The homological spectrum and nilpotence theorems for Lie superalgebra representations

For finite group representations, the detecting nilpotence in cohomology via restriction maps and elementary abelian subgroups is an important idea that was used in the study of support varieties in the work of Quillen, Avrunin and Scott. The spectrum of the cohomology for elementary abelian groups can be described through explicit realizations with polynomial rings and this yields a concrete description of the support varieties through a rank variety description. The cohomological nilpotence theorem plays an essential role in the theory because it allows one to describe these support varieties for a finite group with the support varieties for its elementary abelian subgroups. In the case for restricted Lie algebras, Friedlander and Parshall showed that detecting nilpotence entails the use of one-dimensional $p$-nilpotent subalgebras of the Lie algebras. Friedlander and Pevtsova developed a general theory of $\pi$-points that captures both nilpotence and the realization of supports for arbitrary finite group schemes.

For a small rigid symmetric tensor triangulated category (TTC), $\bf K$, Balmer introduced the concept of homological primes and homological residue fields [Bal20, BC21]. For a TTC, the collection of homological primes, $\text{Spc}^{\text{h}}(\bf K)$, forms a topological space that can potentially realize the Balmer spectrum, $\text{Spc}(\bf K)$, and its support theory in a concrete way. The central problem in this identification for the theory of homological primes is the following “Nerves of Steel” Conjecture (cf. [Bal20]).

The [NoS Conj] was verified by using a deep stratification result (see [Bal20, BIKP18]) for the stable module category for finite group schemes. In this setting, the homological primes play the role of the $\pi$-points.

More recently, Balmer [Bal20] developed a nilpotence theorem for morphisms in a tensor triangulated category and Balmer and Cameron [BC21] investigated properties of homological residue fields. Balmer showed that (i) the homological primes naturally surject via the comparison map onto the categorical spectrum and (ii) if a nilpotence theorem holds for homological residue fields then the comparison map is a bijection. In the case of finite group schemes, the picture is complete: the categorical spectrum identifies with projectivization of the cohomology ring and the homological primes with the $\pi$-points (in a non-trivial way). Recent work on the Nerves of Steel Conjecture in relation to stratification and nilpotence can be found in work by Barthel, Castellana, Heard, Naumann, Sanders, and Pol [BHS23a, BHS23b], [BCHNP23] [BCHS23].

Let ${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}$ be a finite-dimensional classical Lie superalgebra with $\text{Lie }G_{\bar{0}}={\mathfrak{g}}_{\bar{0}}$ and $G_{\bar{0}}$ a reductive algebraic group. Moreover, let ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})}$ be the category of finite-dimensional ${\mathfrak{g}}$-supermodules that are completely reducible over ${\mathfrak{g}}_{\bar{0}}$. In the mid 2000s, Boe, Kujawa and Nakano [BKN10a] introduced the notion of a detecting subalgebra for classical simple Lie superalgebras by using geometric invariant theory applied to the $G_{\bar{0}}$ action on ${\mathfrak{g}}_{\bar{1}}$. The detecting subalgebras are important because they detect the ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})}$-cohomology, and can be regarded as the analogs of elementary abelian subgroups.

There are two families of detecting subalgebras ${\mathfrak{e}}$ and ${\mathfrak{f}}$. The detecting subalgebras, ${\mathfrak{e}}$, were used to provide a geometric interpretation of the well-known combinatorial notion of atypicality due to Kac and Wakimoto for basic classical Lie superalgebras. On the other hand, detecting subalgebras, ${\mathfrak{f}}$, obtained by stable actions, were used to investigate the tensor triangular geometry and describe the Balmer spectrum for $\text{stab}({\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})})$ for ${\mathfrak{g}}=\mathfrak{gl}(m|n)$. For any classical simple Lie superalgebra ${\mathfrak{g}}$, the detecting subalgebras ${\mathfrak{f}}$ can also be used to form a natural triangular decomposition for ${\mathfrak{g}}={\mathfrak{n}}^{+}\oplus{\mathfrak{f}}\oplus{\mathfrak{n}}^{-}$ where ${\mathfrak{b}}={\mathfrak{f}}\oplus{\mathfrak{n}}^{-}$ is a BBW parabolic subalgebra. The BBW parabolic subgroups/subalgebras have well-behaved homological properties as demonstrated in [GGNW21].

