global principle (7.1) which is a necessary condition for stratification. In [BHS23b, Theorem 3.21] it was shown that if $\operatorname{Spc}(\mathscr{T}^{c})$ is Noetherian then $\mathscr{T}$ satisfies the local-to-global principle. We strengthen their result as follows:

Theorem C (7.6).

A rigidly-compactly generated tensor triangulated category $\mathscr{T}$ satisfies the local-to-global principle if the Balmer spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ is Hochster weakly scattered.

The relation between Noetherian and Hochster weakly scattered spectral spaces can be depicted as follows (see 2.16):

\text{Noetherian}\implies\text{Hochster scattered}\iff\begin{gathered}\text{Hochster weakly scattered}\\ +\\ \text{weakly Noetherian}.\end{gathered}

An immediate consequence of B is that the stable homotopy category is not stratified. Nevertheless, we can show that it satisfies the local-to-global principle and hence the detection property. As an application, we give a support theoretical description of the (non-zero) $p$ -local dissonant spectra: they are precisely the $p$ -local spectra supported at the single point at height infinity in the chromatic picture. Note that the Balmer–Favi support does not “see” this point since it is not weakly visible.

In order to prove B, we study relations between the homological support proposed by Balmer [Bal20a] and the tensor triangular support, which were established in [BHS23a] under the weakly Noetherian hypothesis. Our generalizations of the comparison results in [BHS23a] may be of independent interest; see Section 8.

\ast\ast\ast

For a rigidly-compactly generated tensor triangulated category $\mathscr{T}$ , we now have the tensor triangular support $\operatorname{Supp}$ which takes values in the Balmer spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ and the canonical BIK support $\operatorname{Supp}_{\textup{BIK}}$ which takes values in the homogeneous Zariski spectrum $\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ . Moreover, there is a comparison map

\rho\colon\operatorname{Spc}(\mathscr{T}^{c})\to\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))

introduced in [Bal10]. It is then natural to ask how the two support theories are related. We prove the following:

Theorem D (9.3).

Let $\mathscr{T}$ be a rigidly-compactly generated tensor triangulated category such that $\rho$ is a homeomorphism. Then $\rho(\operatorname{Supp}(t))=\operatorname{Supp}_{\textup{BIK}}(t)$ for any $t\in\mathscr{T}$ .

Inspired by (1.1), we say that $\mathscr{T}$ is cohomologically stratified if the map induced by the canonical BIK support

\{\text{localizing ideals of }\mathscr{T}\}\xrightarrow{\operatorname{Supp}_{\textup{BIK}}}\{\text{closed subsets of }\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})\}

is a bijection, where $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$ inherits the localizing topology on $\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ . A corollary of D is that if the comparison map $\rho$ is a homeomorphism then $\mathscr{T}$ is stratified if and only if it is cohomologically stratified; see 9.10. However, a category can be cohomologically stratified without $\rho$ being a homeomorphism. This is the case for the following example:

Theorem E (9.16).

The stable module category $\operatorname{StMod}(kG)$ is cohomologically stratified.

\ast\ast\ast

The paper is organized as follows. In Section 2 we record some basic facts about spectral spaces. In particular, we discuss the localizing topology (and its relation with the constructible topology), which will be used throughout this paper. In Section 3 we define the notion of (small) support for modules over (non-Noetherian) commutative rings. In Section 4 we establish the non-Noetherian BIK support theory and prove A. In Section 5 we study how the tensor triangular support behaves under base-change functors. In Section 6 we prove that the detection property for the tensor triangular support is an algebraically local property. In Section 7 we introduce a local-to-global principle and prove C. In Section 8 we establish B. Finally, we prove D and E in Section 9.

Acknowledgements

The author is grateful to Beren Sanders for inspiring discussions and his constant support. He also thanks Paul Balmer for useful conversations and the organizers of the Oberwolfach workshop Tensor-Triangular Geometry and Interactions for their invitation to present some part of this work.

2. Preliminaries on spectral spaces

We start by recalling some basic concepts concerning spectral spaces.

2.1 Definition.

Let $X$ be a spectral space in the sense of [DST19]. A subset of $X$ is Thomason if it is a union of closed subsets, each of which has quasi-compact complement. The Thomason subsets form the open subsets of a dual spectral topology on $X$ called the Hochster dual topology.¹¹1The Hochster dual topology is called the inverse topology in [DST19]. We write $X^{*}$ for $X$ equipped with the Hochster dual topology.

2.2 Definition.

Let $X$ be a spectral space. A subset $W$ of $X$ is said to be weakly visible if there exist Thomason subsets $U$ and $V$ such that $W=U\cap V^{c}$ . In particular, we say a point $x\in X$ is weakly visible if the singleton $\{x\}$ is weakly visible. The spectral space $X$ is said to be weakly Noetherian if every point of $X$ is weakly visible.

2.3 Example.

Every Noetherian spectral space and every profinite space is weakly Noetherian; see [BHS23b, Remarks 2.2 and 2.4].

2.4 Remark.

The Thomason closed subsets of a spectral space $X$ are precisely the closed subsets whose complements are quasi-compact; see [San13, Lemma 3.3]. Since the quasi-compact open subsets form a basis for the topology, the closure $\overline{\{x\}}$ of a point $x\in X$ is the intersection of all Thomason closed subsets containing $x$ . We write

\operatorname{gen}(x)\coloneqq\big{\{}\,y\in X\,\big{|}\,x\in\overline{\{y\}}\,\big{\}}

for the set of generalizations of $x$ in $X$ . Note that $\operatorname{gen}(x)=\bigcap_{x\notin Z}Z^{c}$ , where $Z$ ranges over all Thomason closed subsets of $X$ , is the complement of a Thomason subset; see [BHS23b, Remark 1.21]. As explained in [BHS23b, Remark 2.8], it follows that if $x\in X$ is weakly visible then

\{x\}=Z\cap\operatorname{gen}(x)

for some Thomason closed subset $Z$ .

2.5 Remark.

The notion of a weakly visible subset leads to the following definition introduced by W. Sanders [San17]:

2.6 Definition.

Let $X$ be a spectral space. The weakly visible subsets of $X$ form a basis of open subsets for a topology on $X$ called the localizing topology of $X$ . We write $X_{\operatorname{loc}}$ for $X$ equipped with the localizing topology, and for any subset $S$ of $X$ we write $\overline{S}^{\operatorname{loc}}$ for the closure of $S$ in $X_{\operatorname{loc}}$ .

2.7 Remark.

Note that a spectral space $X$ is weakly Noetherian if and only if its localizing topology is discrete. We will call a subset of $X$ localizing closed if it is closed with respect to the localizing topology. Thus, $X$ is weakly Noetherian if and only if every subset of $X$ is localizing closed.

2.8 Remark.

Recall that the constructible topology on a spectral space $X$ is the topology generated by the sets $U\cap V^{c}$ with $U$ and $V$ Thomason closed subsets of $X$ . We write $X_{\operatorname{con}}$ for $X$ equipped with the constructible topology. From the definitions we see that the localizing topology is finer than the constructible topology. Note that the constructible topology is discrete if and only if the spectral space is finite; see [DST19, Example 1.3.12 and Theorem 1.3.14]. Therefore, any infinite weakly Noetherian spectral space provides an example whose localizing topology is strictly finer than the constructible topology. An explicit example is:

2.9 Example.

Let $S^{*}$ denote the one-point compactification of a discrete infinite space $S$ . We denote the point at infinity by $\infty$ . Note that the space $S^{*}$ is a Boolean space. We now define a partial order on $S^{*}$ by

s\leq t\iff s=\infty\text{ or }s=t\in S.

This is a spectral order and hence yields a Priestly space whose associated spectral space is denoted by $S_{\infty}$ ; see [DST19, 1.6.13] for details. Note that the constructible topology on $S_{\infty}$ coincides with the original topology on $S^{*}$ . However, by [DST19, 1.6.15(iv) and (v)] the localizing topology on $S_{\infty}$ is discrete and is therefore strictly finer than the constructible topology on $S_{\infty}$ .

2.10 Remark.

On the other hand, there are examples where the localizing and constructible topologies coincide:

2.11 Example.

Consider the space $\mathbb{N}\cup\{\infty\}$ of extended natural numbers whose nonempty open subsets are of the form $[n,\infty]$ for $n\in\mathbb{N}$ . This is a spectral space and we denote its Hochster dual by $X$ . The space $X$ coincides with the Balmer spectrum $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ of the $p$ -local stable homotopy category ${\operatorname{SH}_{(p)}}$ ; see [BHS23b, Theorem 10.7], for example. Since every Thomason subset of $X$ is closed, the localizing topology coincides with the constructible topology.

2.12 Remark.

In general, the localizing topology is not a spectral topology. In fact, the localizing topology is quasi-compact if and only if it coincides with the constructible topology. Indeed, this follows from the fact that a continuous surjection from a quasi-compact space to a Hausdorff space is a topological quotient, in light of the continuous bijection $X_{\operatorname{loc}}\to X_{\operatorname{con}}$ .

2.13 Remark.

To conclude this section, we introduce a class of spectral spaces whose definition is somewhat technical but it turns out that some results which hold for Noetherian spectral spaces extend to this class of spaces; see 7.6.

2.14 Definition.

Let $S$ be a subset of a topological space $X$ . A point $x$ is an isolated point of $S$ if there exists an open subset $U$ of $X$ such that $\{x\}=U\cap S$ . More generally, the point $x$ is weakly isolated if there exists an open subset $U$ of $X$ such that $\{x\}\subseteq U\cap S\subseteq\overline{\{x\}}$ . A topological space $X$ is said to be (weakly) scattered if every nonempty closed subset of $X$ has a (weakly) isolated point. See [NR87] for further discussion.

2.15 Definition.

A spectral space $X$ is Hochster (weakly) scattered if its Hochster dual $X^{*}$ is (weakly) scattered.

2.16 Remark.

A spectral space is Hochster scattered if and only if it is weakly Noetherian and Hochster weakly scattered; see [San17, Lemma 7.16]. All Noetherian spectral spaces are Hochster scattered; see [San17, Lemma 7.17(1)], for example.

2.17 Example.

Let $R$ be a non-Noetherian absolutely flat commutative ring. The Zariski spectrum $\operatorname{Spec}(R)$ is not Noetherian by [Ste14, Lemma 3.6]. Nevertheless, if $R$ is semi-artinian then $\operatorname{Spec}(R)$ is Hochster scattered. Indeed, since $\operatorname{Spec}(R)$ carries the constructible topology (that is, $\operatorname{Spec}(R)=\operatorname{Spec}(R)_{\operatorname{con}}$ ) and has Cantor-Bendixson rank (by the proof of [Ste17, Theorem 6.4]), it then follows from [San17, Lemma 7.17(2)] that $\operatorname{Spec}(R)$ is Hochster scattered.

3. Small support for modules

We now introduce a notion of small support for graded modules over graded-commutative graded rings which extends the usual notion to the non-Noetherian setting. Our definition uses weakly associated primes, which behave better than associated primes in the absence of any Noetherian assumption.

3.1 Notation.

For this section, $R$ will denote a $\mathbb{Z}$ -graded graded-commutative ring. Ideals and modules will always be graded, respectively. The abelian category of (graded) $R$ -modules and degree-zero homomorphisms will be denoted $\operatorname{Mod}(R)$ . We write $\operatorname{Spec^{h}}(R)$ for the homogeneous Zariski spectrum of $R$ . Note that it is a spectral space. For any subset $S$ of $\operatorname{Spec^{h}}(R)$ , we write $\operatorname{cl}(S)$ for its specialization closure. Given any ideal $\mathfrak{a}$ of $R$ and any prime ideal $\mathfrak{p}$ in $\operatorname{Spec^{h}}(R)$ , we write $\cal V(\mathfrak{a})$ for the set of prime ideals containing $\mathfrak{a}$ and write $\operatorname{gen}(\mathfrak{p})$ for the generalization closure of $\mathfrak{p}$ . The complement of $\operatorname{gen}(\mathfrak{p})$ is denoted by $\cal Z(\mathfrak{p})$ . We refer the reader to [BH98, Section 1.5] and [DS13, Section 2] for more on graded commutative algebra.

3.2 Definition.

The big support of an $R$ -module $M$ is defined as

\operatorname{Supp}_{R}M\coloneqq\big{\{}\,\mathfrak{p}\in\operatorname{Spec^{h}}(R)\,\big{|}\,M_{\mathfrak{p}}\neq 0\,\big{\}}

where $M_{\mathfrak{p}}$ is the graded localization of $M$ at $\mathfrak{p}$ .

3.3 Definition.

Let $\cal V\subseteq\operatorname{Spec^{h}}(R)$ be specialization closed. For an $R$ -module $M$ we define

F_{\cal V}M\coloneqq\ker(M\to\prod_{\mathfrak{q}\notin\cal V}M_{\mathfrak{q}})

as in [BIK08, Section 9].

3.4 Definition.

Let $M$ be an $R$ -module. A prime $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ is said to be associated to $M$ if there exists a homogeneous element $m$ such that $\mathfrak{p}=\operatorname{Ann}(m)$ , the annihilator of $m$ . We denote the set of associated primes of $M$ by $\operatorname{Ass}(M)$ . More generally, if $\mathfrak{p}$ is minimal among the primes containing the annihilator of some homogeneous element in $M$ then $\mathfrak{p}$ is said to be weakly associated to $M$ . The set of weakly associated primes of $M$ is denoted by $\operatorname{WeakAss}(M)$ .

3.5 Lemma.

Let $M$ be an $R$ -module, $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ , and $\cal V$ a specialization closed subset of $\operatorname{Spec^{h}}(R)$ . The following hold:

(a)

$\operatorname{Ass}(M)\subseteq\operatorname{WeakAss}(M)\subseteq\operatorname{Supp}_{R}M$ . If $R$ is Noetherian then we have $\operatorname{Ass}(M)=\operatorname{WeakAss}(M)$ .
(b)

$M=0$ if and only if $\operatorname{WeakAss}(M)=\varnothing$ .
(c)

$\operatorname{WeakAss}(M_{\mathfrak{p}})=\operatorname{WeakAss}(M)\cap\operatorname{gen}(\mathfrak{p})$ .
(d)

$\operatorname{cl}(\operatorname{WeakAss}(M))=\operatorname{Supp}_{R}M$ .
(e)

$\operatorname{WeakAss}(F_{\cal V}M)\subseteq\operatorname{WeakAss}(M)\cap\cal V$ . If $\cal V=\cal V(x)$ for some homogeneous element $x\in R$ then equality holds.

Proof.

For parts (a), (b), and (c), the proofs in [Sta23, Lemma 0589, Lemma 058A, Lemma 0588, Lemma 05C9] carry over to the graded setting. Part (d) is a direct consequence of (b) and (c) since $\operatorname{Supp}_{R}M$ is always specialization closed. To show part (e), suppose $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ is minimal over $\operatorname{Ann}(m)$ for some homogeneous element $m\in F_{\cal V}M$ . If $\mathfrak{p}\notin\cal V$ then the image of $m$ under the localization $M\to M_{\mathfrak{p}}$ is zero, which contradicts $\mathfrak{p}\supseteq\operatorname{Ann}(m)$ . This establishes the first statement of (e). Now suppose $\cal V=\cal V(x)$ for some homogeneous element $x\in R$ . The equality then follows from the graded version of [Bou98, IV, Exercise 17(e), page 289] by noting that $F_{\cal V(x)}M=\ker(M\to M_{x})$ . ∎

3.6 Remark.

When $R$ is not Noetherian, there can exist a nonzero $R$ -module $M$ such that $\operatorname{Ass}(M)=\varnothing$ ; see [Sta23, Remark 05BX], for example. We now define the (small) support for modules, extending the one given in [BIK08, Section 2] to the non-Noetherian setting.

3.7 Definition.

The support of an $R$ -module $M$ is the set

\operatorname{supp}_{R}M\coloneqq\bigcup_{i=0}^{\infty}\operatorname{WeakAss}(I^{i})\subseteq\operatorname{Spec^{h}}(R)

where $I^{*}$ is a minimal injective resolution of $M$ .

3.8 Remark.

The support of a module is well-defined since a minimal injective resolution is unique up to isomorphism; see [BH98, pages 99 and 137], for example. From 3.5(b) we see that

(3.9)

\operatorname{supp}_{R}M=\varnothing\iff M=0.

3.10 Remark.

If $R$ is Noetherian then by 3.5(a) we have $\mathfrak{p}\in\operatorname{supp}_{R}M$ if and only if $\mathfrak{p}\in\operatorname{Ass}(I^{i})$ for some $i\geq 0$ , where $I^{i}$ is the $i$ -th term of some minimal injective resolution of $M$ . By [BH98, Theorem 3.6.3] this means that $I^{i}$ has a direct summand isomorphic to a shifted copy of the injective hull $E(R/\mathfrak{p})$ . Therefore, our definition of support recovers the one defined in [BIK08, Section 2] when $R$ is Noetherian.

3.11 Lemma.

Let $M$ be an $R$ -module and $\cal U\subseteq\operatorname{Spec^{h}}(R)$ a specialization closed subset. The following hold:

(a)

$\operatorname{supp}_{R}M\subseteq\operatorname{cl}(\operatorname{supp}_{R}M)=\operatorname{Supp}_{R}M\subseteq\cal V(\operatorname{Ann}(M))$ .
(b)

$\operatorname{supp}_{R}M_{\mathfrak{p}}=\operatorname{supp}_{R}M\cap\operatorname{gen}(\mathfrak{p})$ .
(c)

$\operatorname{supp}_{R}M\subseteq\cal U\iff M_{\mathfrak{q}}=0\text{ for all }\mathfrak{q}\notin\cal U$ .

Proof.

Let $I^{*}$ be a minimal injective resolution of $M$ . We have

\operatorname{cl}(\operatorname{supp}_{R}M)=\bigcup_{i=0}^{\infty}\operatorname{cl}(\operatorname{WeakAss}(I^{i}))=\bigcup_{i=0}^{\infty}\operatorname{Supp}_{R}I^{i}=\operatorname{Supp}_{R}M

where the second equality is due to 3.5(d), and the last equality holds because $I_{\mathfrak{p}}^{*}$ is a minimal injective resolution of $M_{\mathfrak{p}}$ for any $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ . The inclusions in part (a) are clear. Part (b) follows from 3.5(c). Part (c) is immediate from (a). ∎

3.12 Definition.

Let $\cal V$ be a specialization closed subset of $\operatorname{Spec^{h}}(R)$ . Define the full subcategory

\operatorname{Mod}(R)_{\cal V}\coloneqq\big{\{}\,M\in\operatorname{Mod}(R)\,\big{|}\,\operatorname{supp}_{R}M\subseteq\cal V\,\big{\}}.