The focus of this paper will be to study the tensor triangular geometry of the stable module category ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})}$ where ${\mathfrak{g}}$ is a classical Lie superalgebra. In particular, we seek to investigate the homological and Balmer spectrum for $\text{stab}({\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})})$. For ${\mathfrak{g}}=\mathfrak{gl}(m|n)$, the Balmer spectrum was computed by Boe, Kujawa and Nakano using heavy representation theoretic techniques. The approach that we use involves using a circle of ideas developed by Benson, Iyengar, and Krause involving the classification of localizing subcategories and by Balmer on homological primes and nilpotence for tensor triangulated categories.

The strategy first entails classifying localizing subcategories for the detecting subalgebras. This answers a question posed in [BKN17]. This result involving stratification by the cohomology ring leads to a nilpotence thoerem for $\text{stab}({\mathcal{F}}_{({\mathfrak{f}},{\mathfrak{f}}_{\bar{0}})})$. We then employ the recent work of Serganova and Sherman [SS22] on the existence of “splitting subgroups” where one can find a copy of the trivial module in the induction of the trivial module from the splitting subgroup to the ambient supergroup to prove a nilpotence theorem for $\text{stab}({\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})})$.

The nilpotence theorem entails the use of homological residue fields that yield homological primes. In order to show that the homological spectrum identifies with the Balmer spectrum we employ natural assumptions on the realization of supports stated in [BKN17], which was earlier verified for ${\mathfrak{g}}=\mathfrak{gl}(m|n)$. This new approach has the appeal that it is much more conceptual, streamlined, and applicable to a wider class of Lie superalgebras.

The paper is organized as follows. In Section 2, a discussion about classical Lie superalgebras is presented with information about ample detecting subalgebras. Examples are provided that illustrate the $G_{\bar{0}}$-orbit structure on ${\mathfrak{g}}_{\pm 1}$ for Type I Lie superalgebras. In the following section (Section 3) we provide conditions for projectivity for classical Lie superalgebras using the detecting subalgebra for finite-dimensional modules and defining the concept of ample detecting subalgebras. For our purposes, stronger detection theorems will be needed to nilpotence theorems with infinitely generated modules which will be stated using the concept of “splitting subalgebras” as introduced in [SS22].

Section 4 focuses on Lie superalgebras ${\mathfrak{z}}$ whose commutator between the even and odd component is zero, and whose even component is a torus. This class of algebras includes all detecting subalgebras. Our analysis concludes with a classification of localizing subcategories for the stable module category of ${\mathcal{C}}_{({\mathfrak{z}},{\mathfrak{z}}_{\bar{0}})}$.

In Section 5, we present several important results on nilpotence theorems for Lie superalgebras (cf. Theorem 5.4.2 and Theorem 5.5.1). In the following section, Section 6, the homological spectrum is calculated for Lie superalgebras that contain a splitting subalgebra isomorphic to algebras considered in Section 4.

Additional conditions are presented in Section 7, that allow one to verify the Nerves of Steel Conjecture for these Lie superalgebras (with a splitting subalgebra). Finally, the paper concludes by providing computations of the homological spectrum and the Balmer spectrum for Type A classical Lie superalgebras (that include $\mathfrak{sl}(m|n)$) via our nilpotence theorem in Section 8.

Throughout this paper, let ${\mathfrak{g}}$ be a Lie superalgebra over the complex numbers ${\mathbb{C}}$ where ${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}$ is a $\mathbb{Z}_{2}$-graded vector space with a supercommutator $[\;,\;]:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$. Let $U(\mathfrak{g})$ be the universal enveloping superalgebra of ${\mathfrak{g}}$. The representation theory entails (super)-modules which are $\mathbb{Z}_{2}$-graded left $U(\mathfrak{g})$-modules. Given ${\mathfrak{g}}$-modules $M$ and $N$ one can use the antipode and coproduct of $U({\mathfrak{g}})$ to define a ${\mathfrak{g}}$-module structure on the dual $M^{*}$ and the tensor product $M\otimes N$. These constructions are essential for applying the theory of tensor triangular geometry. For definitions and detailed background, the reader is referred to [BKN10a, BKN09, BKN10b, BKN17].

A finite dimensional Lie superalgebra ${\mathfrak{g}}$ is called classical if there is a connected reductive algebraic group $G_{\bar{0}}$ such that $\operatorname{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}},$ and the action of $G_{\bar{0}}$ on $\mathfrak{g}_{\bar{1}}$ differentiates to the adjoint action of $\mathfrak{g}_{\bar{0}}$ on $\mathfrak{g}_{\bar{1}}.$

We will be interested in studying the category of finite-dimensional and infinite-dimensional Lie superalgebra representations. Let ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})}$ (resp. ${\mathcal{C}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})})$ be the full subcategory of finite dimensional (resp. infinite-dimensional) ${\mathfrak{g}}$-modules which are semisimple over ${\mathfrak{g}}_{\bar{0}}$. A ${\mathfrak{g}}_{\bar{0}}$-module is semisimple if it decomposes into a direct sum of finite dimensional simple ${\mathfrak{g}}_{\bar{0}}$-modules. The categories ${\mathcal{F}}_{(\mathfrak{g},\mathfrak{g}_{\bar{0}})}$ and ${\mathcal{C}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})}$ have enough projective (and injective) objects. Furthermore, injectivity is equivalent to projectivity [BKN10b] so they are Frobenius categories. Since these categories are Frobenius, one can form the stable module categories, $\text{stab}({\mathcal{F}}_{(\mathfrak{g},\mathfrak{g}_{\bar{0}})})$ and $\text{Stab}({\mathcal{C}}_{({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}})})$.

In [BKN10a, Section 3,4], Boe, Kujawa and Nakano introduced two families of detecting subalgebras that arise for classical simple Lie superalgebras. One of these families, often denoted by ${\mathfrak{e}},$ arises from a polar action of $G_{\bar{0}}$ on ${\mathfrak{g}}_{\bar{1}}$. These ${\mathfrak{e}}$-detecting subalgebras will not be germane to the results of the paper. The other family of detecting subalgebras arises from $G_{\bar{0}}$ having a stable action on ${\mathfrak{g}}_{\bar{1}}$ (cf. [BKN10a, Section 3.2]). Let $x_{0}$ be a generic point in ${\mathfrak{g}}_{\bar{1}}$ (i.e., $x_{0}$ is semisimple and regular). Set $H=\text{Stab}_{G_{\bar{0}}}x_{0}$ and $N:=N_{G_{\bar{0}}}(H)$. In order to construct a detecting subalgebra, let ${\mathfrak{f}}_{\bar{1}}={\mathfrak{g}}_{\bar{1}}^{H}$, ${\mathfrak{f}}_{\bar{0}}=[{\mathfrak{f}}_{\bar{1}},{\mathfrak{f}}_{\bar{1}}]$, and set

Note our definition is slightly modified from the original construction in [BKN10a].

Then ${\mathfrak{f}}$ is a classical Lie superalgebra and a sub Lie superalgebra of ${\mathfrak{g}}$. The stability of the action of $G_{\bar{0}}$ on ${\mathfrak{g}}_{\bar{1}}$ implies the following properties.