3.13 Remark.

Note that $F_{\cal V}$ in 3.3 defines a functor from $\operatorname{Mod}(R)$ to $\operatorname{Mod}(R)_{\cal V}$ which is right adjoint to the inclusion $i\colon\operatorname{Mod}(R)_{\cal V}\hookrightarrow\operatorname{Mod}(R)$ by 3.11(c).

3.14 Lemma.

If $\cal V\subseteq\operatorname{Spec^{h}}(R)$ is specialization closed then $\operatorname{Mod}(R)_{\cal V}$ is a localizing Serre subcategory of $\operatorname{Mod}(R)$ . That is, $\operatorname{Mod}(R)_{\cal V}$ is closed under arbitrary direct sums, and for any exact sequence $0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$ of $R$ -modules, $M$ is in $\operatorname{Mod}(R)_{\cal V}$ if and only if $M^{\prime}$ and $M^{\prime\prime}$ are in $\operatorname{Mod}(R)_{\cal V}$ .

Proof.

This follows from 3.11(c) and the fact that the localization functor is exact and preserves direct sums. ∎

3.15 Remark.

3.11 and 3.14 generalize [BIK08, Lemmas 2.2] and [BIK08, Lemmas 2.3], respectively.

3.16 Remark.

Let $\cal V\subseteq\operatorname{Spec^{h}}(R)$ be specialization closed. Since $\operatorname{Mod}(R)_{\cal V}$ is a Serre subcategory of $\operatorname{Mod}(R)$ , by [Sta23, Equation 06UR] the inclusion induces a functor

\operatorname{D}(\operatorname{Mod}(R)_{\cal V})\xrightarrow{j}\operatorname{D}_{\cal V}(\operatorname{Mod}(R))\coloneqq\big{\{}\,X\in\operatorname{D}(\operatorname{Mod}(R))\,\big{|}\,H^{i}(X)\in\operatorname{Mod}(R)_{\cal V}\text{ for all }i\,\big{\}}.

Moreover, deriving the adjunction $i:\operatorname{Mod}(R)_{\cal V}\rightleftarrows\operatorname{Mod}(R):F_{\cal V}$ we obtain an adjunction $\mathrm{L}i:\operatorname{D}(\operatorname{Mod}(R)_{\cal V})\rightleftarrows\operatorname{D}(\operatorname{Mod}(R)):\mathrm{R}F_{\cal V}$ such that $\mathrm{L}i$ factors as

\operatorname{D}(\operatorname{Mod}(R)_{\cal V})\xrightarrow{j}\operatorname{D}_{\cal V}(\operatorname{Mod}(R))\hookrightarrow\operatorname{D}(\operatorname{Mod}(R)).

3.17 Lemma.

Let $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ . There exists an exact triangle

\mathrm{L}i(\mathrm{R}F_{\cal Z(\mathfrak{p})}X)\to X\to X_{\mathfrak{p}}

for every $X\in\operatorname{D}^{+}(\operatorname{Mod}(R))$ .

Proof.

An argument similar to [BHS23b, Example 1.35] shows that the extension of scalars $\operatorname{D}(\operatorname{Mod}(R))\to\operatorname{D}(\operatorname{Mod}(R_{\mathfrak{p}}))$ is the smashing localization associated to the idempotent ring object $R_{\mathfrak{p}}$ . The kernel of this localization is $\operatorname{D}_{\cal Z(\mathfrak{p})}(\operatorname{Mod}(R))$ by 3.11(b). Denoting the corresponding colocalization functor by $\varGamma_{\cal Z(\mathfrak{p})}$ , we obtain an exact triangle

\varGamma_{\cal Z(\mathfrak{p})}X\to X\to X_{\mathfrak{p}}

for every object $X$ of $\operatorname{D}(\operatorname{Mod}(R))$ . It then suffices to show that $\varGamma_{\cal Z(\mathfrak{p})}X$ is isomorphic to $\mathrm{L}i(\mathrm{R}F_{\cal Z(\mathfrak{p})}X)$ for any bounded-below complex $X$ . We claim that the functor $j\colon\operatorname{D}(\operatorname{Mod}(R)_{\cal V})\to\operatorname{D}_{\cal V}(\operatorname{Mod}(R))$ in 3.16 restricts to an equivalence $\operatorname{D}^{+}(\operatorname{Mod}(R)_{\cal V})\cong\operatorname{D}^{+}_{\cal V}(\operatorname{Mod}(R))$ . Indeed, this follows from [Har66, Proposition I.4.8] since every $R$ -module supported in $\cal V$ can be embedded into an injective $R$ -module supported in $\cal V$ . In particular, $\mathrm{L}i$ restricts to

\operatorname{D}^{+}(\operatorname{Mod}(R)_{\cal Z(\mathfrak{p})})\cong\operatorname{D}^{+}_{\cal Z(\mathfrak{p})}(\operatorname{Mod}(R))\hookrightarrow\operatorname{D}^{+}(\operatorname{Mod}(R)).

It then follows that for any $X\in\operatorname{D}^{+}(\operatorname{Mod}(R))$ we have $\varGamma_{\cal Z(\mathfrak{p})}X\simeq\mathrm{L}i(\mathrm{R}F_{\cal Z(\mathfrak{p})}X)$ . ∎

3.18 Remark.

Let $M$ be an $R$ -module, $\mathfrak{a}$ an ideal of $R$ , and $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ . The module $M$ is $\mathfrak{p}$ -local if the natural map $M\to M_{\mathfrak{p}}$ is an isomorphism of $R$ -modules. The module $M$ is $\mathfrak{a}$ -torsion if every element of $M$ is annihilated by a power of $\mathfrak{a}$ . The following lemma generalizes [BIK08, Lemma 2.4], whose proof relies on the structure theorem for injective modules over Noetherian rings.

3.19 Lemma.

Let $M$ be an $R$ -module, $\mathfrak{a}$ an ideal of $R$ , and $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ . We have:

(a)

$M$ is $\mathfrak{p}$ -local if and only if $\operatorname{supp}_{R}M\subseteq\operatorname{gen}(\mathfrak{p})$ .
(b)

If $M$ is $\mathfrak{a}$ -torsion then $\operatorname{supp}_{R}M\subseteq\cal V(\mathfrak{a})$ .
(c)

$\operatorname{supp}_{R}M\subseteq\cal V(\mathfrak{a})$ if and only if $M$ is $\mathfrak{b}$ -torsion for any finitely generated ideal $\mathfrak{b}\subseteq\mathfrak{a}$ .

Proof.

The only if part of (a) follows from 3.11(b). For the other direction, let $I^{*}$ be a minimal injective resolution of $M$ . Observe that

$\displaystyle\operatorname{supp}_{R}M\subseteq\operatorname{gen}(\mathfrak{p})$	$\displaystyle\implies\operatorname{WeakAss}(I^{i})\cap\cal Z(\mathfrak{p})=\varnothing\text{ for all }i\geq 0$
	$\displaystyle\implies\operatorname{WeakAss}(F_{\cal Z(\mathfrak{p})}(I^{i}))=\varnothing\text{ for all }i\geq 0$	by 3.5(e)
	$\displaystyle\implies\mathrm{R}F_{\cal Z(\mathfrak{p})}(I^{i})=F_{\cal Z(\mathfrak{p})}I^{i}=0\text{ for all }i\geq 0$	by 3.5(b)
	$\displaystyle\implies I^{i}\text{ is $\mathfrak{p}$-local}\text{ for all }i\geq 0$	by 3.17
	$\displaystyle\implies M\text{ is $\mathfrak{p}$-local}.$

For part (b), suppose $M$ is $\mathfrak{a}$ -torsion. We then have $M_{\mathfrak{p}}=0$ for every $\mathfrak{p}\notin\cal V(\mathfrak{a})$ and thus $\operatorname{supp}_{R}M\subseteq\operatorname{Supp}_{R}M\subseteq\cal V(\mathfrak{a})$ . For part (c), let $I^{*}$ be a minimal injective resolution of $M$ and $m$ a homogeneous element of $I^{0}$ . Note that

\begin{split}\operatorname{supp}_{R}M\subseteq\cal V(\mathfrak{a})&\implies\operatorname{WeakAss}(I^{0})\subseteq\cal V(\mathfrak{a})\\ &\implies\mathfrak{a}\subseteq\bigcap_{\mathfrak{p}\in\operatorname{WeakAss}(I^{0})}\mathfrak{p}\subseteq\sqrt{\operatorname{Ann}(m)}.\end{split}

Hence $M$ is $\mathfrak{b}$ -torsion for any finitely generated ideal $\mathfrak{b}\subseteq\mathfrak{a}$ . The other direction follows from (b). ∎

3.20 Remark.

By the lemma above, we see that for a finitely generated ideal $\mathfrak{a}$ , an $R$ -module $M$ is $\mathfrak{a}$ -torsion if and only if $\operatorname{supp}_{R}M\subseteq\cal V(\mathfrak{a})$ . However, this does not always hold when $\mathfrak{a}$ is not finitely generated. Indeed, in [Roh19] the author studied two torsion functors for an ideal $\mathfrak{a}$ of a commutative ring $R$ (in the ungraded setting): the small $\mathfrak{a}$ -torsion functor $\varGamma_{\mathfrak{a}}$ and the large $\mathfrak{a}$ -torsion functor $\overline{\varGamma}_{\mathfrak{a}}$ , which are defined as

\varGamma_{\mathfrak{a}}M\coloneqq\big{\{}\,m\in M\,\big{|}\,\mathfrak{a}^{n}\subseteq\operatorname{Ann}(m)\text{ for some }n\in\mathbb{N}\,\big{\}}

and

\overline{\varGamma}_{\mathfrak{a}}M\coloneqq\big{\{}\,m\in M\,\big{|}\,\mathfrak{a}\subseteq\sqrt{\operatorname{Ann}(m)}\,\big{\}}.

Hence $M=\varGamma_{\mathfrak{a}}M$ if and only if $M$ is $\mathfrak{a}$ -torsion. On the other hand, $M=\overline{\varGamma}_{\mathfrak{a}}M$ if and only if $\operatorname{supp}_{R}M\subseteq\cal V(\mathfrak{a})$ ; this follows from[Roh19, (3.3)(B)] and 3.11(a). It is clear that $\varGamma_{\mathfrak{a}}$ is a subfunctor of $\overline{\varGamma}_{\mathfrak{a}}$ and $\varGamma_{\mathfrak{a}}=\overline{\varGamma}_{\mathfrak{a}}$ if $\mathfrak{a}$ is finitely generated. However, these two functors do not coincide in general; see [Roh19, Section 4].

4. Non-Noetherian BIK support

Benson, Iyengar, and Krause [BIK08] developed a theory of support for any compactly generated triangulated category equipped with a central action by a Noetherian graded-commutative graded ring. In this section we show how the Noetherian hypothesis can be removed.

4.1 Terminology.

For the rest of the section, we fix a compactly generated $R$ -linear triangulated category $\mathscr{T}$ . That is, a compactly generated triangulated category $\mathscr{T}$ equipped with a homomorphism of graded rings $R\to Z(\mathscr{T})$ where $Z(\mathscr{T})$ is the graded center of $\mathscr{T}$ . The full subcategory of compact objects in $\mathscr{T}$ is denoted by $\mathscr{T}^{c}$ . Given any two objects $x$ and $t$ in $\mathscr{T}$ , the graded abelian group

\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)\coloneqq\coprod_{n\in\mathbb{Z}}\operatorname{Hom}_{\mathscr{T}}(x,\Sigma^{n}t)

has a graded $Z(\mathscr{T})$ -module structure and hence is a graded module over $R$ ; see [BIK08, Section 4] for further details.

4.2 Remark.

Recall that a localizing subcategory $\mathscr{L}$ of $\mathscr{T}$ is strictly localizing if the inclusion $\mathscr{L}\hookrightarrow\mathscr{T}$ admits a right adjoint. This is equivalent to $\mathscr{L}$ being the kernel of a Bousfield localization on $\mathscr{T}$ ; see [Nee01, Proposition 9.1.8], for example.

4.3 Lemma.

Let $\cal V\subseteq\operatorname{Spec^{h}}(R)$ be specialization closed. The subcategory

\mathscr{T}_{\cal V}\coloneqq\big{\{}\,t\in\mathscr{T}\,\big{|}\,\operatorname{supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)\subseteq\cal V\text{ for each }x\in\mathscr{T}^{c}\,\big{\}}

is strictly localizing.

Proof.

Since 3.11(c) generalizes [BIK08, Lemma 2.3(1)], the proof of Proposition 4.5 in loc. cit. carries over verbatim. ∎

4.4 Remark.

For an object $t\in\mathscr{T}$ and a specialization closed subset $\cal V\subseteq\operatorname{Spec^{h}}(R)$ , there exists (by the lemma above) a localization triangle

\varGamma_{\cal V}t\to t\to L_{\cal V}t

where $L_{\cal V}$ and $\varGamma_{\cal V}$ are the corresponding Bousfield localization and colocalization functor, respectively. We think of $\varGamma_{\cal V}t$ as the part of $t$ supported on $\cal V$ and $L_{\cal V}t$ as the part of $t$ supported away from $\cal V$ . By 3.11(a) we have

\mathscr{T}_{\cal V}=\big{\{}\,t\in\mathscr{T}\,\big{|}\,\operatorname{Supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)\subseteq\cal V\text{ for each }x\in\mathscr{T}^{c}\,\big{\}}.

This category is also equal to the one defined in [BIK08, Lemma 4.3] by [BIK08, Lemma 2.2(1)]. Therefore, the localization functor $L_{\cal V}$ is the same as the one in [BIK08, Definition 4.6]. Moreover, these localization functors satisfy the following composition rules:

4.5 Lemma.

Let $\cal V$ and $\cal W$ be specialization closed subsets of $\operatorname{Spec^{h}}(R)$ . The following hold:

(a)

$\varGamma_{\cal V}\varGamma_{\cal W}\cong\varGamma_{\cal V\cap\cal W}\cong\varGamma_{\cal W}\varGamma_{\cal V}$ .
(b)

$L_{\cal V}L_{\cal W}\cong L_{\cal V\cup\cal W}\cong L_{\cal W}L_{\cal V}$ .
(c)

$\varGamma_{\cal V}L_{\cal W}\cong L_{\cal W}\varGamma_{\cal V}$ .

Proof.

See [BIK08, Proposition 6.1]. ∎

4.6 Remark.

In 4.4 we have noted that the construction of the localization functors in [BIK08, Definition 4.6] depends only on the notion of big support of modules, which does not require the ring $R$ to be Noetherian. However, to show that such a localization functor is a finite localization we want to have a more concrete description of the category $\mathscr{T}_{\cal V}$ . For example, the objects in $\mathscr{T}_{\cal V(\mathfrak{a})}$ for a finitely generated ideal $\mathfrak{a}$ should be those $t\in\mathscr{T}$ with the property that $\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)$ is $\mathfrak{a}$ -torsion for any $x\in\mathscr{T}^{c}$ . This was established in [BIK08, Lemma 2.4(2)] under the Noetherian hypothesis on the ring $R$ . Thanks to 3.19, this remains true for general commutative rings. Our next goal is to show that for any finitely generated ideal $\mathfrak{a}$ of $R$ and any prime ideal $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ the localization functors corresponding to $\cal V(\mathfrak{a})$ and $\cal Z(\mathfrak{p})$ are finite localizations, that is, the localizing subcategories $\mathscr{T}_{\cal V(\mathfrak{a})}$ and $\mathscr{T}_{\cal Z(\mathfrak{p})}$ are compactly generated. Let us first recall the notion of Koszul objects.

4.7 Definition.

Let $r\in R$ be a homogeneous element of degree $d$ and let $t$ be an object of $\mathscr{T}$ . We denote by ${t/\!\!/r}$ any object that fits into an exact triangle

t\xrightarrow{r}\Sigma^{d}t\to{t/\!\!/r}.

This is called a Koszul object of $r$ on $t$ . Given a finite sequence $\underline{r}=(r_{1},\ldots,r_{n})$ of homogeneous elements, a Koszul object of $\underline{r}$ on $t$ is defined iteratively and denoted by ${t/\!\!/\underline{r}}$ . For a finitely generated ideal $\mathfrak{a}$ of $R$ , we write ${t/\!\!/\mathfrak{a}}$ for any Koszul object of any finite sequence of homogeneous generators for $\mathfrak{a}$ ; see [BIK08, Definition 5.10] for further discussion. A Koszul object ${t/\!\!/\mathfrak{a}}$ depends on the choice of generating sequence for the ideal $\mathfrak{a}$ . Nevertheless, the thick subcategory generated by ${t/\!\!/\mathfrak{a}}$ depends only on the radical of $\mathfrak{a}$ by [BIK11a, Lemma 3.4(2)]. Note also that ${t/\!\!/\mathfrak{a}}$ is compact if $t$ is compact.

4.8 Lemma.

For every object $t\in\mathscr{T}$ and every finitely generated ideal $\mathfrak{a}$ of $R$ we have ${t/\!\!/\mathfrak{a}}\in\mathscr{T}_{\cal V(\mathfrak{a})}$ .

Proof.

By [BIK08, Lemma 5.11(1)] the $R$ -module $\operatorname{Hom}_{\mathscr{T}}^{*}(x,{t/\!\!/\mathfrak{a}})$ is $\mathfrak{a}$ -torsion for every $x\in\mathscr{T}^{c}$ . 3.19(c) then yields the desired result. ∎

4.9 Proposition.

For any finitely generated ideal $\mathfrak{a}$ of $R$ and any $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ , the categories $\mathscr{T}_{\cal V(\mathfrak{a})}$ and $\mathscr{T}_{\cal Z(\mathfrak{p})}$ are compactly generated:

(a)

$\mathscr{T}_{\cal V(\mathfrak{a})}=\operatorname{Loc}\langle{x/\!\!/\mathfrak{a}}\mid x\in\mathscr{T}^{c}\rangle$ .
(b)

$\mathscr{T}_{\cal Z(\mathfrak{p})}=\operatorname{Loc}\langle{x/\!\!/r}\mid x\in\mathscr{T}^{c}\text{ and }r\in R\setminus\mathfrak{p}\text{ homogeneous}\rangle.$

Proof.