• (2.2.1)

  The restriction homomorphism $S(\mathfrak{g}_{\bar{1}}^{*})\longrightarrow S(\mathfrak{f}_{\bar{1}}^{*})$ induces an isomorphism

  $$\operatorname{res}:\text{H}^{\bullet}({\mathfrak{g}},{\mathfrak{g}}_{\bar{0}},{\mathbb{C}})\rightarrow\text{H}^{\bullet}({\mathfrak{f}},{\mathfrak{f}}_{\bar{0}},{\mathbb{C}})^{N}.$$

• (2.2.2)

  The set $G_{\bar{0}}\cdot{\mathfrak{f}}_{\bar{1}}$ is dense in $\mathfrak{g}_{\bar{1}}.$

Property (2.2.1) will be discussed later in the context of our nilpotence theorem. For much of our work, a stronger version of property (2.2.2) is needed in order to prove the nilpotence theorem and compute the Balmer spectrum.

A Lie superalgebra is Type I if it admits a $\mathbb{Z}$-grading ${\mathfrak{g}}=\mathfrak{g}_{-1}\oplus{\mathfrak{g}}_{0}\oplus{\mathfrak{g}}_{1}$ concentrated in degrees $-1,$ $0,$ and $1$ with ${\mathfrak{g}}_{\bar{0}}={\mathfrak{g}}_{0}$ and ${\mathfrak{g}}_{\bar{1}}=\mathfrak{g}_{-1}\oplus{\mathfrak{g}}_{1}$. Examples of Type I Lie superalgebras include the general linear Lie superalgebra $\mathfrak{gl}(m|n)$ and simple Lie superalgebras of types $A(m,n)$, $C(n)$ and $P(n)$. These all have stable actions of $G_{\bar{0}}$ on ${\mathfrak{g}}_{\bar{1}}$ which yields Type I detecting subalgebras.

For our paper, it will be important to distinguish Type I classical Lie superalgebras that contain a detecting subalgebra with favorable geometric properties.

As we will see in next section, there is an abundance of examples of Type I Lie superalgebras with ample detecting subalgebras that encompass many cases of simple Lie superalgebras over ${\mathbb{C}}$.

In this section, we will provide examples of Type I classical Lie superalgebras that contain an ample detecting subalgebra. Many of these actions involving $G_{0}$ on ${\mathfrak{g}}_{\pm 1}$ arise naturally in the context of linear algebra. For a more detailed description of these actions, the reader is referred to [BKN10b, Section 3.8].

We review the constructions in [BKN10b, Section 3.2] for Type I Lie superalgebras. Let ${\mathfrak{g}}=\mathfrak{g}_{-1}\oplus{\mathfrak{g}}_{\bar{0}}\oplus\mathfrak{g}_{1}$ be a Type I Lie superalgebra. Then $\mathfrak{g}_{\pm 1}$ are abelian Lie superalgebras. Consequently, $U({\mathfrak{g}}_{\pm 1})$ identifies with an exterior algebra, and the cohomology ring for these superalgebras identifies with the symmetric algebra on the dual of $\mathfrak{g}_{\pm 1}$. Set $R_{\pm 1}=\operatorname{H}^{\bullet}(\mathfrak{g}_{\pm 1},\mathbb{C})\cong S^{\bullet}(\mathfrak{g}_{\pm 1}^{*})$. Let $M$ be a finite-dimensional $U({\mathfrak{g}}_{\pm 1}))$-module and let

The (cohomological) support variety of $M$ is defined as

Moreover, the support variety $\mathcal{V}_{\mathfrak{g}_{\pm 1}}(M)$ is canonically isomorphic to the following rank variety:

These varieties satisfy many of the important properties of support theory that include (i) the detection of projectivity over $U({\mathfrak{g}}_{\pm 1})$ and (ii) the tensor product property.

For a detecting subalgebra ${\mathfrak{f}}={\mathfrak{f}}_{-1}\oplus{\mathfrak{f}}_{0}\oplus{\mathfrak{f}}_{1}$, one can apply the prior construction to obtain support varieties for $M\in{\mathcal{F}}_{({\mathfrak{f}},{\mathfrak{f}}_{0})}$, namely $\mathcal{V}_{{\mathfrak{f}}_{\pm 1}}(M)$ and $\mathcal{V}^{\text{rank}}_{{\mathfrak{f}}_{\pm 1}}(M)$.

For Type I classical Lie superalgebras, one can construct Kac and dual Kac modules (cf. [BKN10b, Section 3.1]). A module in ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{0})}$ is tilting if and only if it has both a Kac and a dual Kac filtration. The use of these filtrations was a key idea in proving the following criteria for projectivity in the category ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{0})}$ (see [BKN10b, Section 3]).

It should be noted that for an arbitrary (infinitely generated) module $M\in{\mathcal{C}}_{({\mathfrak{g}},{\mathfrak{g}}_{0})}$, one can have projectivity over $U({\mathfrak{g}}_{\pm 1})$, but $M$ may not be projective in ${\mathcal{C}}_{({\mathfrak{g}},{\mathfrak{g}}_{0})}$. For example, if ${\mathfrak{g}}=\mathfrak{gl}(1|1)$, one can take an infinite coproduct of projective modules in the principal block $P=\oplus_{m\in{\mathbb{Z}}}P(m|-m)$. By making suitable identifications, one can form an (infinite) “zigzag module” (of radical length $2$) that has a Kac and dual Kac filtration, which is projective upon restriction to $U({\mathfrak{g}}_{\pm 1})$. The zigzag module is not projective in ${\mathcal{C}}_{({\mathfrak{g}},{\mathfrak{g}}_{0})}$ because it has radical length less than $4$. This construction can also be performed for projective modules in the principal block for the restricted enveloping algebra of ${\mathfrak{s}l}_{2}$ (cf. [Po68]), and has been observed in other situations by Cline, Parshall and Scott [CPS85, (3.2) Example].

The following theorem allows us connect projectivity of a module in ${\mathcal{F}}_{({\mathfrak{g}},{\mathfrak{g}}_{0})}$ to projectivity when restricting the module to the detecting subalgebra.

For our purposes we will need a stronger projectivity criterion than the one given in Theorem 3.3.1 that can be applied to infinite-dimensional modules. In order to state such a projectivity criterion, one need to employ the concept of splitting subgroups/subalgebras as introduced by Serganova and Sherman [SS22]. The following definition which fits in our context is equivalent to their definition.

The following theorem summarizes results in [SS22, Section 2]. We present a slightly different approach using the work for BBW parabolic subgroups by D. Grantcharov, N. Grantcharov, Nakano and Wu [GGNW21].

Serganova and Sherman proved that the detecting subalgebra ${\mathfrak{f}}$ for classical Lie algebras of Type A are splitting subalgebras. They also present other examples of splitting subalgebras for Type Q classical “simple” Lie superalgebras that are not detecting subalgebras. See [SS22, Theorem 1.1]. An interesting question would be to determine for Type I classical Lie superalgebras with an ample detecting subalgebra whether the detecting subalgebra is a splitting subalgebra.

We will review the basic notions of triangulated categories, as these ideas lie at the foundation of the rest of the paper. Let $\mathbf{T}$ be a triangulated category. Recall that this means $\mathbf{T}$ is an additive category equipped with an auto-equivalence $\Sigma:\mathbf{T}\longrightarrow\mathbf{T}$ called the shift, and a set of distinguished triangles:

all subject to a list of axioms the reader can find in [Nee01, Ch. 1].