By 4.8 we have $\mathscr{S}\coloneqq\operatorname{Loc}\langle{x/\!\!/\mathfrak{a}}\mid x\in\mathscr{T}^{c}\rangle\subseteq\mathscr{T}_{\cal V(\mathfrak{a})}$ . Since $\mathscr{S}$ is a compactly generated subcategory of $\mathscr{T}_{\cal V(\mathfrak{a})}$ , it is strictly localizing [Nee01, Proposition 9.1.19]. Thus there exists a functor $F\colon\mathscr{T}_{\cal V(\mathfrak{a})}\to\mathscr{S}$ such that the composite $\mathscr{T}_{\cal V(\mathfrak{a})}\xrightarrow{F}\mathscr{S}\hookrightarrow\mathscr{T}_{\cal V(\mathfrak{a})}$ is the corresponding colocalization functor. Now for any $t\in\mathscr{T}_{\cal V(\mathfrak{a})}$ we have an exact triangle

Ft\to t\to s

for some $s\in\mathscr{S}^{\perp}$ . It remains to prove $s=0$ . Let $r_{1},\ldots,r_{n}$ be a sequence of homogeneous generators for $\mathfrak{a}$ . Note that for any $x\in\mathscr{T}^{c}$ we have

\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/(r_{1},\ldots,r_{n})},s)=0.

We claim

\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/(r_{1},\ldots,r_{n-1})},s)=0.

Let $m\in\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/(r_{1},\ldots,r_{n-1})},s)$ . Since $s\in\mathscr{T}_{\cal V(\mathfrak{a})}=\bigcap_{i=1}^{n}\mathscr{T}_{\cal V(r_{i})}$ , there exists a positive integer $k$ with $r_{n}^{k}m=0$ by 3.19. Let $d$ be the degree of $r_{n}$ . Applying $\operatorname{Hom}_{\mathscr{T}}^{*}(-,s)$ to the exact triangle

{x/\!\!/(r_{1},\ldots,r_{n-1})}\\ \xrightarrow{r_{n}}\Sigma^{d}{x/\!\!/(r_{1},\ldots,r_{n-1})}\to{x/\!\!/(r_{1},\ldots,r_{n})}

yields an exact sequence

0=\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},s)\to\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/(r_{1},\ldots,r_{n-1})},s)[-d]\\ \xrightarrow{r_{n}}\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/(r_{1},\ldots,r_{n-1})},s)\to\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},s)[-1]=0.

Thus multiplying by $r_{n}$ is an isomorphism, so $m=0$ . The claim follows. An induction yields $\operatorname{Hom}_{\mathscr{T}}^{*}(x,s)=0$ . This is true for all $x\in\mathscr{T}^{c}$ . Therefore $s=0$ , which establishes (a).

For part (b), set $\mathscr{S}\coloneqq\operatorname{Loc}\langle{x/\!\!/r}\mid x\in\mathscr{T}^{c}\text{ and }r\in R\setminus\mathfrak{p}\text{ is homogeneous}\rangle\subseteq\mathscr{T}_{\cal Z(\mathfrak{p})}$ . A similar argument as above shows that there exists a functor $F\colon\mathscr{T}_{\cal Z(\mathfrak{p})}\to\mathscr{S}$ such that for any $t\in\mathscr{T}_{\cal Z(\mathfrak{p})}$ we have an exact triangle

Ft\to t\to s

with $s\in\mathscr{S}^{\perp}$ . Since $s\in\mathscr{T}_{\cal Z(\mathfrak{p})}$ , we have $\operatorname{supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}(x,s)\subseteq\cal Z(\mathfrak{p})$ and hence $\operatorname{Hom}_{\mathscr{T}}^{*}(x,s)_{\mathfrak{p}}=0$ by 3.11(b) and (3.9), for every $x\in\mathscr{T}^{c}$ . It follows that for any homogeneous element $m\in\operatorname{Hom}_{\mathscr{T}}^{*}(x,s)$ there exists some homogeneous element $r\notin\mathfrak{p}$ with $rm=0$ . Let $d$ be the degree of $r$ . The exact sequence

0=\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/r},s)[-1]\to\operatorname{Hom}_{\mathscr{T}}^{*}(x,s)[-d]\xrightarrow{r}\operatorname{Hom}_{\mathscr{T}}^{*}(x,s)\to\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/r},s)=0

implies that $m=0$ . Therefore $s=0$ , which completes the proof. ∎

4.10 Remark.

Now we are ready to define the BIK support for objects in $\mathscr{T}$ . Recall that the Thomason closed subsets of $\operatorname{Spec^{h}}(R)$ are exactly subsets of the form $\cal V(\mathfrak{a})$ for some finitely generated ideal $\mathfrak{a}$ . Also recall that the subset $\cal Z(\mathfrak{p})$ is the largest Thomason subset not containing $\mathfrak{p}$ .

4.11 Definition.

The BIK support of an object $t$ in $\mathscr{T}$ is defined as

\operatorname{Supp}_{\textup{BIK}}(t)\coloneqq\left\{\mathfrak{p}\in\operatorname{Spec^{h}}(R)\left|{\begin{gathered}\varGamma_{\cal V(\mathfrak{a})}L_{\cal Z(\mathfrak{p})}t\neq 0\text{ for any finitely}\\ \text{ generated ideal }\mathfrak{a}\text{ contained in }\mathfrak{p}\end{gathered}}\right.\right\}

4.12 Remark.

If $R$ is Noetherian then every $\mathfrak{p}\in\operatorname{Spec^{h}}(R)$ is finitely generated. 4.5(a) then implies that $\mathfrak{p}\in\operatorname{Supp}_{\textup{BIK}}(t)$ if and only if $\varGamma_{\cal V(\mathfrak{p})}L_{\cal Z(\mathfrak{p})}t\neq 0$ . We thus obtain the following:

4.13 Theorem.

Let $\mathscr{T}$ be a compactly generated triangulated category equipped with an action by a graded-commutative graded ring $R$ . There is an associated support $\operatorname{Supp}_{\textup{BIK}}(t)\subseteq\operatorname{Spec^{h}}(R)$ for every object $t\in\mathscr{T}$ which recovers the one in [BIK08] when $R$ is Noetherian.

4.14 Remark.

Recall that a Thomason subset of a spectral space is a union of Thomason closed subsets. By 4.10 and 4.5 we have

\operatorname{Supp}_{\textup{BIK}}(t)=\left\{\mathfrak{p}\in\operatorname{Spec^{h}}(R)\left|{\begin{gathered}\varGamma_{\cal V}L_{\cal Z}t\neq 0\text{ for any Thomason}\\ \text{ subsets }\cal V,\cal Z\text{ such that }\mathfrak{p}\in\cal V\cap\cal Z^{c}\end{gathered}}\right.\right\}

for any $t\in\mathscr{T}$ .

4.15 Notation.

For any $R$ -module $M$ we write $\min M$ for the set of prime ideals in $\operatorname{supp}_{R}M$ that are minimal (with respect to inclusion) among the prime ideals in $\operatorname{supp}_{R}M$ .

4.16 Theorem.

For any $t\in\mathscr{T}$ we have

\bigcup_{x\in\mathscr{T}^{c}}\min\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)\subseteq\operatorname{Supp}_{\textup{BIK}}(t)\subseteq\bigcup_{x\in\mathscr{G}}\operatorname{Supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)

where $\mathscr{G}$ is any set of compact generators for $\mathscr{T}$ .

Proof.

Suppose $\mathfrak{p}\in\min\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)$ for some $x\in\mathscr{T}^{c}$ . By 3.11(a) we have $\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)_{\mathfrak{p}}\neq 0$ and hence $\operatorname{supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)_{\mathfrak{p}}=\{\mathfrak{p}\}$ by the minimality of $\mathfrak{p}$ and 3.11(b). It then follows from 3.19(c) that $\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)_{\mathfrak{p}}$ is $a$ -torsion for any homogeneous element $a\in\mathfrak{p}$ . Now let $r\in\mathfrak{p}$ be a homogeneous element of degree $d$ . Applying $\operatorname{Hom}_{\mathscr{T}}^{*}(-,t)_{\mathfrak{p}}$ to the exact triangle

x\xrightarrow{r}\Sigma^{d}x\to{x/\!\!/r}

yields a long exact sequence

\cdots\to\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)_{\mathfrak{p}}[-1]\to\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/r},t)_{\mathfrak{p}}\\ \to\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)_{\mathfrak{p}}[-d]\xrightarrow{r}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)_{\mathfrak{p}}\to\cdots.

Hence $\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/r},t)_{\mathfrak{p}}$ is also $a$ -torsion for any homogeneous element $a\in\mathfrak{p}$ and $\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/r},t)_{\mathfrak{p}}\neq 0$ . An induction shows that $\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},t)_{\mathfrak{p}}\neq 0$ for any finitely generated ideal $\mathfrak{a}\subseteq\mathfrak{p}$ . On the other hand, since $\varGamma_{\cal Z(\mathfrak{p})}t\in\mathscr{T}_{\cal Z(\mathfrak{p})}$ it follows that

\operatorname{supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},\varGamma_{\cal Z(\mathfrak{p})}t)\subseteq\cal Z(\mathfrak{p}),

which implies $\operatorname{supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},\varGamma_{\cal Z(\mathfrak{p})}t)_{\mathfrak{p}}=\varnothing$ by 3.11(b). In view of (3.9), we then have $\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},\varGamma_{\cal Z(\mathfrak{p})}t)_{\mathfrak{p}}=0$ and thus

\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},L_{\cal Z(\mathfrak{p})}t)_{\mathfrak{p}}\cong\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},t)_{\mathfrak{p}}\neq 0

by 4.4. Since ${x/\!\!/\mathfrak{a}}\in\mathscr{T}_{\cal V(\mathfrak{a})}$ (4.8), we have

\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},\varGamma_{\cal V(\mathfrak{a})}L_{\cal Z(\mathfrak{p})}t)_{\mathfrak{p}}\cong\operatorname{Hom}_{\mathscr{T}}^{*}({x/\!\!/\mathfrak{a}},L_{\cal Z(\mathfrak{p})}t)_{\mathfrak{p}}\neq 0

and therefore $\varGamma_{\cal V(\mathfrak{a})}L_{\cal Z(\mathfrak{p})}t\neq 0$ . This is true for every finitely generated ideal $\mathfrak{a}\subseteq\mathfrak{p}$ , so $\mathfrak{p}\in\operatorname{Supp}_{\textup{BIK}}(t)$ , which establishes the first inclusion. For the second, note that

\mathfrak{p}\notin\bigcup_{x\in\mathscr{G}}\operatorname{Supp}_{R}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)\iff t\in\mathscr{T}_{\cal Z(\mathfrak{p})}\iff L_{\cal Z(\mathfrak{p})}t=0,

which implies $\mathfrak{p}\notin\operatorname{Supp}_{\textup{BIK}}(t)$ . ∎

4.17 Remark.

The theorem above generalizes [BIK08, Corollary 5.3]. In [BIK08, Theorem 5.2] it was shown that the first inclusion is an equality under the assumption that $R$ is Noetherian. It is not clear if the equality still holds in our non-Noetherian setting.

4.18 Definition.

We say that the BIK support satisfies the detection property if $\operatorname{Supp}_{\textup{BIK}}(t)=\varnothing$ implies $t=0$ for each object $t\in\mathscr{T}$ .

4.19 Corollary.

If the descending chain condition holds for the prime ideals of $R$ then the BIK support satisfies the detection property.

Proof.

The hypothesis implies that if a subset $S$ of $\operatorname{Spec^{h}}(R)$ is nonempty then $\min S$ is nonempty. The statement then follows from 4.16 and (3.9). ∎

4.20 Remark.

If $R$ is Noetherian then the descending chain condition on the prime ideals of $R$ holds (see [DST19, Corollary 12.4.5(1)], for example) and thus the BIK support has the detection property. However, we do not know if the detection property is always satisfied when $R$ is not Noetherian.

4.21 Remark.

In the following we record some basic properties of the BIK support, which are inspired by [San17, Theorems 4.2 and 4.7].

4.22 Proposition.

The following hold:

(a)

$\operatorname{Supp}_{\textup{BIK}}(0)=\varnothing$ .
(b)

$\operatorname{Supp}_{\textup{BIK}}(t)=\operatorname{Supp}_{\textup{BIK}}(\Sigma t)$ for every $t\in\mathscr{T}$ .
(c)

$\operatorname{Supp}_{\textup{BIK}}(c)\subseteq\operatorname{Supp}_{\textup{BIK}}(a)\cup\operatorname{Supp}_{\textup{BIK}}(b)$ if $a\to b\to c\to\Sigma a$ is an exact triangle in $\mathscr{T}$ .
(d)

$\operatorname{Supp}_{\textup{BIK}}(t_{1}\oplus t_{2})=\operatorname{Supp}_{\textup{BIK}}(t_{1})\cup\operatorname{Supp}_{\textup{BIK}}(t_{2})$ for any $t_{1},t_{2}\in\mathscr{T}$ .
(e)

$\operatorname{Supp}_{\textup{BIK}}(\varGamma_{\cal V}t)=\cal V\cap\operatorname{Supp}_{\textup{BIK}}(t)$ and $\operatorname{Supp}_{\textup{BIK}}(L_{\cal V}t)=\cal V^{c}\cap\operatorname{Supp}_{\textup{BIK}}(t)$ for any $t\in\mathscr{T}$ and any specialization closed subset $\cal V$ of $\operatorname{Spec^{h}}(R)$ .

Proof.

Part (a) is clear. Parts (b) and (d) follow from the fact that the localization and colocalization functors preserve finite direct sums and are exact.

For part (c), if $\mathfrak{p}\notin\operatorname{Supp}_{\textup{BIK}}(a)\cup\operatorname{Supp}_{\textup{BIK}}(b)$ , then there exist finitely generated ideals $\mathfrak{a},\mathfrak{b}\subseteq\mathfrak{p}$ such that $\varGamma_{\cal V(\mathfrak{a})}L_{\cal Z(\mathfrak{p})}a=0$ and $\varGamma_{\cal V(\mathfrak{b})}L_{\cal Z(\mathfrak{p})}b=0$ . Thus by 4.5 we have

\varGamma_{\cal V(\mathfrak{a}+\mathfrak{b})}L_{\cal Z(\mathfrak{p})}a=\varGamma_{\cal V(\mathfrak{a})}\varGamma_{\cal V(\mathfrak{b})}L_{\cal Z(\mathfrak{p})}a=0=\varGamma_{\cal V(\mathfrak{a})}\varGamma_{\cal V(\mathfrak{b})}L_{\cal Z(\mathfrak{p})}b=\varGamma_{\cal V(\mathfrak{a}+\mathfrak{b})}L_{\cal Z(\mathfrak{p})}b.

It then follows from the exactness of localization functors that $\varGamma_{\cal V(\mathfrak{a}+\mathfrak{b})}L_{\cal Z(\mathfrak{p})}c=0$ . Therefore $\mathfrak{p}\notin\operatorname{Supp}_{\textup{BIK}}(c)$ , which establishes (c).

For (e), note that $\varGamma_{\operatorname{Spec^{h}}(R)}L_{\cal V}(\varGamma_{\cal V}t)=0=\varGamma_{\cal V}L_{\varnothing}(L_{\cal V}t)$ . Thus $\operatorname{Supp}_{\textup{BIK}}(\varGamma_{\cal V}t)\subseteq\cal V$ and $\operatorname{Supp}_{\textup{BIK}}(L_{\cal V}t)\subseteq\cal V^{c}$ . Applying (c) to the exact triangle $\varGamma_{\cal V}t\to t\to L_{\cal V}t$ we obtain

\operatorname{Supp}_{\textup{BIK}}(\varGamma_{\cal V}t)\subseteq\operatorname{Supp}_{\textup{BIK}}(t)\cup\operatorname{Supp}_{\textup{BIK}}(L_{\cal V}t)\subseteq\operatorname{Supp}_{\textup{BIK}}(t)\cup\cal V^{c}.

Intersecting with $\cal V$ leads to $\operatorname{Supp}_{\textup{BIK}}(\varGamma_{\cal V}t)=\operatorname{Supp}_{\textup{BIK}}(t)\cap\cal V$ . A similar argument shows that $\operatorname{Supp}_{\textup{BIK}}(L_{\cal V}t)=\operatorname{Supp}_{\textup{BIK}}(t)\cap\cal V^{c}$ , which completes the proof. ∎

4.23 Notation.

For any class $\mathscr{E}$ of objects in $\mathscr{T}$ we write $\operatorname{Loc}\langle\mathscr{E}\rangle$ for the localizing subcategory generated by $\mathscr{E}$ and define $\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})\coloneqq\bigcup_{t\in\mathscr{E}}\operatorname{Supp}_{\textup{BIK}}(t)$ .

4.24 Proposition.

The following hold:

(a)

$\operatorname{Supp}_{\textup{BIK}}(t)$ is localizing closed (recall 2.7) for any $t\in\mathscr{T}$ .
(b)

$\operatorname{Supp}_{\textup{BIK}}(\mathscr{L})$ is localizing closed for any localizing subcategory $\mathscr{L}$ of $\mathscr{T}$ .

(c)

If the BIK support satisfies the detection property then for any class of objects $\mathscr{E}$ of $\mathscr{T}$ we have

\operatorname{Supp}_{\textup{BIK}}(\operatorname{Loc}\langle\mathscr{E}\rangle)=\overline{\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})}^{\operatorname{loc}}

where $\overline{\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})}^{\operatorname{loc}}$ denotes the closure of $\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})$ in $\operatorname{Spec^{h}}(R)$ with respect to the localizing topology.

(d)

If the BIK support satisfies the detection property then for any set of objects $\{t_{i}\}_{i\in I}$ of $\mathscr{T}$ we have

(4.25)

\operatorname{Supp}_{\textup{BIK}}(\coprod_{i\in I}t_{i})=\overline{\bigcup_{i\in I}\operatorname{Supp}_{\textup{BIK}}(t_{i})}^{\operatorname{loc}}.

Proof.