A non-empty, full, additive subcategory $\mathbf{S}$ of $\mathbf{T}$ is called a triangulated subcategory if (i) $M\in\mathbf{S}$ implies that $\Sigma^{n}M\in\mathbf{S}$ for all $n\in\mathbb{Z}$ and (ii) if $M\longrightarrow N\longrightarrow Q\longrightarrow\Sigma M$ is a distinguished triangle in $\mathbf{T},$ and two of $\{M,N,Q\}$ are objects in $\mathbf{S},$ then the third object is also in $\mathbf{S}.$ A triangulated subcategory $\mathbf{S}$ of $\mathbf{T}$ is called thick if $\mathbf{S}$ is closed under taking direct summands.

Assume that the triangulated category $\mathbf{T}$ admits set-indexed coproducts. A triangulated subcategory $\mathbf{S}$ of $\mathbf{T}$ is called a localizing subcategory if $\mathbf{S}$ is closed under taking set-indexed coproducts. It follows from the Eilenberg swindle that localizing subcategories are necessarily thick.

An object $C\in\mathbf{T}$ is called compact if $\operatorname{Hom}_{\mathbf{T}}(C,-)$ commutes with set-indexed coproducts. The full subcategory of compact objects in $\mathbf{T}$ is denoted by $\mathbf{T}^{c},$ and the triangulated category $\mathbf{T}$ is said to be compactly generated if the isomorphism classes of compact objects form a set, and if for each non-zero object $M\in\mathbf{T}$ there is an object $C\in\mathbf{T}^{c}$ such that $\operatorname{Hom}_{\mathbf{T}}(C,M)\neq 0.$

Let $\mathbf{T}$ be a triangulated category with suspension $\Sigma,$ and let $R$ be a graded-commutative ring. Recall that $R$ being graded-commutative means that $R$ admits a $\mathbb{Z}$ grading and that for homogeneous elements $x,y\in R,$ $xy=(-1)^{|x||y|}yx.$ In this section we recall what it means for $R$ to act on $\mathbf{T}.$

Let $M$ and $N$ be objects in $\mathbf{T},$ and set

Then $\operatorname{Hom}^{\bullet}_{\mathbf{T}}(M,N)$ is a graded abelian group, and $\operatorname{End}^{\bullet}_{\mathbf{T}}(M):=\operatorname{Hom}^{\bullet}_{\mathbf{T}}(M,M)$ is a graded ring where the multiplication is “shift and compose”. Notice that $\operatorname{Hom}^{\bullet}_{\mathbf{T}}(M,N)$ is a right $\operatorname{End}^{\bullet}_{\mathbf{T}}(M)$ and a left $\operatorname{End}^{\bullet}_{\mathbf{T}}(N)$-bimodule.

The graded center of $\mathbf{T}$ is the graded-commutative ring denoted by $Z^{\bullet}(\mathbf{T})$ whose degree $n$ component is given by

With this setup, an action of $R$ on $\mathbf{T}$ is a homomorphism $\phi:R\longrightarrow Z^{\bullet}(\mathbf{T})$ of graded rings, and if $\mathbf{T}$ admits such an action, $\mathbf{T}$ is called an $R$-linear triangulated category. It follows that if $\mathbf{T}$ is $R$-linear, then for each object $M$ in $\mathbf{T}$ there is an induced homomorphism of graded rings $\phi_{M}:R\longrightarrow\operatorname{End}^{\bullet}_{\mathbf{T}}(M)$ such that the induced $R$-module structures on $\operatorname{Hom}^{\bullet}_{\mathbf{T}}(M,N)$ by $\phi_{M}$ and $\phi_{N}$ agree up to sign.

Our paper will concern triangulated categories that have an additional monoidal structure. Namely, we work with tensor triangulated categories as defined in [Bal05]. A tensor triangulated category, or TTC, is a triple $(\mathbf{K},\otimes,\mathbf{1})$ where $\mathbf{K}$ is a triangulated category, $\otimes:\mathbf{K}\times\mathbf{K}\longrightarrow\mathbf{K}$ is a symmetric monoidal (tensor) product which is exact in each variable, and a monoidal unit $\mathbf{1}.$

Let $\mathbf{K}$ be a tensor triangulated category. With the additional structure of the tensor product on $\mathbf{K}$ one can define analogues to the usual notions in commutative algebra like prime ideal and spectrum. A tensor ideal in $\mathbf{K}$ is a triangulated subcategory $\mathbf{I}$ of $\mathbf{K}$ such that $M\otimes N\in\mathbf{I}$ for all $M\in\mathbf{I}$ and $N\in\mathbf{K}.$ A proper, thick tensor ideal $\mathbf{P}$ of $\mathbf{K}$ is said to be a prime ideal if $M\otimes N\in\mathbf{P}$ implies that either $M\in\mathbf{P}$ or $N\in\mathbf{P}.$ The Balmer spectrum [Bal05, Definition 2.1] is defined as

The Zariski topology on $\operatorname{Spc}(\mathbf{K})$ has as closed sets

where $\mathcal{C}$ is any family of objects in $\mathbf{K}.$

The tensor triangulated category $\mathbf{K}$ is said to be a compactly generated TTC if $\mathbf{K}$ is closed under set indexed coproducts, the tensor product preserves set indexed coproducts, $\mathbf{K}$ is compactly generated as a triangulated category, the tensor product of compact objects is compact, $\mathbf{1}$ is a compact object, and every compact object is rigid (i.e., strongly dualizable).

Given a tensor triangulated category $\mathbf{K},$ it is a difficult problem to classify its localizing subcategories and its thick tensor ideals. However, there are some circumstances in which such classifications are known, particularly in the case when $\mathbf{K}$ is some category of representations.

Stratification was introduced for triangulated categories in [BIK11a], and for tensor triangulated categories in [BIK11b]. The property of stratification allows for the classification of the localizing subcategories in terms of the $R$-support. Importantly, stratification is also easily tracked under change-of-categories, which plays a key role in these arguments. Since our work deals with tensor triangulated categories, we review that setting in this section. For a more detailed exposition the reader is referred to the original sources.

Let $\mathbf{K}$ be a tensor triangulated category. Furthermore, assume that the unit object $\mathbf{1}$ is a compact generator for $\mathbf{K}$. Then $\operatorname{End}_{\mathbf{K}}^{\bullet}(\mathbf{1})$ is a graded commutative ring which induces an action on $\mathbf{K}$ by the homomorphisms

for each object $M\in\mathbf{K}.$ Let $R$ be a graded-commutative ring. A canonical action of $R$ on $\mathbf{K}$ is an action of $R$ on $\mathbf{K}$ which is induced by a homomorphism of graded-commutative rings $R\longrightarrow\operatorname{End}_{\mathbf{K}}^{\bullet}(\mathbf{1})$.

Let $\operatorname{Proj}(R)$ denote the set of homogeneous prime ideals in $R.$ One can construct a notion of support for objects of $\mathbf{K}$ by considering localization and colocalization functors. This support is described in detail in [BIK11a, Section 3.1]. Recall that the specialization closure of a subset $U\subseteq\operatorname{Proj}(R)$ is the set of all $\mathfrak{p}\in\operatorname{Proj}(R)$ for which there exists $\mathfrak{q}\in U$ with $\mathfrak{q}\subseteq\mathfrak{p}.$ A subset $V\subseteq\operatorname{Proj}(R)$ is specialization closed if it equals its specialization closure. Given a specialization closed subset $V\subseteq\operatorname{Proj}(R)$, the full subcategory of $V$-torsion objects $\mathbf{T}_{V}$ is defined as

It can be checked that $\mathbf{T}_{V}$ is a localizing subcategory of $\mathbf{T},$ so there exists a localization functor $L_{V}:\mathbf{T}\longrightarrow\mathbf{T},$ and an induced colocalization functor $\mathit{\Gamma_{V}},$ called the local cohomology functors of $V,$ which for each object $M\in\mathbf{T}$ give a functorial triangle