(a): Suppose $\mathfrak{p}\not\in\operatorname{Supp}_{\textup{BIK}}(t)$ . By 4.14 there exist Thomason subsets $\cal V$ and $\cal Z$ such that $\mathfrak{p}\in\cal V\cap\cal Z^{c}$ and $\varGamma_{\cal V}L_{\cal Z}t=0$ . Hence $\cal V\cap\cal Z^{c}$ is an open neighborhood of $\mathfrak{p}$ (with respect to the localizing topology) for which $\cal V\cap\cal Z^{c}\cap\operatorname{Supp}_{\textup{BIK}}(t)=\varnothing$ . Therefore, $\operatorname{Supp}_{\textup{BIK}}(t)$ is localizing closed.

(b): For any $\mathfrak{p}\in\operatorname{Supp}_{\textup{BIK}}(\cal L)\eqqcolon S$ we choose an object $t(\mathfrak{p})\in\mathscr{L}$ such that $\mathfrak{p}\in\operatorname{Supp}_{\textup{BIK}}(t(\mathfrak{p}))$ . Now we have

S=\bigcup_{\mathfrak{p}\in S}\operatorname{Supp}_{\textup{BIK}}(t(\mathfrak{p}))=\operatorname{Supp}_{\textup{BIK}}(\coprod_{\mathfrak{p}\in S}t(\mathfrak{p})),

where the second equality follows from 4.22(c) and that $\coprod_{\mathfrak{p}\in S}t(\mathfrak{p})$ belongs to $\mathscr{L}$ . Therefore, $\operatorname{Supp}_{\textup{BIK}}(\mathscr{L})$ is localizing closed by (a).

(c): First note that (b) implies

\operatorname{Supp}_{\textup{BIK}}(\operatorname{Loc}\langle\mathscr{E}\rangle)\supseteq\overline{\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})}^{\operatorname{loc}}.

If $\mathfrak{p}\notin\overline{\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})}^{\operatorname{loc}}$ then there exist Thomason subsets $\cal V$ and $\cal Z$ with $\mathfrak{p}\in\cal V\cap\cal Z^{c}$ and $\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})\cap\cal V\cap\cal Z^{c}=\varnothing$ . By 4.22(e) we have

\operatorname{Supp}_{\textup{BIK}}(\varGamma_{\cal V}L_{\cal Z}(\mathscr{E}))=\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})\cap\cal V\cap\cal Z^{c}=\varnothing.

The detection property then implies $\varGamma_{\cal V}L_{\cal Z}(\mathscr{E})=0$ and hence $\varGamma_{\cal V}L_{\cal Z}(\operatorname{Loc}\langle\mathscr{E}\rangle)=0$ , which implies $\mathfrak{p}\notin\operatorname{Supp}_{\textup{BIK}}(\operatorname{Loc}\langle\mathscr{E}\rangle)$ . We thus obtain

\operatorname{Supp}_{\textup{BIK}}(\operatorname{Loc}\langle\mathscr{E}\rangle)=\overline{\operatorname{Supp}_{\textup{BIK}}(\mathscr{E})}^{\operatorname{loc}}.

(d): Let $\{t_{i}\}_{i\in I}$ be a set of objects in $\mathscr{T}$ . Observe that

	$\displaystyle\overline{\bigcup_{i\in I}\operatorname{Supp}_{\textup{BIK}}(t_{i})}^{\operatorname{loc}}$	$\displaystyle=\operatorname{Supp}_{\textup{BIK}}(\operatorname{Loc}\langle t_{i}\mid i\in I\rangle)=\operatorname{Supp}_{\textup{BIK}}(\operatorname{Loc}\langle\coprod_{i\in I}t_{i}\rangle)$
		$\displaystyle=\overline{\operatorname{Supp}_{\textup{BIK}}(\coprod_{i\in I}t_{i})}^{\operatorname{loc}}=\operatorname{Supp}_{\textup{BIK}}(\coprod_{i\in I}t_{i}).\qed$

4.26 Example.

If $\mathscr{T}$ is a rigidly-compactly generated tensor triangulated category then the graded endomorphism ring $\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1})$ of the unit object is graded-commutative (see [DS13, Example 2.5(2)]) and it canonically acts on the $\mathscr{T}$ (see [BIK11b, Section 7]). Therefore, we can always use this canonical action to obtain a BIK support theory on $\mathscr{T}$ . In this case we have

(4.27)

\operatorname{Supp}_{\textup{BIK}}(t_{1}\otimes t_{2})\subseteq\operatorname{Supp}_{\textup{BIK}}(t_{1})\cap\operatorname{Supp}_{\textup{BIK}}(t_{2})

for any objects $t_{1},t_{2}\in\mathscr{T}$ . It follows that the space of supports $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$ coincides with $\operatorname{Supp}_{\textup{BIK}}(\mathbb{1})$ . Moreover, 4.22(a)(b)(c), (4.25), and (4.27) are equivalent to the statement that $\big{\{}\,t\in\mathscr{T}\,\big{|}\,\operatorname{Supp}_{\textup{BIK}}(t)\subseteq Y\,\big{\}}$ is a localizing ideal of $\mathscr{T}$ for any localizing closed subset $Y\subseteq\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ ; cf. [BHS23b, Remark 2.12]. In Section 9 we will use the canonical BIK support to give a notion of stratification; see 9.6. Note that $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$ may not be the whole homogeneous Zariski spectrum $\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ . See, however, 9.7 and 9.8.

5. Tensor triangular support

In this section we summarize some basic properties of the tensor triangular support given in [San17]. We assume some familiarity with [BHS23b] and follow their the terminology and notation.

5.1 Hypothesis.

For the rest of the paper, we fix a rigidly-compactly generated tensor triangulated category $\mathscr{T}$ .

5.2 Notation.

Let $W=U\cap V^{c}$ be a weakly visible subset of $\operatorname{Spc}(\mathscr{T}^{c})$ (recall 2.6). Define $g_{W}\coloneqq e_{U}\otimes f_{V}$ , where $e_{U}$ and $f_{V}$ are the corresponding idempotent objects. By [BF11, Remark 7.6] the idempotent object $g_{W}$ does not depend on the choice of $U$ and $V$ up to isomorphism. Moreover, it follows from [BHS23b, Lemma 1.27] that

(5.3)

g_{W}=0\iff W=\varnothing.

5.4 Remark.

Let $F\colon\mathscr{C}\to\mathscr{D}$ be a geometric functor between rigidly-compactly generated tensor triangulated categories and let $\varphi\colon\operatorname{Spc}(\mathscr{D}^{c})\to\operatorname{Spc}(\mathscr{C}^{c})$ be the induced spectral map. By [BS17, Proposition 5.11] we have

(5.5)

F(g_{W})\simeq g_{\varphi^{-1}(W)}

for any weakly visible subset $W$ of $\operatorname{Spc}(\mathscr{C}^{c})$ .

5.6 Remark.

A point $\mathscr{P}$ in the Balmer spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ is not necessarily weakly visible. However, since the Thomason closed subsets of $\operatorname{Spc}(\mathscr{T}^{c})$ form a basis of closed subsets it follows that

\overline{\{\mathscr{P}\}}=\bigcap_{\begin{subarray}{c}a\in\mathscr{T}^{c}\\ a\notin\mathscr{P}\end{subarray}}\operatorname{supp}(a)\quad\text{and hence}\quad\{\mathscr{P}\}=\bigcap_{\begin{subarray}{c}a\in\mathscr{T}^{c}\\ a\notin\mathscr{P}\end{subarray}}\operatorname{supp}(a)\cap\operatorname{gen}(\mathscr{P}).

In other words, $\{\mathscr{P}\}$ is an intersection of weakly visible subsets. This leads to the following:

5.7 Definition (W. Sanders).

The tensor triangular support of an object $t\in\mathscr{T}$ is

\operatorname{Supp}(t)\coloneqq\left\{\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})\left|\begin{gathered}g_{W}\otimes t\neq 0\text{ for every weakly}\\ \text{ visible subset }W\text{ containing }\mathscr{P}\end{gathered}\right.\right\}

5.8 Remark.

By [BHS23b, Lemma 1.27] we have

(5.9)

g_{W_{1}\cap W_{2}}=g_{W_{1}}\otimes g_{W_{2}}

for any weakly visible subsets $W_{1}$ and $W_{2}$ . By 2.4 every Thomason subset of $\operatorname{Spc}(\mathscr{T}^{c})$ is a union of Thomason closed subsets and $Y_{\mathscr{P}}\coloneqq\operatorname{gen}(\mathscr{P})^{c}$ is the largest Thomason subset not containing $\mathscr{P}$ . Therefore, a Balmer prime $\mathscr{P}$ is in $\operatorname{Supp}(t)$ if and only if $e_{\operatorname{supp}a}\otimes f_{Y_{\mathscr{P}}}\otimes t\neq 0$ for every Thomason closed subset $\operatorname{supp}a$ that contains $\mathscr{P}$ (cf. 4.11).

5.10 Remark.

If $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian, that is, every point of $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly visible, then the tensor triangular support coincides with the Balmer–Favi support in [BHS23b, Definition 2.11]. Indeed, if $\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})$ is weakly visible then (5.9) implies that $\mathscr{P}\in\operatorname{Supp}(t)$ if and only if $g_{\mathscr{P}}\otimes t\neq 0$ .

5.11 Proposition.

The tensor triangular support has the following basic properties:

(a)

$\operatorname{Supp}(0)=\varnothing$ and $\operatorname{Supp}(\mathbb{1})=\operatorname{Spc}(\mathscr{T}^{c})$ .
(b)

$\operatorname{Supp}(\Sigma t)=\operatorname{Supp}(t)$ for every $t\in\mathscr{T}$ .
(c)

$\operatorname{Supp}(c)\subseteq\operatorname{Supp}(a)\cup\operatorname{Supp}(b)$ for any exact triangle $a\to b\to c\to\Sigma a$ in $\mathscr{T}$ .
(d)

$\operatorname{Supp}(t_{1}\oplus t_{2})=\operatorname{Supp}(t_{1})\cup\operatorname{Supp}(t_{2})$ for any $t_{1},t_{2}\in\mathscr{T}$ .
(e)

$\operatorname{Supp}(t_{1}\otimes t_{2})\subseteq\operatorname{Supp}(t_{1})\cap\operatorname{Supp}(t_{2})$ for any $t_{1},t_{2}\in\mathscr{T}$ .
(f)

$\operatorname{Supp}(t\otimes e_{Y})=\operatorname{Supp}(t)\cap Y$ and $\operatorname{Supp}(t\otimes f_{Y})=\operatorname{Supp}(t)\cap Y^{c}$ for every object $t\in\mathscr{T}$ and every Thomason subset $Y\subseteq\operatorname{Spc}(\mathscr{T}^{c})$ .

Proof.

Parts (a), (b), (d), and (e) are immediate from the definitions. The proofs for parts (c) and (f) can be found in [San17, Theorem 4.2]. ∎

5.12 Remark.

The half-tensor formula [BHS23b, Lemma 2.18] still holds without the weakly Noetherian assumption:

5.13 Lemma.

For any compact object $x\in\mathscr{T}^{c}$ and arbitrary object $t\in\mathscr{T}$ we have

\operatorname{Supp}(x\otimes t)=\operatorname{supp}(x)\cap\operatorname{Supp}(t).

In particular, for any compact object $x\in\mathscr{T}^{c}$ , the tensor triangular support coincides with the usual notion of support: $\operatorname{Supp}(x)=\operatorname{supp}(x)$ .

Proof.

Observe that

	$\displaystyle\mathscr{P}\notin\operatorname{Supp}(x\otimes t)$	$\displaystyle\iff x\otimes t\otimes g_{W}=0\text{ for some weakly visible }W\ni\mathscr{P}$
		$\displaystyle\iff e_{\operatorname{supp}(x)}\otimes t\otimes g_{W}=0\quad(\operatorname{Loc}_{\otimes}\langle x\rangle=e_{\operatorname{supp}(x)}\otimes\mathscr{T})$
		$\displaystyle\iff\mathscr{P}\notin\operatorname{Supp}(e_{\operatorname{supp}(x)}\otimes t)=\operatorname{supp}(x)\cap\operatorname{Supp}(t)$

where the last equality is due to 5.11(f). ∎

5.14 Remark.

Recall from 2.7 that $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian if and only if its localizing topology is discrete. Thus the localizing topology becomes relevant if $\operatorname{Spc}(\mathscr{T}^{c})$ is not weakly Noetherian, as the following shows.

5.15 Proposition.

For any object $t\in\mathscr{T}$ and any localizing subcategory $\mathscr{L}$ of $\mathscr{T}$ we have

(a)

$\operatorname{Supp}(t)$ is localizing closed.
(b)

$\operatorname{Supp}(\mathscr{L})$ is localizing closed.

Proof.

See [San17, Theorems 4.2]. ∎

5.16 Remark.

We now study how the tensor triangular support behaves under base-change functors.

5.17 Proposition.

Let $F\colon\mathscr{C}\to\mathscr{D}$ be a geometric functor between rigidly-compactly generated tt-categories, $U$ its right adjoint, and $\varphi\colon\operatorname{Spc}(\mathscr{D}^{c})\to\operatorname{Spc}(\mathscr{C}^{c})$ the induced spectral map. We have

\operatorname{Supp}(U(\mathbb{1}))=\overline{\operatorname{im}\varphi}^{\operatorname{loc}}

where $\overline{\operatorname{im}\varphi}^{\operatorname{loc}}$ denotes the closure of $\operatorname{im}\varphi$ in $\operatorname{Spc}(\mathscr{C}^{c})$ with respect to the localizing topology.

Proof.

If $\mathscr{Q}\notin\operatorname{Supp}(U(\mathbb{1}))$ then there exists some weakly visible subset $W\ni\mathscr{Q}$ such that $g_{W}\otimes U(\mathbb{1})=0$ and thus $U(F(g_{W}))=0$ by the projection formula in [BDS16, Proposition 2.15]. It follows that $0=\operatorname{Hom}(g_{W},UF(g_{W}))\simeq\operatorname{Hom}(F(g_{W}),F(g_{W}))$ and hence $F(g_{W})=0$ . We then have $\varphi^{-1}(W)=\varnothing$ in view of (5.3) and (5.5). Therefore $\mathscr{Q}\notin\operatorname{im}\varphi$ . We have established $\operatorname{im}\varphi\subseteq\operatorname{Supp}(U(\mathbb{1}))$ and it follows from 5.15 that $\overline{\operatorname{im}\varphi}^{\operatorname{loc}}\subseteq\operatorname{Supp}(U(\mathbb{1}))$ . For the other inclusion, if $\mathscr{Q}\notin\overline{\operatorname{im}\varphi}^{\operatorname{loc}}$ then there exists a weakly visible subset $W\ni\mathscr{Q}$ such that $W\cap\operatorname{im}\varphi=\varnothing$ . This means $\varphi^{-1}(W)=\varnothing$ , which implies $F(g_{W})=0$ . By the projection formula we obtain $g_{W}\otimes U(\mathbb{1})=0$ and hence $\mathscr{Q}\notin\operatorname{Supp}(U(\mathbb{1}))$ . ∎

5.18 Remark.

In [BCHS23, Corollary 13.15] it was shown that if $\operatorname{Spc}(\mathscr{C}^{c})$ is weakly Noetherian then $\operatorname{Supp}(U(\mathbb{1}))=\operatorname{im}\varphi$ , which is a special case of the proposition above.

5.19 Proposition.

Let $F$ , $U$ , and $\varphi$ be as in 5.17. The following hold:

(a)

$\operatorname{Supp}(F(c))\subseteq\varphi^{-1}(\operatorname{Supp}(c))$ for any $c\in\mathscr{C}$ .
(b)

$\varphi(\operatorname{Supp}(d))\subseteq\operatorname{Supp}(U(d))$ for any $d\in\mathscr{D}$ if $U$ is conservative.

Proof.

For part (a), let $\mathscr{P}\in\operatorname{Supp}(F(c))$ . If $\mathscr{Q}\coloneqq\varphi(\mathscr{P})\notin\operatorname{Supp}(c)$ then there exists a weakly visible subset $W\ni\mathscr{Q}$ with $g_{W}\otimes c=0$ . Hence

0=F(g_{W}\otimes c)\simeq F(g_{W})\otimes F(c)\simeq g_{\varphi^{-1}(W)}\otimes F(c)

which contradicts $\mathscr{P}\in\operatorname{Supp}(F(c))$ . This establishes (a).

To prove part (b), suppose that $\mathscr{P}\in\operatorname{Supp}(d)$ but $\mathscr{Q}\coloneqq\varphi(\mathscr{P})\notin\operatorname{Supp}(U(d))$ . By definition there exists a weakly visible subset $W\ni\mathscr{Q}$ such that $g_{W}\otimes U(d)=0$ . We then have $U(F(g_{W})\otimes d)=0$ by the projection formula and thus $g_{\varphi^{-1}(W)}\otimes d=0$ by the conservativity of $U$ . 5.11(f) then implies $\varphi^{-1}(W)\cap\operatorname{Supp}(d)=\varnothing$ , which contradicts $\mathscr{P}\in\operatorname{Supp}(d)$ . ∎

5.20 Corollary.

Let $F$ , $U$ , and $\varphi$ be as in 5.17. If $F$ is a finite localization then we have

(a)

$\varphi(\operatorname{Supp}(d))=\operatorname{Supp}(U(d))$ for any $d\in\mathscr{D}$ .
(b)

$\operatorname{Supp}(F(c))=\varphi^{-1}(\operatorname{Supp}(c))$ for any $c\in\mathscr{C}$ .

Proof.

For part (a), it suffices to show that $\operatorname{Supp}(U(d))\subseteq\varphi(\operatorname{Supp}(d))$ by 5.19(b). To this end, suppose $\mathscr{Q}=\varphi(\mathscr{P})\in\operatorname{Supp}(U(d))$ but $\mathscr{P}\notin\operatorname{Supp}(d)$ . By definition there exists a weakly visible subset $W\subseteq\operatorname{Spc}(\mathscr{D}^{c})$ such that $\mathscr{P}\in W$ and $g_{W}\otimes d=0$ . The map $\varphi\colon\operatorname{Spc}(\mathscr{D}^{c})\hookrightarrow\operatorname{Spc}(\mathscr{C}^{c})$ exhibits $\operatorname{Spc}(\mathscr{D}^{c})$ as a spectral subspace of $\operatorname{Spc}(\mathscr{C}^{c})$ since $F$ is a finite localization. We thus have $W=\varphi^{-1}(Z)$ for some weakly visible subset $Z\subseteq\operatorname{Spc}(\mathscr{C}^{c})$ by [DST19, Theorem 2.1.3]. It follows from the projection formula and (5.5) that $g_{Z}\otimes U(d)\simeq U(F(g_{Z})\otimes d)\simeq U(g_{W}\otimes d)=0$ , which contradicts $\mathscr{Q}\in\operatorname{Supp}(U(d))$ . This establishes (a).

For part (b), suppose that $F$ is the finite localization associated to a Thomason subset $Y\subseteq\operatorname{Spc}(\mathscr{C}^{c})$ . For any $c\in\mathscr{C}$ we have $\operatorname{Supp}(F(c))=\varphi^{-1}(\operatorname{Supp}(UF(c)))$ by part (a). Observe that

\begin{split}\varphi^{-1}(\operatorname{Supp}(UF(c)))&=\varphi^{-1}(\operatorname{Supp}(U(\mathbb{1})\otimes c))=\varphi^{-1}(\operatorname{Supp}(f_{Y}\otimes c))\\ &=\varphi^{-1}(Y^{c}\cap\operatorname{Supp}(c))=\varphi^{-1}(\operatorname{Supp}(c))\end{split}

which completes the proof. ∎

6. The detection property

Our next goal is to show that the detection property can be checked at the algebraic localizations at the closed points of the Zariski spectrum of the graded endomorphism ring of the unit.

6.1 Recollection.

Let $\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})$ . The finite localization $\mathscr{T}\to\mathscr{T}_{\mathscr{P}}\coloneqq\mathscr{T}(\operatorname{gen}(\mathscr{P}))$ induces a spectral map $\varphi_{\mathscr{P}}\colon\operatorname{Spc}(\mathscr{T}_{\mathscr{P}}^{c})\hookrightarrow\operatorname{Spc}(\mathscr{T}^{c})$ , which identifies $\operatorname{Spc}(\mathscr{T}_{\mathscr{P}}^{c})$ with its image $\operatorname{gen}(\mathscr{P})$ . The category $\mathscr{T}_{\mathscr{P}}$ is called the localization of $\mathscr{T}$ at $\mathscr{P}$ . We write $t_{\mathscr{P}}$ for the image of an object $t\in\mathscr{T}$ in $\mathscr{T}_{\mathscr{P}}$ . See [BHS23b, Remark 1.23 and Definition 1.25] for further discussion.

6.2 Example (Graded algebraic localization).

Recall from 4.26 that the graded endomorphism ring $\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1})$ of the unit object canonically acts on $\mathscr{T}$ . Thus the machinery in Section 4 applies. Denote by $\operatorname{End}_{\mathscr{T}}^{\mathrm{hom}}(\mathbb{1})$ the set of homogeneous elements in $\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1})$ . There is a natural continuous map

\begin{split}\rho\colon\operatorname{Spc}(\mathscr{T}^{c})&\to\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))\\ \mathscr{P}&\mapsto\langle f\in\operatorname{End}^{\mathrm{hom}}_{\mathscr{T}}(\mathbb{1})\mid\operatorname{cone}(f)\notin\mathscr{P}\rangle\end{split}

such that $\rho^{-1}(\cal V(s))=\operatorname{supp}(\operatorname{cone}(s))$ for any $s\in\operatorname{End}^{\mathrm{hom}}_{\mathscr{T}}(\mathbb{1})$ [Bal10, Theorem 5.3]. Let $S\subseteq\operatorname{End}^{\mathrm{hom}}_{\mathscr{T}}(\mathbb{1})$ be a multiplicative subset of homogeneous elements. We denote the finite localization of $\mathscr{T}$ associated to the Thomason subset

\rho^{-1}(\bigcup_{s\in S}\cal V(s))=\bigcup_{s\in S}\operatorname{supp}(\operatorname{cone}(s))

by $S^{-1}\mathscr{T}$ , which is called the algebraic localization of $\mathscr{T}$ with respect to $S$ . The corresponding localization functor and colocalization functor are denoted by $L_{S}$ and $\varGamma_{S}$ , respectively.

In particular, for a prime ideal $\mathfrak{p}\in\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ we define the multiplicative subset $S_{\mathfrak{p}}\coloneqq\big{\{}\,s\in\operatorname{End}_{\mathscr{T}}^{\mathrm{hom}}(\mathbb{1})\,\big{|}\,s\notin\mathfrak{p}\,\big{\}}$ and call $\mathscr{T}_{\mathfrak{p}}\coloneqq S_{\mathfrak{p}}^{-1}\mathscr{T}$ the algebraic localization of $\mathscr{T}$ at $\mathfrak{p}$ . Observe that

\operatorname{Loc}_{\otimes}\langle\operatorname{cone}(s)\mid s\in S_{\mathfrak{p}}\rangle=\operatorname{Loc}\langle{x/\!\!/s}\mid x\in\mathscr{T}^{c},s\in S_{\mathfrak{p}}\rangle=\mathscr{T}_{\cal Z(\mathfrak{p})}

where the last equality follows from 4.9(b). Therefore, the localization functor $L_{\cal Z(\mathfrak{p})}$ in 4.11 is identical to the algebraic localization functor $L_{S_{\mathfrak{p}}}$ which is associated to the Thomason subset $\rho^{-1}(\bigcup_{s\in S_{\mathfrak{p}}}\cal V(s))=\rho^{-1}(\cal Z(\mathfrak{p}))$ .

6.3 Notation.

For any $t\in\mathscr{T}$ we write $\pi_{*}(t)$ for

\operatorname{Hom}_{\mathscr{T}}^{-*}(\mathbb{1},t)=\coprod_{n\in\mathbb{Z}}\operatorname{Hom}_{\mathscr{T}}(\mathbb{1},\Sigma^{-n}t).

6.4 Remark.

The following proposition generalizes both [HPS97, Theorem 3.3.7] and [Bal10, Corollary 3.10].

6.5 Proposition.

Let $S\subseteq\operatorname{End}^{\mathrm{hom}}_{\mathscr{T}}(\mathbb{1})$ be a multiplicative subset of homogeneous elements. There is a natural isomorphism

\operatorname{Hom}_{\mathscr{T}}^{*}(x,L_{S}(t))\cong S^{-1}\operatorname{Hom}_{\mathscr{T}}^{*}(x,t)

for $x\in\mathscr{T}^{c}$ and $t\in\mathscr{T}$ . In particular, we have

\pi_{*}(L_{S}(t))\cong S^{-1}\pi_{*}(t)

for any $t\in\mathscr{T}$ .

Proof.

If $x$ is compact then $S^{-1}\operatorname{Hom}^{*}_{\mathscr{T}}(x,-)$ is a coproduct-preserving homological functor on $\mathscr{T}$ which vanishes on $t\otimes\operatorname{cone}(s)$ for all $t\in\mathscr{T}$ and $s\in S$ and hence on $\operatorname{Loc}\langle\mathscr{T}\otimes\operatorname{cone}(s)\mid s\in S\rangle=\operatorname{Loc}_{\otimes}\langle\operatorname{cone}(s)\mid s\in S\rangle=\varGamma_{S}\mathscr{T}$ . It follows from the localization triangle

\varGamma_{S}(t)\to t\to L_{S}(t)

that there is a natural isomorphism

S^{-1}\operatorname{Hom}^{*}_{\mathscr{T}}(x,t)\cong S^{-1}\operatorname{Hom}^{*}_{\mathscr{T}}(x,L_{S}(t))

for any $x\in\mathscr{T}^{c}$ and $t\in\mathscr{T}$ . Let $d$ be the degree of $s$ . Applying $\operatorname{Hom}^{*}_{\mathscr{T}}(-,t)$ to the exact triangle

x\xrightarrow{s}\Sigma^{d}x\to\operatorname{cone}(s)\otimes x

yields an exact sequence

0=\operatorname{Hom}^{*}_{\mathscr{T}}(\operatorname{cone}(s)\otimes x,L_{S}(t))\to\operatorname{Hom}^{*}_{\mathscr{T}}(x,L_{S}(t))[-d]\\ \xrightarrow{s}\operatorname{Hom}^{*}_{\mathscr{T}}(x,L_{S}(t))\to\operatorname{Hom}^{*}_{\mathscr{T}}(\operatorname{cone}(s)\otimes x,L_{S}(t))[1]=0.

Thus multiplying by $s$ is an isomorphism on $\operatorname{Hom}^{*}_{\mathscr{T}}(x,L_{S}(t))$ . Therefore,

S^{-1}\operatorname{Hom}^{*}_{\mathscr{T}}(x,L_{S}(t))\cong\operatorname{Hom}^{*}_{\mathscr{T}}(x,L_{S}(t)),

which completes the proof. ∎

6.6 Example.

Recall from [Bal10, Corollary 9.5] that the Balmer spectrum $\operatorname{Spc}(\operatorname{SH}^{c})$ of the stable homotopy category together with its comparison map can be depicted as follows:

where $\mathscr{P}_{p,n}$ is the kernel in ${\operatorname{SH}^{c}}$ of the $p$ -local $n$ -th Morava K-theory. In particular, $\mathscr{P}_{p,0}=\operatorname{SH}_{\mathrm{tor}}^{c}$ is the subcategory of finite torsion spectra, which is independent of $p$ . For any prime number $p$ , the $p$ -local stable homotopy category ${\operatorname{SH}_{(p)}}$ , which is defined as the Bousfield localization of the stable homotopy category $\operatorname{SH}$ with respect to the mod- $p$ Moore spectrum, can be realized as the algebraic localization at $p\mathbb{Z}\in\operatorname{Spec^{h}}(\operatorname{End}_{\operatorname{SH}}^{*}(\mathbb{1}))\cong\operatorname{Spec}(\mathbb{Z})$ of $\operatorname{SH}$ by 6.5. Moreover, since we have $\mathscr{P}_{p,\infty}=\operatorname{thick}_{\otimes}\langle\operatorname{cone}(s)\mid s\notin p\mathbb{Z}\rangle$ by [Bal10, Corollary 9.5(c)], the algebraic localization at $p\mathbb{Z}$ coincides with the localization at $\mathscr{P}_{p,\infty}$ in the sense of 6.1. Therefore, the Balmer spectrum $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ can be identified with

\operatorname{gen}(\mathscr{P}_{p,\infty})=\{\mathscr{P}_{p,n}\mid 0\leq n\leq\infty\}\subset\operatorname{Spc}({\operatorname{SH}^{c}})

where $\mathscr{P}_{p,0}$ denotes $\operatorname{SH}_{\mathrm{tor}}^{c}$ for each prime $p$ . Given a spectrum $t\in\operatorname{SH}$ , the corresponding $p$ -local spectrum is denoted by $t_{(p)}\coloneqq t_{p\mathbb{Z}}=t_{\mathscr{P}_{p,\infty}}$ .

6.7 Example.

The space $\operatorname{Spc}({\operatorname{SH}^{c}})$ is not weakly Noetherian. Indeed, by [Bal10, Corollary 9.5] any Thomason subset of $\operatorname{Spc}({\operatorname{SH}^{c}})$ is the union of subsets of the form $\overline{\{\mathscr{P}_{p,n_{p}}\}}$ where $p$ is a prime number and $0\leq n_{p}<\infty$ . Thus the closed point $\mathscr{P}_{p,\infty}$ is not weakly visible for every prime number $p$ . In particular, any Thomason subset of $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ is of the form $\overline{\{\mathscr{P}_{p,n}\}}$ for $0\leq n<\infty$ and therefore $\mathscr{P}_{p,\infty}$ is the only point in $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ that is not weakly visible.

6.8 Remark.

In [BHS23b, Remark 11.11] the authors considered the Balmer–Favi support for ${\operatorname{SH}_{(p)}}$ that excludes $\mathscr{P}_{p,\infty}$ since it is not weakly visible. More precisely, for any $t_{(p)}\in{\operatorname{SH}_{(p)}}$ they defined

\operatorname{Supp}_{<\infty}(t_{(p)})\coloneqq\big{\{}\,\mathscr{P}_{p,n}\,\big{|}\,0\leq n<\infty\text{ and }g_{\mathscr{P}_{p,n}}\otimes t_{(p)}\neq 0\,\big{\}}.

They further extended this support to the point $\mathscr{P}_{p,\infty}$ by declaring that $\mathscr{P}_{p,\infty}$ is in the support of a $p$ -local spectrum $t_{(p)}$ if and only if ${\operatorname{H}\mathbb{F}_{p}}\otimes t_{(p)}\neq 0$ . We denote this extended support function by $\operatorname{Supp}_{\leq\infty}$ . Let $I\in\operatorname{SH}$ be the Brown-Comenetz dual of the sphere spectrum. Note that the $p$ -local Brown-Comenetz dual of the sphere spectrum $I_{(p)}\in{\operatorname{SH}_{(p)}}$ is isomorphic to the Brown-Comenetz dual of the $p$ -local sphere spectrum as defined in [HP99, Section 7]. In [BHS23b, Remark 11.11] it was explained that $\operatorname{Supp}_{<\infty}(I_{(p)})=\varnothing$ and ${\operatorname{H}\mathbb{F}_{p}}\otimes I_{(p)}=0$ . Hence $\operatorname{Supp}_{\leq\infty}(I_{(p)})=\varnothing$ . This motivates the following:

6.9 Definition (The detection property).

We say that $\mathscr{T}$ has the detection property if $\operatorname{Supp}(t)=\varnothing$ implies $t=0$ for every $t\in\mathscr{T}$ .

6.10 Remark.

If $\operatorname{Spc}(\mathscr{T}^{c})$ is Noetherian then $\mathscr{T}$ has the detection property by [BHS23b, Theorem 3.22 and Remark 3.9]. For example, the derived category $\operatorname{D}(R)$ of a commutative Noetherian ring has the detection property. However, we do not know in what generality the detection property holds; see, for example, the discussion in [San17, Section 8.1] and [BCHS23, Remark 6.6].

6.11 Example.

In 6.8 we see that the function $\operatorname{Supp}_{\leq\infty}$ does not detect vanishing of objects in ${\operatorname{SH}_{(p)}}$ . However, the tensor triangular support $\operatorname{Supp}$ does:

6.12 Proposition.

The category ${\operatorname{SH}_{(p)}}$ satisfies the detection property.

Proof.

Suppose $\operatorname{Supp}(t_{(p)})=\varnothing$ for some $t_{(p)}\in{\operatorname{SH}_{(p)}}$ . Since $\mathscr{P}_{p,\infty}\notin\operatorname{Supp}(t_{(p)})$ , we have $e_{\overline{\{\mathscr{P}_{p,n}\}}}\otimes t_{(p)}=0$ for some $0\leq n<\infty$ and thus $t_{(p)}\simeq f_{\overline{\{\mathscr{P}_{p,n}\}}}\otimes t_{(p)}$ . If $n=0$ then $t_{(p)}=0$ as desired. Now suppose $n>0$ . By $\mathscr{P}_{p,n-1}\notin\operatorname{Supp}(t_{(p)})$ we have

0=e_{\overline{\{\mathscr{P}_{p,n-1}\}}}\otimes f_{\overline{\{\mathscr{P}_{p,n}\}}}\otimes t_{(p)}\simeq e_{\overline{\{\mathscr{P}_{p,n-1}\}}}\otimes t_{(p)}.

An induction shows that $0=e_{\overline{\{\mathscr{P}_{p,0}\}}}\otimes t_{(p)}\simeq t_{(p)}$ , which completes the proof. ∎

6.13 Example.

By 6.8 and 6.12, the $p$ -local Brown-Comenetz dual of the sphere spectrum has support $\operatorname{Supp}(I_{(p)})=\{\mathscr{P}_{p,\infty}\}$ . In fact, the detection property of ${\operatorname{SH}_{(p)}}$ tells us more:

6.14 Corollary.

The following are equivalent for a spectrum $t_{(p)}\in{\operatorname{SH}_{(p)}}$ :

(a)

$t_{(p)}$ is a nonzero dissonant spectrum.
(b)

$\operatorname{Supp}(t_{(p)})=\{\mathscr{P}_{p,\infty}\}$ .

Proof.

By definition the dissonant spectra are precisely the objects in the localizing ideal $\mathscr{D}\coloneqq\bigcap_{0\leq n<\infty}\operatorname{Loc}_{\otimes}\langle\mathscr{P}_{p,n}\rangle\subset{\operatorname{SH}_{(p)}}$ . Note that

\operatorname{Supp}(\mathscr{D})\subseteq\bigcap_{0\leq n<\infty}\overline{\{\mathscr{P}_{p,n}\}}=\{\mathscr{P}_{p,\infty}\}.

By the detection property any nonzero dissonant spectrum is then supported at the single point $\mathscr{P}_{p,\infty}$ .

On the other hand, suppose $t_{(p)}\in{\operatorname{SH}_{(p)}}$ has support $\{\mathscr{P}_{p,\infty}\}$ . We then have $\operatorname{Supp}(t_{(p)}\otimes f_{Y_{\mathscr{P}_{p,n}}})=\operatorname{Supp}(t_{(p)})\cap\operatorname{gen}(\mathscr{P}_{p,n})=\varnothing$ for each $0\leq n<\infty$ . It follows from the detection property that $t_{(p)}\simeq t_{(p)}\otimes e_{Y_{\mathscr{P}_{p,n}}}$ is nonzero for each $0\leq n<\infty$ and therefore $t_{(p)}\in\bigcap_{0\leq n<\infty}\operatorname{Loc}_{\otimes}\langle\mathscr{P}_{p,n}\rangle=\mathscr{D}$ . ∎

6.15 Remark.

The following corollary says that the tensor triangular support of an object can be computed at its localizations at all closed points in the Balmer spectrum. Keep in mind the notation introduced in 6.1.

6.16 Corollary.

For every object $t\in\mathscr{T}$ we have

\operatorname{Supp}(t)=\bigcup_{\begin{subarray}{c}\mathscr{M}\in\operatorname{Spc}(\mathscr{T}^{c})\\ \mathscr{M}\text{ closed}\end{subarray}}\varphi_{\mathscr{M}}(\operatorname{Supp}(t_{\mathscr{M}})).

Proof.

By 5.20(b), for any $t\in\mathscr{T}$ we have $\operatorname{Supp}(t_{\mathscr{M}})=\varphi_{\mathscr{M}}^{-1}(\operatorname{Supp}(t))$ and therefore $\varphi_{\mathscr{M}}(\operatorname{Supp}(t_{\mathscr{M}}))=\operatorname{Supp}(t)\cap\operatorname{im}\varphi_{\mathscr{M}}=\operatorname{Supp}(t)\cap\operatorname{gen}(\mathscr{M})$ . The result then follows from [Bal05, Proposition 2.11]. ∎

6.17 Example.

From 6.6, 6.13, and 6.16 we obtain that $\operatorname{Supp}(I)=\big{\{}\,\mathscr{P}_{p,\infty}\,\big{|}\,p\text{ prime}\,\big{\}}$ . In other words, the Brown-Comenetz dual of the sphere spectrum $I$ is supported on the line at infinity in the picture of $\operatorname{Spc}({\operatorname{SH}^{c}})$ .

6.18 Theorem (Detection property is algebraically local).

The category $\mathscr{T}$ has the detection property if and only if for every closed point $\mathfrak{m}\in\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ the algebraic localization $\mathscr{T}_{\mathfrak{m}}$ has the detection property .

Proof.

If $\mathscr{T}$ satisfies the detection property then $\mathscr{T}_{\mathfrak{p}}$ has the detection property for every $\mathfrak{p}\in\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ by 5.20(a). Conversely, suppose that $\mathscr{T}_{\mathfrak{m}}$ satisfies the detection property for every closed point $\mathfrak{m}\in\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ . Let $t$ be an object in $\mathscr{T}$ with $\operatorname{Supp}(t)=\varnothing$ . By 5.19(a) we have $\operatorname{Supp}(t_{\mathfrak{m}})=\varnothing$ and thus $t_{\mathfrak{m}}=0$ . It follows from 6.5 that $\operatorname{Hom}(x,t)_{\mathfrak{m}}\cong\operatorname{Hom}(x,t_{\mathfrak{m}})=0$ for every $x\in\mathscr{T}^{c}$ . Since $\mathfrak{m}$ ranges over all the closed points, $t=0$ , which completes the proof. ∎

6.19 Example.

The stable homotopy category ${\operatorname{SH}_{(p)}}$ of $p$ -local spectra satisfies the detection property (6.12). The theorem above thus implies that the stable homotopy category $\operatorname{SH}$ of all spectra satisfies the detection property.

7. The local-to-global principle

We now introduce a local-to-global principle for a rigidly-compactly generated tensor triangulated category which does not require any topological hypothesis on the Balmer spectrum.

7.1 Definition (The local-to-global principle).

We say that $\mathscr{T}$ satisfies the local-to-global principle if

\operatorname{Loc}_{\otimes}\langle t\rangle=\operatorname{Loc}_{\otimes}\langle t\otimes g_{W_{i}}\mid i\in I\rangle

for every object $t\in\mathscr{T}$ and every cover of $\operatorname{Spc}(\mathscr{T}^{c})$ by weakly visible subsets $W_{i}$ .

7.2 Remark.

By [BHS23b, Lemma 3.6] the definition above is equivalent to

\mathbb{1}\in\operatorname{Loc}_{\otimes}\langle g_{W_{i}}\mid i\in I\rangle

for every collection $\{W_{i}\}_{i\in I}$ of weakly visible subsets such that $\bigcup_{i\in I}W_{i}=\operatorname{Spc}(\mathscr{T}^{c})$ . If $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian then every point $\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})$ is weakly visible and thus we can always consider the cover $\bigcup_{\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})}\{\mathscr{P}\}=\operatorname{Spc}(\mathscr{T}^{c})$ . In this case, the local-to-global principle is equivalent to

\operatorname{Loc}_{\otimes}\langle t\rangle=\operatorname{Loc}_{\otimes}\langle t\otimes g_{\mathscr{P}}\mid\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})\rangle,

which recovers [BHS23b, Definition 3.8].

7.3 Proposition.

The $p$ -local stable homotopy category $\mathscr{S}\coloneqq{\operatorname{SH}_{(p)}}$ satisifies the local-to-global principle.

Proof.

For any weakly visible cover $\bigcup_{i\in I}W_{i}=\operatorname{Spc}(\mathscr{S}^{c})$ , there exists some $j\in I$ such that $\mathscr{P}_{p,\infty}\in W_{j}$ . If $W_{j}=\operatorname{Spc}(\mathscr{S}^{c})$ then $\mathbb{1}=g_{W_{j}}\in\operatorname{Loc}_{\otimes}\langle g_{W_{i}}\mid i\in I\rangle$ . Now we assume $W_{j}$ is proper in $\operatorname{Spc}(\mathscr{S}^{c})$ . We then have $W_{j}=\overline{\{\mathscr{P}_{p,n+1}\}}=\operatorname{supp}(\mathscr{P}_{p,n})$ for some $0\leq n<\infty$ and therefore $g_{W_{j}}\simeq e_{\overline{\{\mathscr{P}_{p,n+1}\}}}$ . Let $F_{n}\colon\mathscr{S}\to\mathscr{S}_{n}$ denote the finite localization associated to the Thomason subset $W_{j}$ . We write $e_{n}$ for $e_{\overline{\{\mathscr{P}_{p,n+1}\}}}$ and $f_{n}$ for $f_{\overline{\{\mathscr{P}_{p,n+1}\}}}$ . Since $\operatorname{Spc}(\mathscr{S}^{c}_{n})$ consists of only finitely many points, $\mathscr{S}_{n}$ satisfies the local-to-global principle by [BHS23b, Theorem 3.22]. Therefore

F_{n}(\mathbb{1})\in\operatorname{Loc}_{\otimes}\langle F_{n}(f_{n}\otimes g_{W_{i}})\mid i\in I\rangle.

By [BHS23b, Lemma 3.16] we obtain

\mathbb{1}\in\operatorname{Loc}_{\otimes}\langle e_{n},\{f_{n}\otimes g_{W_{i}}\}_{i\in I}\rangle=\operatorname{Loc}_{\otimes}\langle g_{W_{i}}\mid i\in I\rangle.\qed

7.4 Remark.

The local-to-global principle implies the detection property: Let $t\in\mathscr{T}$ be an object with $\operatorname{Supp}(t)=\varnothing$ . For every point $\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})$ there exists a weakly visible subset $W_{\mathscr{P}}\ni\mathscr{P}$ with $t\otimes g_{W_{\mathscr{P}}}=0$ . By the local-to-global principle we have

t\in\operatorname{Loc}_{\otimes}\langle t\otimes g_{W_{\mathscr{P}}}\mid\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})\rangle=0

which forces $t=0$ .

7.5 Remark.

Recall from 2.15 that the Balmer spectrum $\operatorname{Spc}(\mathscr{T}^{c})$ is said to be Hochster weakly scattered if its Hochster dual is weakly scattered. This means that for every proper Thomason subset $Y\subsetneq\operatorname{Spc}(\mathscr{T}^{c})$ , there exist a point $\mathscr{P}\notin Y$ and a Thomason subset $U\subseteq\operatorname{Spc}(\mathscr{T}^{c})$ such that

\mathscr{P}\in U\cap Y^{c}\subseteq\operatorname{gen}(\mathscr{P}).

7.6 Theorem.

If $\operatorname{Spc}(\mathscr{T}^{c})$ is Hochster weakly scattered then $\mathscr{T}$ satisfies the local-to-global principle .

Proof.

Let $\{W_{i}\}_{i\in I}$ be a cover of $\operatorname{Spc}(\mathscr{T}^{c})$ by weakly visible subsets. Consider the localizing ideal $\mathscr{L}\coloneqq\operatorname{Loc}_{\otimes}\langle g_{W_{i}}\mid i\in I\rangle$ and define $Y\coloneqq\bigcup_{a\in\mathscr{L}\cap\mathscr{T}^{c}}\operatorname{supp}(a)$ . Note that $Y$ is Thomason subset and $e_{Y}\in\operatorname{Loc}_{\otimes}\langle e_{\operatorname{supp}(a)}\mid a\in\mathscr{L}\cap\mathscr{T}^{c}\rangle$ by [BHS23b, Remark 1.26]. Note also that for any $a\in\mathscr{T}^{c}$ we have $e_{\operatorname{supp}(a)}\in\mathscr{L}$ if and only if $a\in\mathscr{L}$ . Thus $e_{Y}\in\mathscr{L}$ . It then remains to show $Y=\operatorname{Spc}(\mathscr{T}^{c})$ . Suppose ab absurdo that $Y\subsetneq\operatorname{Spc}(\mathscr{T}^{c})$ . By assumption, there exist $\mathscr{P}\in Y^{c}$ and $b\in\mathscr{T}^{c}$ such that

\mathscr{P}\in\operatorname{supp}(b)\cap Y^{c}\subseteq\operatorname{gen}(\mathscr{P}).

Choose $j\in I$ such that $\mathscr{P}\in W_{j}=U\cap V^{c}$ , where $U$ and $V$ are Thomason subsets. By intersecting with $U$ we may assume that $\operatorname{supp}(b)$ is contained in $U$ . Hence

\operatorname{supp}(b)\cap Y^{c}\subseteq U\cap\operatorname{gen}(\mathscr{P})\subseteq U\cap V^{c}.

It follows that $e_{\operatorname{supp}(b)}\otimes f_{Y}\in\operatorname{Loc}_{\otimes}\langle g_{W_{j}}\rangle\subseteq\mathscr{L}$ . Now the exact triangle

e_{\operatorname{supp}(b)}\otimes e_{Y}\to e_{\operatorname{supp}(b)}\to e_{\operatorname{supp}(b)}\otimes f_{Y}

implies $e_{\operatorname{supp}(b)}\in\mathscr{L}$ since the other two terms are in $\mathscr{L}$ . Thus $\mathscr{P}\in\operatorname{supp}(b)\subseteq Y$ by the definition of $Y$ , which is absurd. ∎

7.7 Remark.

7.6 strengthens [BHS23b, Theorem 3.22] which states that if $\operatorname{Spc}(\mathscr{T}^{c})$ is Noetherian then $\mathscr{T}$ satisfies the local-to-global principle; see 2.16. 7.6 also strengthens [San17, Theorems 7.9 and 7.18] which prove:

(a)

If $\operatorname{Spc}(\mathscr{T}^{c})$ is Hochster weakly scattered then $\mathscr{T}$ has the detection property.
(b)

If $\operatorname{Spc}(\mathscr{T}^{c})$ is Hochster scattered and $\mathscr{T}$ admits a monoidal model then $\mathscr{T}$ satisfies the local-to-global principle.

7.8 Remark.

The Balmer spectrum $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ satisfies the Hochster weakly scattered condition except for the $Y=\varnothing$ case (in the notation of 7.5). Nevertheless, for this example the proof of 7.6 still goes through because the Thomason subset $Y$ constructed in the proof is nonempty, as shown in 7.3. The stable homotopy category $\operatorname{SH}$ satisfies the local-to-global principle for similar reasons.

7.9 Remark.

We end this section with a few words on the theory of cosupport. Dual to the tensor triangular support, a theory of tensor triangular cosupport was systematically developed (under the assumption that the Balmer spectrum is weakly Noetherian) in [BCHS23], building on prior work in [HS99, Nee11, BIK12]. In particular, one can define the notions of costratification, colocal-to-global principle, and codetection property in terms of cosupport. Their work demonstrates that to completely understand a big tt-category one needs to consider both the support and the cosupport. Moreover, they discovered surprising relations between the theories of support and cosupport. For example, for a rigidly-compactly generated tt-category $\mathscr{T}$ with weakly Noetherian spectrum, the colocal-to-global principle, the codetection property, and the local-to-global principle are all equivalent [BCHS23, Theorem 6.4].

We propose here a notion of cosupport which works beyond the weakly Noetherian setting. The tensor triangular cosupport of an object $t\in\mathscr{T}$ is defined to be the set

\operatorname{Cosupp}(t)\coloneqq\left\{\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})\left|\begin{gathered}\neq 0\text{ for any weakly}\\ \text{ visible subset }W\text{ containing }\mathscr{P}\end{gathered}\right.\right\}

where $[-,-]$ denotes the internal hom. This recovers the notion of cosupport in [BCHS23, Definition 4.23] when $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian. Similarly to 7.1, we say that $\mathscr{T}$ satisfies the colocal-to-global principle if we have an equality of colocalizing coideals

\operatorname{Colocid}\langle t\rangle=\operatorname{Colocid}\langle[g_{W_{i}},t]\mid i\in I\rangle

for every object $t\in\mathscr{T}$ and every cover of $\operatorname{Spc}(\mathscr{T}^{c})$ by weakly visible subsets $W_{i}$ . Again, this notion of colocal-to-global principle specializes to the one in [BCHS23, Definition 4.23] if $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian. With these definitions, several results in [BCHS23] still hold without the weakly Noetherian hypothesis. For example:

7.10 Theorem (Barthel–Castellana–Heard–Sanders).

The following statements are equivalent for a rigidly-compactly generated tt-category $\mathscr{T}$ :

(a)

$\mathscr{T}$ satisfies the codetection property.
(b)

$\mathscr{T}$ satisfies the local-to-global principle.
(c)

$\mathscr{T}$ satisfies the colocal-to-global principle.

Proof.

The proof of (a) $\implies$ (b) $\implies$ (c) $\implies$ (a) in [BCHS23, Theorem 6.4] carries over, mutatis mutandis. ∎

8. Stratification implies weakly Noetherian

It is natural to ask whether the tensor triangular support can be used to classify the localizing ideals of $\mathscr{T}$ . More precisely:

8.1 Definition.

We say that $\mathscr{T}$ is stratified if the map

\begin{split}\big{\{}\text{localizing ideals of $\mathscr{T}$}\big{\}}&\to\big{\{}\text{localizing closed subsets of $\operatorname{Spc}(\mathscr{T}^{c})$}\big{\}}\\ \mathscr{L}&\mapsto\operatorname{Supp}(\mathscr{L})\end{split}

is a bijection.

8.2 Remark.

The map above is well-defined by 5.15(b). If $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian then 8.1 recovers [BHS23b, Definition 4.4]. In fact, we will see that if $\mathscr{T}$ is stratified in the sense of 8.1 then $\operatorname{Spc}(\mathscr{T}^{c})$ is necessarily weakly Noetherian. Our proof is based on the comparison between the tensor triangular support and the homological support, which was studied in detail in the weakly Noetherian context in [BHS23a].

8.3 Recollection.

Recall that each homological prime $\mathscr{B}\in\operatorname{Spc}^{\mathrm{h}}(\mathscr{T}^{c})$ gives rise to a pure-injective object $E_{\mathscr{B}}\in\mathscr{T}$ and the homological support of an object $t\in\mathscr{T}$ is given by

\operatorname{Supp}^{\mathrm{h}}(t)\coloneqq\big{\{}\,\mathscr{B}\in\operatorname{Spc}^{\mathrm{h}}(\mathscr{T}^{c})\,\big{|}\,[t,E_{\mathscr{B}}]\neq 0\,\big{\}}

where $[-,-]$ denotes the internal hom. For every homological prime $\mathscr{B}\in\operatorname{Spc}^{\mathrm{h}}(\mathscr{T}^{c})$ we have $\operatorname{Supp}^{\mathrm{h}}(E_{\mathscr{B}})=\{\mathscr{B}\}$ . The homological support satisfies the tensor product formula [Bal20a, Theorem 1.2]

\operatorname{Supp}^{\mathrm{h}}(t_{1}\otimes t_{2})=\operatorname{Supp}^{\mathrm{h}}(t_{1})\cap\operatorname{Supp}^{\mathrm{h}}(t_{2})\quad\text{for any }t_{1},t_{2}\in\mathscr{T}.

Moreover, there exists a surjective continuous map $\phi\colon\operatorname{Spc}^{\mathrm{h}}(\mathscr{T}^{c})\to\operatorname{Spc}(\mathscr{T}^{c})$ ; see [Bal20b, Corollary 3.9].

8.4 Lemma.

If $W\subseteq\operatorname{Spc}(\mathscr{T}^{c})$ is weakly visible then $\operatorname{Supp}^{\mathrm{h}}(g_{W})=\phi^{-1}(W)$ .

Proof.

Apply [BHS23a, Lemma 3.8] and the tensor-product formula. ∎

8.5 Lemma.

For every $t\in\mathscr{T}$ we have $\phi(\operatorname{Supp}^{\mathrm{h}}(t))\subseteq\operatorname{Supp}(t)$ .

Proof.

If $\mathscr{B}\in\operatorname{Supp}^{\mathrm{h}}(t)$ then for any weakly visible subset $W\ni\mathscr{P}\coloneqq\phi(\mathscr{B})$ we have $\mathscr{B}\in\phi^{-1}(W)=\operatorname{Supp}^{\mathrm{h}}(g_{W})$ by 8.4 and hence $\mathscr{B}\in\operatorname{Supp}^{\mathrm{h}}(t\otimes g_{W})$ in view of the tensor-product formula. In particular, $t\otimes g_{W}\neq 0$ . This is true for every weakly visible subset $W$ containing $\mathscr{P}$ , so $\mathscr{P}\in\operatorname{Supp}(t)$ . ∎

8.6 Lemma.

For every $\mathscr{B}\in\operatorname{Spc}^{\mathrm{h}}(\mathscr{T}^{c})$ we have $\operatorname{Supp}(E_{\mathscr{B}})=\{\phi(\mathscr{B})\}$ .

Proof.

Let $\mathscr{P}\coloneqq\phi(\mathscr{B})$ . First we show that $E_{\mathscr{B}}$ is $\mathscr{P}$ -local, i.e., $E_{\mathscr{B}}\simeq E_{\mathscr{B}}\otimes f_{Y_{\mathscr{P}}}$ . For any object $x\in\mathscr{P}$ we have $\mathscr{P}\notin\operatorname{supp}(x)$ . By 8.5 we have $\mathscr{B}\notin\operatorname{Supp}^{\mathrm{h}}(x)$ , that is, $[x,E_{\mathscr{B}}]=0$ . This holds for every $x\in\mathscr{P}$ , so $E_{\mathscr{B}}\in\operatorname{Loc}_{\otimes}\langle e_{Y_{\mathscr{P}}}\rangle^{\perp}=f_{Y_{\mathscr{P}}}\otimes\mathscr{T}$ is $\mathscr{P}$ -local. Thus $\operatorname{Supp}(E_{\mathscr{B}})\subseteq\operatorname{gen}(\mathscr{P})$ . On the other hand, we claim that $E_{\mathscr{B}}\in\operatorname{Loc}_{\otimes}\langle e_{\operatorname{supp}(a)}\rangle$ for any object $a\notin\mathscr{P}$ . Indeed,

$\displaystyle a\notin\mathscr{P}$	$\displaystyle\iff\mathscr{P}\in\operatorname{supp}(a)$
	$\displaystyle\iff\mathscr{B}\notin\phi^{-1}(\operatorname{supp}(a)^{c})=\operatorname{Supp}^{\mathrm{h}}(f_{\operatorname{supp}(a)})$	by 8.4
	$\displaystyle\iff E_{\mathscr{B}}\otimes f_{\operatorname{supp}(a)}=0$	by [Bal20a, Theorem 1.8]
	$\displaystyle\iff E_{\mathscr{B}}\in\operatorname{Loc}_{\otimes}\langle e_{\operatorname{supp}(a)}\rangle.$

Hence $E_{\mathscr{B}}\otimes e_{\operatorname{supp}(a)}\simeq E_{\mathscr{B}}$ for any $a\notin\mathscr{P}$ and $\operatorname{Supp}(E_{\mathscr{B}})\subseteq\bigcap_{a\notin\mathscr{P}}\operatorname{supp}(a)=\overline{\{\mathscr{P}\}}$ . Therefore $\operatorname{Supp}(E_{\mathscr{B}})\subseteq\overline{\{\mathscr{P}\}}\cap\operatorname{gen}(\mathscr{P})=\{\mathscr{P}\}$ . It remains to prove $\mathscr{P}\in\operatorname{Supp}(E_{\mathscr{B}})$ . From what we have shown it follows that $0\neq E_{\mathscr{B}}\simeq E_{\mathscr{B}}\otimes f_{Y_{\mathscr{P}}}\simeq E_{\mathscr{B}}\otimes f_{Y_{\mathscr{P}}}\otimes e_{\operatorname{supp}(a)}$ for any $a\notin\mathscr{P}$ . By 5.8 we conclude that $\mathscr{P}\in\operatorname{Supp}(E_{\mathscr{B}})$ . ∎

8.7 Example.

By [BC21, Corollary 3.6], the Morava K-theory spectrum $K(p,n)$ , for a prime $p$ and $0\leq n\leq\infty$ , is isomorphic to $E_{\mathscr{B}_{p,n}}$ , where $\mathscr{B}_{p,n}\in\operatorname{Spc}^{\mathrm{h}}({\operatorname{SH}^{c}})$ is the homological prime corresponding to the Balmer prime $\mathscr{P}_{p,n}\in\operatorname{Spc}({\operatorname{SH}^{c}})$ . We then have $\operatorname{Supp}(K(p,n))=\{\mathscr{P}_{p,n}\}$ by 8.6. In particular, the mod- $p$ Eilenberg-Maclane spectrum ${\operatorname{H}\mathbb{F}_{p}}=K(p,\infty)$ is supported at the singleton $\{\mathscr{P}_{p,\infty}\}$ . This also follows from 6.14.

8.8 Corollary.

If $\mathscr{T}$ satisfies the detection property then the map in 8.1 is surjective.

Proof.

Let $F\subseteq\operatorname{Spc}(\mathscr{T}^{c})$ be a localizing closed subset. Choosing a homological prime $\mathscr{B}_{\mathscr{P}}\in\phi^{-1}(\{\mathscr{P}\})$ for every $\mathscr{P}\in F$ , by 8.6 and [San17, Theorem 4.7(4)] we have

\operatorname{Supp}(\operatorname{Loc}_{\otimes}\langle E_{\mathscr{B}_{\mathscr{P}}}\mid\mathscr{P}\in F\rangle)=\overline{\bigcup_{\mathscr{P}\in F}\operatorname{Supp}(E_{\mathscr{B}_{\mathscr{P}}})}^{\operatorname{loc}}=\overline{F}^{\operatorname{loc}}=F.\qed

8.9 Remark.

In [BHS23b, Lemma 3.4] it was proved that if $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian then the map above is surjective (without assuming the detection property).

8.10 Example.

The Balmer spectrum $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ is not weakly Noetherian (6.7) but ${\operatorname{SH}_{(p)}}$ satisfies the detection property (6.12). Thus every localizing closed subset of $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ can be realized as the support of some localizing ideal of $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ . Moreover, a subset $S\subseteq\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ is localizing closed if and only if either $\mathscr{P}_{p,\infty}\in S$ or $\mathscr{P}_{p,\infty}\notin S$ and $S$ is finite. This follows from that fact that the Thomason subsets of $\operatorname{Spc}({\operatorname{SH}^{c}_{(p)}})$ are of the form $\overline{\{\mathscr{P}_{p,i}\}}$ for $0\leq i<\infty$ ; see 6.7.

8.11 Remark.

The following result indicates that the weakly Noetherian hypothesis in [BHS23a, Theorem 4.7] is unnecessary.

8.12 Proposition.

If $\mathscr{T}$ is stratified then $\phi(\operatorname{Supp}^{\mathrm{h}}(t))=\operatorname{Supp}(t)$ for any $t\in\mathscr{T}$ .

Proof.

By 8.5 the inclusion $\phi(\operatorname{Supp}^{\mathrm{h}}(t))\subseteq\operatorname{Supp}(t)$ always holds. To prove the other inclusion, let $\mathscr{P}\in\operatorname{Supp}(t)$ and choose any $\mathscr{B}\in\phi^{-1}(\{\mathscr{P}\})$ . In light of 8.6, we have $\operatorname{Supp}(E_{\mathscr{B}})=\{\mathscr{P}\}\subseteq\operatorname{Supp}(t)$ and thus $E_{\mathscr{B}}\in\operatorname{Loc}_{\otimes}\langle t\rangle$ due to stratification, which implies $[t,E_{\mathscr{B}}]\neq 0$ since $[E_{\mathscr{B}},E_{\mathscr{B}}]\neq 0$ . This completes the proof. ∎

8.13 Theorem.

If $\mathscr{T}$ is stratified then $\operatorname{Spc}(\mathscr{T}^{c})$ is weakly Noetherian.

Proof.

Let $W$ be any subset of $\operatorname{Spc}(\mathscr{T}^{c})$ . It suffices to show that $W$ is localizing closed by 2.7. To see this, we choose a $\mathscr{B}_{\mathscr{P}}\in\phi^{-1}(\{\mathscr{P}\})$ for every $\mathscr{P}\in W$ and consider the object $t_{W}\coloneqq\coprod_{\mathscr{P}\in W}E_{\mathscr{B}_{\mathscr{P}}}$ . By [Bal20a, Proposition 4.3(b)] we have $\operatorname{Supp}^{\mathrm{h}}(t_{W})=\big{\{}\,\mathscr{B}_{\mathscr{P}}\,\big{|}\,\mathscr{P}\in W\,\big{\}}$ . It then follows from 8.12 and 5.15(a) that $W=\operatorname{Supp}(t_{W})$ is localizing closed. ∎

8.14 Remark.

The theorem above shows that the generalization of the Balmer–Favi support to the tensor triangular support does not broaden the scope of the stratification theory developed in [BHS23b].

8.15 Example.

The Balmer spectrum $\operatorname{Spc}(\operatorname{SH}^{c})$ is not weakly Noetherian and therefore the stable homotopy category $\operatorname{SH}$ is not stratified; cf. 7.8.

9. Comparison of support theories

Our finial goal is to study the relation between the canonical BIK support and the tensor triangular support for a rigidly-compactly generated tensor triangulated category $\mathscr{T}$ , via the comparison map $\rho:\operatorname{Spc}(\mathscr{T}^{c})\to\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ introduced in [Bal10].

9.1 Remark.

Recall from 6.2 that for a prime ideal $\mathfrak{p}\in\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ the BIK localization functor $L_{\cal Z(\mathfrak{p})}$ is the finite localization functor associated to the Thomason subset $\rho^{-1}(\cal Z(\mathfrak{p}))$ . Let $\mathfrak{a}=(x_{1},\ldots,x_{n})$ be a finitely generated homogeneous ideal of $\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1})$ with homogeneous generators $\{x_{i}\}_{1\leq i\leq n}$ . By 4.9(a) we have

\mathscr{T}_{\cal V(\mathfrak{a})}=\operatorname{Loc}\langle{x/\!\!/\mathfrak{a}}\mid x\in\mathscr{T}^{c}\rangle=\operatorname{Loc}_{\otimes}\langle{\mathbb{1}/\!\!/\mathfrak{a}}\rangle=\operatorname{Loc}_{\otimes}\langle\bigotimes_{1\leq i\leq n}\operatorname{cone}(x_{i})\rangle.

Hence the BIK colocalization functor $\varGamma_{\cal V(\mathfrak{a})}$ is the finite colocalization functor associated to the Thomason subset

\bigcap_{1\leq i\leq n}\operatorname{supp}(\operatorname{cone}(x_{i}))=\bigcap_{1\leq i\leq n}\rho^{-1}(\cal V(x_{i}))=\rho^{-1}(\cal V(\mathfrak{a})).

Therefore

(9.2)

\varGamma_{\cal V(\mathfrak{a})}L_{\cal Z(\mathfrak{p})}t\simeq e_{\rho^{-1}(\cal V(\mathfrak{a}))}\otimes f_{\rho^{-1}(\cal Z(\mathfrak{p}))}\otimes t

for every $t\in\mathscr{T}$ .

9.3 Theorem.

Let $\mathscr{T}$ be a rigidly-compactly generated tensor triangulated category. Consider the comparison map $\rho:\operatorname{Spc}(\mathscr{T}^{c})\to\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ . Then:

(a)

$\rho(\operatorname{Supp}(t))\subseteq\operatorname{Supp}_{\textup{BIK}}(t)$ for every $t\in\mathscr{T}$ .
(b)

If $\rho$ is a homeomorphism then $\rho(\operatorname{Supp}(t))=\operatorname{Supp}_{\textup{BIK}}(t)$ for every $t\in\mathscr{T}$ .

Proof.

Let $\mathscr{P}\in\operatorname{Supp}(t)$ . If $\mathfrak{p}\coloneqq\rho(\mathscr{P})\notin\operatorname{Supp}_{\textup{BIK}}(t)$ then by (9.2) there exists a finitely generated homogeneous ideal $\mathfrak{a}\subseteq\mathfrak{p}$ such that

0=\varGamma_{\cal V(\mathfrak{a})}L_{\cal Z(\mathfrak{p})}t=e_{\rho^{-1}(\cal V(\mathfrak{a}))}\otimes f_{\rho^{-1}(\cal Z(\mathfrak{p}))}\otimes t.

Since the point $\mathscr{P}$ is contained in the weakly visible subset $\rho^{-1}(\cal V(\mathfrak{a}))\cap\rho^{-1}(\cal Z(\mathfrak{p}))^{c}$ , we have $\mathscr{P}\notin\operatorname{Supp}(t)$ , which establishes (a). To show (b), let $\mathfrak{p}\in\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ correspond to some $\mathscr{P}\in\operatorname{Spc}(\mathscr{T}^{c})$ . Note that $\rho^{-1}(\cal Z(\mathfrak{p}))=Y_{\mathscr{P}}$ because $\rho$ is a homeomorphism. Then observe that

	$\displaystyle\mathfrak{p}\notin\operatorname{Supp}_{\textup{BIK}}(t)$	$\displaystyle\iff\exists\text{ a Thomason closed }\cal V\ni\mathfrak{p}:e_{\rho^{-1}(\cal V)}\otimes f_{\rho^{-1}(\cal Z(\mathfrak{p}))}\otimes t=0$
		$\displaystyle\iff\exists\text{ a Thomason closed }V\ni\mathscr{P}:e_{V}\otimes f_{Y_{\mathscr{P}}}\otimes t=0$
		$\displaystyle\iff\mathscr{P}\notin\operatorname{Supp}(t).\qed$

9.4 Corollary.

For any compact object $x\in\mathscr{T}^{c}$ we have

\rho(\operatorname{supp}(x))\subseteq\operatorname{Supp}_{\textup{BIK}}(x)\subseteq\operatorname{Supp}_{\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1})}\operatorname{End}^{*}_{\mathscr{T}}(x).

Moreover, if $\rho$ is a homeomorphism then these inclusions are equalities.

Proof.

The first inclusion is a special case of 9.3(a). By [Lau21, (2.2)] we have $\mathfrak{p}\notin\operatorname{Supp}_{R}\operatorname{End}^{*}_{\mathscr{T}}(x)$ if and only if $L_{\cal Z(\mathfrak{p})}(x)=0$ , which implies $\mathfrak{p}\notin\operatorname{Supp}_{\textup{BIK}}(x)$ . This establishes the second inclusion. The equalities follow from [Lau21, Proposition 2.10]. ∎

9.5 Remark.

We now give a notion of stratification with respect to the canonical BIK support function which takes values in $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})\subseteq\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ :

9.6 Definition.

We say that $\mathscr{T}$ is cohomologically stratified if the map

\begin{split}\big{\{}\text{localizing ideals of $\mathscr{T}$}\big{\}}&\to\big{\{}\text{closed subsets of $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$}\big{\}}\\ \mathscr{L}&\mapsto\operatorname{Supp}_{\textup{BIK}}(\mathscr{L})\end{split}

is a bijection, where $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$ is equipped with the subspace topology of the localizing topology on $\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ .

9.7 Remark.

The map above is well-defined by 4.24(b). Moreover, if the comparison map $\rho\colon\operatorname{Spc}(\mathscr{T}^{c})\to\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ is surjective then we have $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})=\operatorname{Supp}_{\textup{BIK}}(\mathbb{1})=\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ by part (a) of 9.3.

9.8 Example.

If $\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1})$ is Noetherian then $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})=\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ by [Bal10, Theorem 7.3].

9.9 Example.

For a commutative ring $A$ , the unbounded derived category $\operatorname{D}(A)$ is rigidly-compactly generated and the derived category $\operatorname{D^{perf}}(A)$ of perfect complexes is its subcategory of rigid-compact objects. The associated comparison map $\operatorname{Spc}(\operatorname{D^{perf}}(A))\to\operatorname{Spec}(A)$ is a homeomorphism [Bal10, Proposition 8.1], so we have $\operatorname{Supp}_{\textup{BIK}}(\operatorname{D}(A))=\operatorname{Spec}(A)$ .

9.10 Corollary.

If the comparison map $\rho$ is a homeomorphism then $\mathscr{T}$ is cohomologically stratified if and only if it is stratified.

Proof.

By 9.7 the BIK space of supports $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$ is $\operatorname{Spec^{h}}(\operatorname{End}^{*}_{\mathscr{T}}(\mathbb{1}))$ . The result thus follows from 9.3(b). ∎

9.11 Example.

Let $A$ be a commutative ring. By 9.9 we can identify $\operatorname{Spc}(\operatorname{D^{perf}}(A))$ with $\operatorname{Spec}(A)$ via the comparison map, under which the tensor triangular support and the canonical BIK support coincide, according to 9.3. For any prime $\mathfrak{p}=\rho(\mathscr{P})\in\operatorname{Spec}(A)$ and any finitely generated ideal $\mathfrak{a}\subseteq\mathfrak{p}$ , it follows from [San17, Lemma 5.1] that

f_{\rho^{-1}(\cal Z(\mathfrak{p}))}=f_{Y_{\mathscr{P}}}\simeq A_{\mathfrak{p}}\text{ and }e_{\rho^{-1}(\cal V(\mathfrak{a}))}=e_{\operatorname{supp}({\mathbb{1}/\!\!/\mathfrak{a}})}\simeq K^{\infty}(\mathfrak{a})

where $K^{\infty}(\mathfrak{a})$ is the stable Koszul complex of $\mathfrak{a}$ . Hence for a complex $X\in\operatorname{D}(A)$ we have

\operatorname{Supp}(X)=\left\{\mathfrak{p}\in\operatorname{Spec}(A)\left|{\begin{gathered}K^{\infty}(\mathfrak{a})\otimes X_{\mathfrak{p}}\neq 0\text{ for any finitely}\\ \text{generated ideal $\mathfrak{a}$ contained in $\mathfrak{p}$}\end{gathered}}\right.\right\}.

This notion of support for complexes over a (non-Noetherian) commutative ring was first proposed and studied in [San17]. The condition $K^{\infty}(\mathfrak{a})\otimes X_{\mathfrak{p}}\neq 0$ is equivalent to $A/\mathfrak{a}\otimes X_{\mathfrak{p}}\neq 0$ since $\operatorname{Loc}_{\otimes}\langle K^{\infty}(\mathfrak{a})\rangle=\operatorname{Loc}_{\otimes}\langle A/\mathfrak{a}\rangle$ by [Gre01, Proposition 5.6]. If the ideal $\mathfrak{p}$ itself is finitely generated then this condition amounts to $\kappa(\mathfrak{p})\otimes X\neq 0$ . Therefore, when $A$ is Noetherian the tensor triangular support recovers the support theory defined in [Fox79].

9.12 Remark.

Neeman proved that $\operatorname{D}(A)$ is (cohomologically) stratified whenever $A$ is Noetherian; see [Nee92, Theorem 2.8]. This result was extended to the absolutely flat approximations of topologically Noetherian commutative rings by Stevenson; see [Ste14, Theorem 4.23]. On the other hand, Neeman [Nee00] gave an example of a non-Noetherian commutative ring such that the stratification fails. It remains an open question to determine for which commutative rings stratification holds. However, our 9.10 and 8.13 show that for any commutative ring $A$ , if $\operatorname{D}(A)$ is stratified then $\operatorname{Spec}(A)$ is necessarily weakly Noetherian.

9.13 Remark.

In [San17, Theorem 5.5(3)] it was shown that if the prime ideals of a commutative ring $A$ satisfy the descending chain condition holds, then $\operatorname{D}(A)$ has the detection property. This can also be deduced from 9.3 and 4.19. In fact, our 4.19 generalizes [San17, Theorem 5.5(2)].

9.14 Example.

For a Noetherian scheme $X$ . The derived category $\operatorname{D_{qc}}(X)$ of complexes of $\mathscr{O}_{X}$ -modules with quasi-coherent cohomology is stratified; see [BHS23b, Corollary 5.10]. However, $\operatorname{D_{qc}}(X)$ is not cohomologically stratified in general, since the graded endomorphism ring of the unit object in this category, that is, the sheaf cohomology ring $H(X,\mathscr{O}_{X})$ , may not have enough prime ideals when $X$ is nonaffine; see [Bal10, Remark 8.2], for example.

9.15 Example.

Let $G$ be a finite group and $k$ a field of characteristic $p>0$ such that $p$ divides the order of $G$ . The big stable module category $\operatorname{StMod}(kG)$ is BIK-stratified by $H^{*}(G,k)$ ([BIK11a, Theorem 10.3]). The associated Balmer spectrum $\operatorname{Spc}(\operatorname{stmod}(kG))$ is homeomorphic to $\operatorname{Proj}(H^{*}(G,k))$ , which is a Noetherian space since $H^{*}(G,k)$ is a Noetherian ring by the Evens-Venkov theorem [Ben98, II(3.10)]. Note that this BIK-stratification for $\operatorname{StMod}(kG)$ is not canonical since the graded endomorphism ring of the unit for $\operatorname{StMod}(kG)$ is not $H^{*}(G,k)$ but rather the Tate cohomology ring $\hat{H}^{*}(G,k)$ ; see [BK02, page 26]. The original statement of Benson–Iyengar–Krause cannot be applied to the canonical action of $\hat{H}^{*}(G,k)$ on $\operatorname{StMod}(kG)$ since $\hat{H}^{*}(G,k)$ is rarely Noetherian. In fact, $\hat{H}^{*}(G,k)$ is Noetherian if and only if the $p$ -rank of $G$ is $1$ if and only if $\hat{H}^{*}(G,k)$ is periodic; see [BIK08, Lemma 10.1].

On the other hand, $\operatorname{StMod}(kG)$ is stratified in the sense of 8.1 by [BHS23b, Example 7.12]. In the following we will show that $\operatorname{StMod}(kG)$ is also cohomologically stratified in the sense of 9.6. In other words, it is canonically stratified by $\hat{H}^{*}(G,k)$ . Note that this does not follow directly from 9.3(b) because in this example the comparison map $\rho$ is not a homeomorphism in general, as we shall see below.

9.16 Theorem.

Let $G$ be a finite group and $k$ a field of characteristic $p>0$ such that $p$ divides the order of $G$ . The stable module category $\operatorname{StMod}(kG)$ is cohomologically stratified.

Proof.

By Rickard [Ric89] the small stable module category $\operatorname{stmod}(kG)$ is equivalent to the quotient $\operatorname{D^{b}}(\operatorname{mod}(kG))/\operatorname{D^{perf}}(kG)$ . Note that the graded endomorphism ring of the unit object of $\mathscr{K}\coloneqq\operatorname{D^{b}}(\operatorname{mod}(kG))$ is isomorphic to the group cohomology ring. Therefore, we can identify $H^{*}(G,k)$ with $\operatorname{End}_{\mathscr{K}}^{*}(\mathbb{1})$ and $\hat{H}^{*}(G,k)$ with $\operatorname{End}_{\operatorname{stmod}(kG)}^{*}(\mathbb{1})$ ; see 9.15.

Now consider the functor $q\colon\mathscr{K}\to\mathscr{K}/\operatorname{D^{perf}}(kG)\simeq\operatorname{stmod}(kG)$ . The naturality of the comparision map gives us the following commutative diagram:

(9.17)

where $\operatorname{Spc}(q)$ is an open embedding and $\rho_{\mathscr{K}}$ is a homeomorphism by [Bal10, Proposition 8.5]. Moreover, $\iota\colon H^{*}(G,k)\hookrightarrow\hat{H}^{*}(G,k)$ is the first map that appears in [BIK11a, (10.2)], which views $H^{*}(G,k)$ as a subring of $\hat{H}^{*}(G,k)$ ; see also [BK02, (2.1)]. It follows that $\rho$ is injective. On the other hand, we have the following commutative diagram from the proof of [Bal10, Proposition 8.5]:

(9.18)

in which $\varphi\colon\operatorname{Proj}(H^{*}(G,k))\xrightarrow{\sim}\operatorname{Spc}(\operatorname{stmod}(kG))$ is the homeomorphism described in [Bal05, Corollary 5.10] and $\operatorname{Proj}(H^{*}(G,k))\hookrightarrow\operatorname{Spec^{h}}(H^{*}(G,k))$ is the canonical open embedding which misses the unique closed point $H^{+}(G,k)$ in $\operatorname{Spec^{h}}(H^{*}(G,k))$ . Combining (9.17) and (9.18) we obtain a commutative diagram:

(9.19)

If $\hat{H}^{*}(G,k)$ is Noetherian then [Bal10, Theorem 7.3] implies that $\rho$ is surjective and hence a bijection. By [BIK08, Lemma 10.1] $\hat{H}^{*}(G,k)$ being Noetherian is equivalent to that the $p$ -rank of $G$ equals $1$ , which implies that the Krull dimension of $\operatorname{Spec^{h}}(H^{*}(G,k))$ is equal to $1$ by Quillen stratification theorem [Qui71]. It follows that $\operatorname{Spc}(\operatorname{stmod}(kG))\cong\operatorname{Proj}(H^{*}(G,k))$ has zero Krull dimension, that is, $\operatorname{Spc}(\operatorname{stmod}(kG))$ is a discrete space. Moreover, since the trivial representation is indecomposable, $\operatorname{Spc}(\operatorname{stmod}(kG))$ is a singleton by [Bal07, Theorem 2.11], which forces $\rho$ to be a homeomorphism. Therefore, $\operatorname{StMod}(kG)$ is cohomologically stratified by 9.10.

If $\hat{H}^{*}(G,k)$ is not Noetherian (i.e., the $p$ -rank of $G$ is at least $2$ ) then the negative part $\hat{H}^{-}(G,k)$ is nilpotent by [BK02, Proposition 2.4]. It follows that

	$\displaystyle\operatorname{Spec^{h}}(\iota)\colon\operatorname{Spec^{h}}(\hat{H}^{*}(G,k))$	$\displaystyle\to\operatorname{Spec^{h}}(H^{*}(G,k))$
	$\displaystyle\mathfrak{p}=\hat{H}^{-}(G,k)\oplus\mathfrak{p}^{\geq 0}$	$\displaystyle\mapsto\mathfrak{p}^{\geq 0}$

is a homeomorphism where $\mathfrak{p}^{\geq 0}$ denotes the nonnegative part of a graded prime $\mathfrak{p}$ . On the other hand, by (9.19) the map $\rho\colon\operatorname{Spc}(\operatorname{stmod}(kG))\hookrightarrow\operatorname{Spec^{h}}(\hat{H}^{*}(G,k))$ is an open embedding which misses the unique closed point $\mathfrak{m}\coloneqq\hat{H}^{i\neq 0}(G,k)$ . Since $\operatorname{Spec^{h}}(\hat{H}^{*}(G,k))\cong\operatorname{Spec^{h}}(H^{*}(G,k))$ is Noetherian, $\cal V(\mathfrak{m})=\{\mathfrak{m}\}$ is Thomason closed. We thus have $e_{\rho^{-1}(\cal V(\mathfrak{m}))}\otimes f_{\rho^{-1}(\cal Z(\mathfrak{m}))}=e_{\varnothing}\otimes f_{\operatorname{Spc}(\operatorname{stmod}(kG))}\simeq 0\otimes\mathbb{1}=0$ and hence $\mathfrak{m}\notin\operatorname{Supp}_{\textup{BIK}}(t)$ for every $t\in\operatorname{StMod}(kG)$ . It then follows from 9.3(a) that

(9.20)

\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})=\operatorname{Supp}_{\textup{BIK}}(\mathbb{1})=\operatorname{im}\rho.

Now suppose that $\mathfrak{p}=\rho(\mathscr{P})\in\operatorname{im}\rho$ is a nonclosed point in $\operatorname{Spec^{h}}(\hat{H}^{*}(G,k))$ . Note that $\{\mathfrak{p}\}=\cal V(\mathfrak{p})\cap\cal Z(\mathfrak{p})^{c}$ , where $\cal V(\mathfrak{p})$ is Thomason closed since $\operatorname{Spec^{h}}(\hat{H}^{*}(G,k))$ is Noetherian. It follows that for every $t\in\operatorname{StMod}(kG)$ we have

\mathfrak{p}\in\operatorname{Supp}_{\textup{BIK}}(t)\iff e_{\rho^{-1}(\cal V(\mathfrak{p}))}\otimes f_{\rho^{-1}(\cal Z(\mathfrak{p}))}\otimes t\neq 0\iff\mathscr{P}\in\operatorname{Supp}(t).

Therefore $\rho(\operatorname{Supp}(t))=\operatorname{Supp}_{\textup{BIK}}(t)$ for all $t\in\operatorname{StMod}(kG)$ . From (9.20) and the fact that $\operatorname{StMod}(kG)$ is stratified, we conclude that $\operatorname{StMod}(kG)$ is cohomologically stratified. ∎

9.21 Remark.

Our definition of cohomological stratification (9.6) generalizes the one given by [BCH⁺23, Definition 2.21] which requires $\mathscr{T}$ to be Noetherian ([BCH⁺23, Definition 2.9]) and the comparison map $\rho$ to be a homeomorphism. Indeed, if $\mathscr{T}$ is Noetherian then $\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1})$ is Noetherian and thus every subset of $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})$ is localizing closed. Moreover, if $\rho$ is a homeomorphism then $\operatorname{Supp}_{\textup{BIK}}(\mathscr{T})=\operatorname{Spec^{h}}(\operatorname{End}_{\mathscr{T}}^{*}(\mathbb{1}))$ by 9.7. Note, however, that our 9.6 does not put any restriction on $\rho$ . As 9.16 shows, requiring $\rho$ to be a homeomorphism would eliminate interesting examples of cohomologically stratified categories in the non-Noetherian context.

References

[Bal05] Paul Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588:149–168, 2005.
[Bal07] Paul Balmer. Supports and filtrations in algebraic geometry and modular representation theory. Amer. J. Math., 129(5):1227–1250, 2007.
[Bal10] Paul Balmer. Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebr. Geom. Topol., 10(3):1521–1563, 2010.
[Bal20a] Paul Balmer. Homological support of big objects in tensor-triangulated categories. J. Éc. polytech. Math., 7:1069–1088, 2020.
[Bal20b] Paul Balmer. Nilpotence theorems via homological residue fields. Tunis. J. Math., 2(2):359–378, 2020.
[BC21] Paul Balmer and James C. Cameron. Computing homological residue fields in algebra and topology. Proc. Amer. Math. Soc., 149(8):3177–3185, 2021.
[BCH⁺23] Tobias Barthel, Natalia Castellana, Drew Heard, Niko Naumann, and Luca Pol. Quillen stratification in equivariant homotopy theory. Preprint, 56 pages, available online at arXiv:2301.02212, 2023.
[BCHS23] Tobias Barthel, Natalia Castellana, Drew Heard, and Beren Sanders. Cosupport in tensor triangular geometry. Preprint, 87 pages, available online at arXiv:2303.13480, 2023.
[BCR97] Dave Benson, Jon F. Carlson, and Jeremy Rickard. Thick subcategories of the stable module category. Fund. Math., 153(1):59–80, 1997.
[BDS16] Paul Balmer, Ivo Dell’Ambrogio, and Beren Sanders. Grothendieck–Neeman duality and the Wirthmüller isomorphism. Compos. Math., 152(8):1740–1776, 2016.
[Ben98] David J. Benson. Representations and cohomology I & II, volume 30 & 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1998.
[BF11] Paul Balmer and Giordano Favi. Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. (3), 102(6):1161–1185, 2011.
[BH98] Winfried Bruns and H. Jürgen Herzog. Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 1998.
[BHS23a] Tobias Barthel, Drew Heard, and Beren Sanders. Stratification and the comparison between homological and tensor triangular support. Q. J. Math., 74(2):747–766, 2023.
[BHS23b] Tobias Barthel, Drew Heard, and Beren Sanders. Stratification in tensor triangular geometry with applications to spectral Mackey functors. Camb. J. Math., 11(4):829–915, 2023.
[BIK08] Dave Benson, Srikanth B. Iyengar, and Henning Krause. Local cohomology and support for triangulated categories. Ann. Sci. Éc. Norm. Supér. (4), 41(4):573–619, 2008.
[BIK11a] Dave Benson, Srikanth B. Iyengar, and Henning Krause. Stratifying modular representations of finite groups. Ann. of Math. (2), 174(3):1643–1684, 2011.
[BIK11b] Dave Benson, Srikanth B. Iyengar, and Henning Krause. Stratifying triangulated categories. J. Topol., 4(3):641–666, 2011.
[BIK12] David J. Benson, Srikanth B. Iyengar, and Henning Krause. Colocalizing subcategories and cosupport. J. Reine Angew. Math., 673:161–207, 2012.
[BIKP18] Dave Benson, Srikanth B. Iyengar, Henning Krause, and Julia Pevtsova. Stratification for module categories of finite group schemes. J. Amer. Math. Soc., 31(1):265–302, 2018.
[BK02] David Benson and Henning Krause. Pure injectives and the spectrum of the cohomology ring of a finite group. J. Reine Angew. Math., 542:23–51, 2002.
[BKS19] Paul Balmer, Henning Krause, and Greg Stevenson. Tensor-triangular fields: ruminations. Selecta Math. (N.S.), 25(1):Paper No. 13, 36, 2019.
[Bou98] Nicolas Bourbaki. Commutative algebra. Chapters 1–7. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998. Reprint of the 1989 English translation.
[BS17] Paul Balmer and Beren Sanders. The spectrum of the equivariant stable homotopy category of a finite group. Invent. Math., 208(1):283–326, 2017.
[DP08] W. G. Dwyer and J. H. Palmieri. The Bousfield lattice for truncated polynomial algebras. Homology Homotopy Appl., 10(1):413–436, 2008.
[DS13] Ivo Dell’Ambrogio and Greg Stevenson. On the derived category of a graded commutative Noetherian ring. J. Algebra, 373:356–376, 2013.
[DST19] Max Dickmann, Niels Schwartz, and Marcus Tressl. Spectral spaces, volume 35 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2019.
[Fox79] Hans-Bjørn Foxby. Bounded complexes of flat modules. J. Pure Appl. Algebra, 15(2):149–172, 1979.
[Gre01] J. P. C. Greenlees. Tate cohomology in axiomatic stable homotopy theory. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 149–176. Birkhäuser, Basel, 2001.
[Har66] Robin Hartshorne. Residues and duality. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, With an appendix by P. Deligne.
[HP99] Mark Hovey and John H. Palmieri. The structure of the Bousfield lattice. In Homotopy invariant algebraic structures (Baltimore, MD, 1998), volume 239 of Contemp. Math., pages 175–196. Amer. Math. Soc., Providence, RI, 1999.
[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc., 128(610), 1997.
[HS98] Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory. II. Ann. of Math. (2), 148(1):1–49, 1998.
[HS99] Mark Hovey and Neil P. Strickland. Morava $K$ -theories and localisation. Mem. Amer. Math. Soc., 139(666):viii+100, 1999.
[Lau21] Eike Lau. The Balmer spectrum of certain Deligne-Mumford stacks. Preprint, 34 pages, available online at arXiv:2101.01446, 2021.
[Nee92] Amnon Neeman. The chromatic tower for $D(R)$ . Topology, 31(3):519–532, 1992.
[Nee00] Amnon Neeman. Oddball Bousfield classes. Topology, 39(5):931–935, 2000.
[Nee01] Amnon Neeman. Triangulated categories, volume 148 of Annals of Mathematics Studies. Princeton University Press, 2001.
[Nee11] Amnon Neeman. Colocalizing subcategories of $\mathbf{D}(R)$ . J. Reine Angew. Math., 653:221–243, 2011.
[NR87] S. B. Niefield and K. I. Rosenthal. Spatial sublocales and essential primes. Topology Appl., 26(3):263–269, 1987.
[Qui71] Daniel Quillen. The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2), 94:549–572; ibid. (2) 94 (1971), 573–602, 1971.
[Ric89] Jeremy Rickard. Derived categories and stable equivalence. J. Pure Appl. Algebra, 61(3):303–317, 1989.
[Roh19] Fred Rohrer. Torsion functors, small or large. Beitr. Algebra Geom., 60(2):233–256, 2019.
[San13] Beren Sanders. Higher comparison maps for the spectrum of a tensor triangulated category. Adv. Math., 247:71–102, 2013.
[San17] William T. Sanders. Support and vanishing for non-Noetherian rings and tensor triangulated categories. Preprint, 33 pages, available online at arXiv:1710.10199, 2017.
[Sta23] The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu, 2023.
[Ste14] Greg Stevenson. Derived categories of absolutely flat rings. Homology Homotopy Appl., 16(2):45–64, 2014.
[Ste17] Greg Stevenson. The local-to-global principle for triangulated categories via dimension functions. J. Algebra, 473:406–429, 2017.
[Tho97] R. W. Thomason. The classification of triangulated subcategories. Compositio Math., 105(1):1–27, 1997.

Support theories for non-Noetherian tensor triangulated categories

Abstract.

1. Introduction

Theorem A (4.13).

Definition.

Theorem B (8.13).

Theorem C (7.6).

Theorem D (9.3).

Theorem E (9.16).

Acknowledgements

2. Preliminaries on spectral spaces

2.1 Definition.

2.2 Definition.

2.3 Example.

2.4 Remark.

2.5 Remark.

2.6 Definition.

2.7 Remark.

2.8 Remark.

2.9 Example.

2.10 Remark.

2.11 Example.

2.12 Remark.

2.13 Remark.

2.14 Definition.

2.15 Definition.

2.16 Remark.

2.17 Example.

3. Small support for modules

3.1 Notation.

3.2 Definition.

3.3 Definition.

3.4 Definition.

3.5 Lemma.

Proof.

3.6 Remark.

3.7 Definition.

3.8 Remark.

3.10 Remark.

3.11 Lemma.

Proof.

3.12 Definition.

3.13 Remark.

3.14 Lemma.

Proof.

3.15 Remark.

3.16 Remark.

3.17 Lemma.

Proof.

3.18 Remark.

3.19 Lemma.

Proof.

3.20 Remark.

4. Non-Noetherian BIK support

4.1 Terminology.

4.2 Remark.

4.3 Lemma.

Proof.

4.4 Remark.

4.5 Lemma.

Proof.

4.6 Remark.

4.7 Definition.

4.8 Lemma.

Proof.

4.9 Proposition.

Proof.

4.10 Remark.

4.11 Definition.

4.12 Remark.

4.13 Theorem.

4.14 Remark.

4.15 Notation.

4.16 Theorem.

Proof.

4.17 Remark.

4.18 Definition.

4.19 Corollary.

Proof.

4.20 Remark.

Support theories for non-Noetherian
tensor triangulated categories