For each $\mathfrak{p}\in\operatorname{Proj}(R)$, let $V$ and $W$ be specialization closed subsets of $\operatorname{Proj}(R)$ such that $\{\mathfrak{p}\}=V\setminus W$. There is a local-cohomology functor $\mathit{\Gamma_{\mathfrak{p}}}:=\mathit{\Gamma_{V}}L_{W}$ independent of the choice of $V$ and $W$. These functors give rise to a notion of support for objects $x\in\mathbf{T},$ and a notion of support for $\mathbf{T}$ itself. In particular, set

and

For each $\mathfrak{p}\in\operatorname{Proj}(R)$, the full subcategory $\mathit{\Gamma_{\mathfrak{p}}}(\mathbf{T})$ is a tensor ideal and localizing. The main definition of this section is the concept of stratification.

For each localizing subcategory $\mathbf{C}$ of $\mathbf{T},$ set

and for each subset $V$ of $\operatorname{Proj}(R)$, set

The following theorems [BIK11a, Theorem 3.8] classify localizing subcategories and thick tensor ideals via the aforementioned maps.

In the case when $\mathbf{K}$ is stratified by a noetherian ring $R$, the maps $\sigma$ and $\tau$ also give a classification of thick tensor ideals in $\mathbf{K}^{c}:$

The following result, whose proof of which can be found in [Nee01], demonstrates a close relationship between stratification of the derived category of a commutative ring with its canonical action, and is needed for our classifications.

Throughout this section $\mathfrak{z}=\mathfrak{z}_{\overline{0}}\oplus\mathfrak{z}_{\overline{1}}$ denotes a Lie superalgebra where $\mathfrak{z}_{\overline{0}}=\mathfrak{t}$ is a torus, and $[\mathfrak{z}_{\overline{0}},\mathfrak{z}_{\overline{1}}]=0$. This is a classical Lie superalgebra. As an example, $\mathfrak{z}$ could be any of the detecting subalgebras introduced in [BKN10a, Section 4.4]. Let $\mathbf{K}=\operatorname{Stab}(\mathcal{C}_{(\mathfrak{z},\mathfrak{z}_{\overline{0}})})$ be the stable module category whose objects are all $\mathfrak{z}$-modules which are finitely semi-simple as $\mathfrak{z}_{\overline{0}}$-modules. Then $\mathbf{K}$ is a compactly generated tensor-triangulated category with full subcategory of compact objects $\mathbf{K}^{c}=\operatorname{stab}(\mathcal{F}_{(\mathfrak{z},\mathfrak{z}_{\overline{0}})})$. Note that the objects of $\mathbf{K}^{c}$ consist precisely of the finite-dimensional objects in $\mathbf{K}$. Let $R$ denote the cohomology ring $\operatorname{H}^{*}(\mathfrak{z},\mathfrak{z}_{\overline{0}};\mathbb{C})$. According to [BKN17, Section 4.3],

and thus $R$ is a polynomial algebra. The following theorem was established in [BKN17, Theorem 4.5.4].

In [BKN17] the authors point out that $\mathbf{K}$ is an $R$-linear triangulated category, and that the local-global principle holds. It was conjectured that $R$ stratifies $\mathbf{K},$ a result which would recover the theorem. The goal of this section is to pursue the stratification avenue, and to prove the following theorem.

Theorems 4.4.2 and 4.4.3 yield the following result on localizing subcategories of thick tensor ideals for the stable module categories associated with ${\mathfrak{z}}$.

The Grothendieck abelian category of right $\mathbf{K}$-modules denoted $\mathbf{A}=\operatorname{Mod}$-$\mathbf{K}$ has as objects the additive functors $M:\mathbf{K}^{\text{op}}\longrightarrow\text{Ab}$ from the opposite category of $\mathbf{K}$ to the category of abelian groups, and has as morphisms the natural transformations between functors. Let h denote the Yoneda embedding: