TENSOR TRIANGULATED GEOMETRY FOR CLASSICAL LIE SUPERALGEBRAS

BRIAN D. BOE, JONATHAN R. KUJAWA, AND DANIEL K. NAKANO

Abstract. Tensor triangulated geometry as introduced by Balmer [Bal05] is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra 𝔤 = 𝔤¯0 ⊕ 𝔤¯1, we construct a Zariski space from a detecting subalgebra of 𝔤 and demonstrate that this topological space governs the tensor triangulated geometry for the category of finite dimensional 𝔤-modules which are semisimple over 𝔤¯0.

1. Introduction

1.1. A predominant theme in representation theory is the utilization of ambient geometric structures to study a given module category. For a given a symmetric monoidal tensor triangulated category, K, Balmer [Bal05] first introduced the idea of tensor triangulated geometry and used it to show that the underlying geometry can be revealed through the use of the tensor structure. He defined the notion of a prime ideal and constructed the (Balmer) spectrum, Spc(K), thus allowing one to study these categories from the viewpoint of commutative algebra and algebraic geometry. In particular he showed that a scheme can be reconstructed from an associated tensor triangulated category. Another important example occurs when K = Stab(mod(G)) is the stable module category of finitely generated modules for a finite group scheme G over a field k. In this setting there exists a homeomorphism between Spc(K) and Proj(Spec(R)) where R = H^{2•}(G, k) is the cohomology ring for G. There are many support variety theories (support data) for G, with Spc(K) being the universal (or “final”) support variety theory. Determining Spc(K) is closely connected to the classification of thick tensor ideals and recovers geometry hidden within K. Indeed, computing the spectrum of interesting tensor triangulated categories remains one of the most basic questions in the subject. See [Bal10] for further discussion of the spectrum.

Methods for classifying thick tensor ideals originated in the work of Hopkins [Hop87] in the context of the derived category of bounded complexes of finitely generated projective modules over a commutative Noetherian ring. Benson, Carlson, and Rickard [BCR97] later studied this question for the stable module category of a group algebra. In these results it became apparent that infinitely generated modules would play a key role in the classification. Specifically, they provide a systematic treatment using ideas from homotopy theory and Rickard’s idempotent modules to demonstrate that the thick tensor ideals for the stable module category of a finite group are in correspondence with the specialization closed sets in Proj(Spec(R)). The aforementioned description of Spc(K) can then be deduced from this fact. In [FP07] Friedlander and Pevtsova introduced a new approach using so-called π-points and extended the above description of Spc(K) to arbitrary finite group schemes. A fundamental new idea recently introduced by Benson, Iyengar and Krause [BIK11, BIK12] is that of having a commutative ring R stratify the category K. This allows them to further develop the theory while recovering the aforementioned results.

Date: February 20, 2014.

2000 Mathematics Subject Classification. Primary 17B56, 17B10; Secondary 13A50.

Research of the second author was partially supported by NSF grant DMS-1160763 and NSA grant H98230-11-1-0127.

Research of the third author was partially supported by NSF grant DMS-1002135.

1.2. The authors initiated a study of classical Lie superalgebras via cohomology and support varieties in [BKN10, BKN09, BKN11, BKN12]. Given a classical Lie superalgebra 𝔤 = 𝔤¯0 ⊕ 𝔤¯1 over ℂ let G¯0 be the connected reductive algebraic group with Lie (G¯0) = 𝔤¯0. It is natural to consider the category F = F_{(𝔤,𝔤¯0)} of finite dimensional 𝔤-modules which admit a compatible action by G¯0 and are completely reducible as G¯0-modules. The category F enjoys many of the features found for finite dimensional group schemes except the blocks in F can have infinitely many irreducible representations. One can use the fact that F is self injective together with the coproduct and counit in the enveloping algebra U(𝔤) to prove that K = Stab(F) is a tensor triangulated category. A natural problem is to classify the thick tensor ideals and to compute the Balmer spectrum for K.

The cohomology ring R = H^{•}(𝔤, 𝔤¯0; ℂ) identifies with S^{•}(𝔤¯1)^{G¯0}, and is thus finitely generated. Using this fact one can construct a cohomological support variety V_{(𝔤,𝔤¯0)}(M) for M ∈ F. In [BKN10] using classical invariant theory we constructed two types of (classical) detecting Lie subsuperalgebras of 𝔤, denoted by 𝔢 and 𝔣, with the property that 𝔢 ≤ 𝔣 ≤ 𝔤. These subalgebras have the striking property that they detect the cohomology for F; that is, the restriction maps

H^{•}(𝔤, 𝔤¯0; ℂ) ⟶^{res} H^{•}(𝔣, 𝔣¯0; ℂ)^{N} ⟶^{res} H^{•}(𝔢, 𝔢¯0; ℂ)^{W}

yield isomorphisms as rings. Here N is a non-connected reductive group and W is a finite pseudoreflection group.

For Type I classical Lie superalgebras, Lehrer, Nakano and Zhang [LNZ11] proved a remarkable fact about the detecting subalgebra 𝔣. For M ∈ F, the restriction maps in cohomology, res : H^{n}(𝔤, 𝔤¯0; M) → H^{n}(𝔣, 𝔣¯0; M) are monomorphisms for all n ≥ 0. Two consequences of this theorem are (i) given M ∈ F the restriction map in cohomology induces the following isomorphisms of support varieties,

V_{(𝔢,𝔢¯0)}(M)//W → V_{(𝔣,𝔣¯0)}(M)//N → V_{(𝔤,𝔤¯0)}(M),

and (ii) from these isomorphisms one can prove that V_{(𝔤,𝔤¯0)}(−) is a support data (see Example 4.6.3).

1.3. It has been a long standing question to find a natural geometric object for a given Lie superalgebra which governs the underlying representation theory in a similar way as the nilpotent cone controls the representation theory for Lie algebras and quantum groups. The main goal of this paper is to reveal the ambient geometry for classical Lie superalgebras given by the Balmer spectrum. In particular, a complete picture will be given for 𝔤 = 𝔤𝔩(m|n) through the use of the aforementioned invariant theory and detecting subalgebras.

In earlier work by Goetz, Quella and Schmerus [GQS07] in the case of 𝔤𝔩(1|1) and 𝔰𝔩(2|1) they study the classification of finite dimensional indecomposable representations and the direct computation of their tensor products. From their calculations one can readily determine the thick tensor ideals of F in this case. For the general case of 𝔤𝔩(m|n) when min(m, n) ≥ 2 these computational methods are not feasible because the category F has wild representation type.

Furthermore, unlike in the work of [BIK12], the cohomology ring of F fails to stratify K and indeed the spectrum of the cohomology ring fails to provide the Balmer spectrum (see Example 4.6.3). In particular, we do not know of a commutative ring which stratifies K. This forces us to adopt a different setup in order to the classify the thick tensor ideals and compute the Balmer spectrum. We anticipate that our approach will be useful in other contexts where a stratifying ring is not readily available.

The classification of thick tensor ideals for classical Lie superalgebras entails establishing powerful theoretical and computational techniques. In Section 2 we follow the ideas of [BIK12, Bal05] by introducing and relating three main inputs: (i) a compactly generated tensor triangulated category (TTC) K, (ii) a Zariski topological space X and (iii) a support data V which takes compact objects (i.e., ones in K^{c}) to closed sets in X. In Section 3 with these inputs we establish the general machinery for the classification of thick tensor ideals. It is shown that under suitable conditions one can use V and X to classify thick tensor ideals in K^{c} and compute the Balmer spectrum Spc(K^{c}). As in [BCR97] and [BIK12] the driving forces behind the scenes are the localization functors constructed for each thick tensor ideal of the subcategory of compact objects. Our classification theorem was proven in somewhat greater generality by Dell’Ambrogio [Del10] (and was also announced by Pevtsova and Smith [PS08]). For the convenience of the reader we include a condensed discussion of these classification results in order to introduce our conventions and notation, and to keep the paper self contained.

In Section 4 we examine the representation theory of classical Lie superalgebras. We first show that K^{c} = Stab(F) is the set of compact objects in a certain compactly generated TTC, K. Examples are given in a number of cases where one can construct a final support data V and calculate the thick tensor ideals and Spc(K^{c}). These examples include the situation when 𝔤 is a detecting subalgebra 𝔢 or 𝔣, and the case (𝔮^{+}, 𝔤¯0) where 𝔮^{+} is the standard parabolic subalgebra of 𝔤𝔩(m|n). Furthermore, in Example 4.6.3 we show for 𝔤 = 𝔤𝔩(m|n) that the cohomological support V_{(𝔤,𝔤¯0)}(−) is a support data but does not contain enough information to classify thick tensor ideals. In particular this shows that the cohomology ring fails to stratify K. The main reason for this is that the support data does not detect projectivity for objects of F_{(𝔤,𝔤¯0)}. We also show the associated varieties introduced by Duflo and Serganova [DS05] provides a support data but fail to classify thick tensor ideals (see Example 4.6.4).

In Section 5, we focus on the case 𝔤 = 𝔤𝔩(m|n). By using the detecting subalgebra 𝔣 we construct a new support data which involves the cohomological spectrum of H^{•}(𝔣, 𝔣¯0; ℂ) and, thanks to a result of [LNZ11], does detects projectivity in F_{(𝔤,𝔤¯0)}. It is then shown that to classify thick tensor ideals with this support data, it suffices to realize the N-stable closed conical subvarieties of 𝔣¯1 as supports of modules in F_{(𝔤,𝔤¯0)}. Once this realization result is established one obtains the classification, along with a determination of Spc(K^{c}), via specialization closed N-stable conical subsets of 𝔣¯1 (or equivalently specialization closed subvarieties of Proj(N-Spec(S^{•}(𝔣∗ ; ¯1)))). The analogues of the above results are also true for the type C Lie superalgebras 𝔬𝔰𝔭(2|2n).

We note that for 𝔤𝔩(m|n) there is a consequence of determining the thick tensor ideals which one does not have in the previously studied settings. Namely, the Groethendieck group of F_{(𝔤,𝔤¯0)} is a module for 𝔤𝔩(∞) where the action is by translation functors [Bru03, Theorem 4.29]. The thick tensor ideals then yield canonical submodules in the Grothendieck group.

Finally, in Section 6 and Section 7 we study the structure of modules obtained by geometric induction from a parabolic subalgebra and use this information to prove the necessary realization result used in Section 5. Normally the question of realization is not a major obstacle because one can use the standard construction via Carlson modules to realize closed conical sets in the spectrum of the cohomology ring. However, in our case we are looking at closed conical sets in 𝔣¯1 but need to realize these using 𝔤-modules. To address this problem we apply the machinery of geometric induction introduced by Penkov and Serganova to construct the required modules. Our analysis involves proving new results involving geometric induction and support varieties, examining carefully the group action of N on 𝔣¯1, and applying intricate computational properties of the induction functor in order to solve this problem.

2. Preliminaries

2.1. Triangulated Categories and Compactness. Let T be a triangulated category; then T is additive and has an equivalence Σ : T → T called the shift. Moreover, T is equipped with a set of distinguished triangles:

M → N → Q → ΣM

which satisfy the axioms as described in, for example, [BIK12, Section 1.3] or [Nee01].

If T is a triangulated category, an additive subcategory S of T is a triangulated subcategory if (i) S is non-empty and full, (ii) for M ∈ S, Σ^{n}M ∈ S for all n ∈ ℤ, and (iii) if M → N → Q → ΣM is distinguished triangle in T and if two objects in {M, N, Q} are in S then the third is in S. A triangulated subcategory S is called thick if M = M_{1} ⊕ M_{2} in S implies M_{j} ∈ S for j = 1, 2 (i.e., S is closed under taking direct summands). For C a collection of objects of T, define Thick(C) as the smallest thick subcategory containing C.

Assume that the triangulated category T admits set indexed coproducts. A localizing subcategory S of T is a triangulated subcategory which is closed under taking set indexed coproducts. Localizing subcategories are necessarily thick. For C a collection of objects of T, let Loc_{T}(C) = Loc(C) be the smallest localizing subcategory containing C.

An object C in T is compact if Hom_{T}(C, −) commutes with set indexed coproducts. Let T^{c} denoted the full subcategory of compact objects in T. The triangulated category T is compactly generated if the isomorphism classes of compact objects form a set and if for each nonzero M ∈ T there is a compact object C such that Hom_{K}(C, M) ≠ 0. It follows from [BIK12, Proposition 1.46] that when T is compactly generated one can find a set of compact objects, C, such that Loc_{T}(C) = T.

2.2. Tensor Triangulated Categories. For the purposes of this paper we will work with tensor triangulated categories as defined in [Bal05, Definition 1.1]. A tensor triangulated category (TTC) is a triple (K, ⊗, 1) such that (i) K is a triangulated category, and (ii) K has a symmetric monodial tensor product ⊗ : K × K → K which is exact in each variable with unit object 1.

If K is a TTC one can define notions prevalent in commutative algebra like prime ideal and spectrum. A (tensor) ideal in K is a triangulated subcategory I of K such that M ⊗ N ∈ I for all M ∈ I and N ∈ K. Following [Bal05, Definition 2.1], a prime ideal P of K is a proper thick tensor ideal such that if M ⊗ N ∈ P then either M ∈ P or N ∈ P. The Balmer spectrum [Bal05, Definition 2.1] is defined as

Spc(K) = {P ⊂ K | P is a prime ideal}.

The topology on Spc(K) is given by closed sets of the form

Z(C) = {P ∈ Spc(K) | C ∩ P = ∅}

where C is a family of objects in K.

When we say K is a compactly generated TTC we mean that K is closed under arbitrary set indexed coproducts, K is compactly generated, that 1 is a compact object, and that every compact object is rigid (i.e. strongly dualizable) as in, for example, [HPS97]. In particular we have an exact contravariant duality functor (−)^{∗} : K^{c} → K^{c} such that

Hom_{K}(N ⊗ M, Q) = Hom_{K}(N, M^{∗} ⊗ Q)

for M ∈ K^{c} and N, Q ∈ K.

2.3. Zariski Spaces. Assume throughout this paper that X is a Noetherian topological space. In this case any closed set in X is the union of finitely many irreducible closed sets. We say that X is a Zariski space if in addition any irreducible closed set Y of X has a unique generic point (i.e., y ∈ Y such that Y = \overline{\{y\}}). The prototypical example of such a topological space is X = Spec(R) where R is a commutative Noetherian ring. If R is graded one can also consider X = Proj(Spec(R)).

Let X be the collection of closed subsets of X and X_{irr} be the set of irreducible closed sets. Let Y be a collection of irreducible closed subsets of X. Following [BIK12, Definition 2.33], the collection Y is specialization closed if for any B ∈ Y and A ⊆ B where A ∈ X_{irr} then A ∈ Y. Alternatively, a set W in X is specialization closed if W = ∪_{j∈J}W_{j} with W_{j} ∈ X. One can translate between these definitions and obtain a bijective correspondence in the following way. If Y is a collection which is specialization closed, one can take W = ∪_{Y ∈Y}Y to obtain a specialization closed set in X. On the other hand, if W = ∪_{j∈J}W_{j} is a specialization closed subset of X, let Y be the collection {A ∈ X_{irr} | A ⊆ W_{j} for some j ∈ J}. The collection of all specialization closed subsets of X will be denoted by X_{sp}.

We will be interested in cases where we have an algebraic group G acting rationally on a graded commutative ring R by automorphisms which preserve the grading. This action induces an action of G on X = Proj(Spec(R)). Following [Lor09], we can consider X_{G} = Proj(G-Spec(R)) which is the set of homogeneous G-prime ideals of R. There exists a canonical map ρ : X ↠ X_{G} with ρ(P) = ∩_{g∈G} gP =: ∩_{g} gP. The topology on X_{G} is given by declaring W ⊆ X_{G} closed if and only if ρ^{−1}(W) is closed in X. An important property to note for us is that ∩_{g} gP_{1} = ∩_{g} gP_{2} for P_{1}, P_{2} ∈ X if and only if \overline{G · P_{1}} = \overline{G · P_{2}} in X [Lor09, (12)].

One can verify that if W is a closed set in X_{G} then ρ^{−1}(W) is a G-invariant closed set in X. Furthermore, the surjectivity of ρ implies that ρ(ρ^{−1}(W)) = W. Moreover, if V is a G-invariant closed set in X then V = ρ^{−1}(ρ(V)) and ρ(V) is closed. We can conclude that the map ρ induces a one-to-one correspondence between closed sets in X_{G} and G-invariant closed sets in X. We say that a subset of X_{G} is specialization closed if it is the union of closed sets; equivalently, its preimage under ρ is the union of G-invariant closed sets. Moreover, a closed subset W of X_{G} is irreducible if and only if ρ^{−1}(W) is not the union of proper G-invariant closed subsets.

We claim that X_{G} is a Zariski space. The fact that X_{G} is Noetherian immediately follows from the fact that X is Noetherian along with the discussion in the previous paragraph. Now let W be an irreducible closed set in X_{G}. If V = ρ^{−1}(W) then V = \overline{G · Y} for some irreducible closed set Y because X is Noetherian. Since X is a Zariski space there exists P ∈ X with Y = \overline{\{P \}}, and so V = \overline{G · P}. Using the properties of ρ above it follows that

W = ρ(\overline{G · P}) = \overline{ρ(\{G · P \})} = \overline{\{∩_{g} gP \}} = \overline{\{ρ(P)\}}.

Therefore W has a generic point. In order to prove uniqueness suppose that W = \overline{\{∩_{g} gP_{1}\}} = \overline{\{∩_{g} gP_{2}\}}. It follows that ρ^{−1}(W) = \overline{G · P_{1}} = \overline{G · P_{2}}. By our earlier remark this implies ∩_{g} gP_{1} = ∩_{g} gP_{2} and so the generic point is unique.

2.4. Support data. We recall the definition of support data as given in [Bal05], and will view support data as a method to relate objects in a TTC to subsets in a given Zariski space. Let K be a TTC, X be a Zariski space and X̂ be a set of subsets of X (e.g. X̂ could be the set of closed subsets or the set of specialization closed subsets). A support data is an assignment V : K → X̂ which satisfies the following six properties (for M, M_{i}, N, Q ∈ K): (2.4.1) V (0) = ∅, V (1) = X;

(2.4.2) V (⊕_{i∈I}M_{i}) = ⋃_{i∈I} V (M_{i}) whenever ⊕_{i∈I}M_{i} is an object of K;

(2.4.3) V (ΣM) = V (M);

(2.4.4) for any distinguished triangle M → N → Q → ΣM we have

V (N) ⊆ V (M) ∪ V (Q);

(2.4.5) V (M ⊗ N) = V (M) ∩ V (N);

(2.4.6) V (M) = V (M^{∗}) for M ∈ K^{c}. We will be interested in support data which satisfy an additional two properties: (2.4.7) V (M) = ∅ if and only if M = 0;

(2.4.8) for any W ∈ X̂ there exists an M ∈ K such that V (M) = W (Realization Property).

We note the following lemma which says roughly that “passing to a localizing subcategory does not increase supports.”

Lemma 2.4.1. Let K a TTC which is closed under set indexed coproducts and admits a support data V : K → X̂. Let C be collection of objects in K and suppose W is a subset of X such that V(M) ⊆ W for all M ∈ C. Then V(M) ⊆ W for all M ∈ Loc(C).

Proof. In forming Loc(C), we iteratively adjoin new objects to the collection by applying the following constructions:

• apply Σ and Σ^{−1} to objects in our collection

• if two objects in a distinguished triangle are in our collection, add the third object to our collection

• take direct summands of objects in our collection

• take set indexed coproducts of objects in our collection. The properties of a support data imply that the support of any such new object is still contained in W. □

It is useful for support computations (see Section 7) to also note that when working with closed sets in the spectrum we can instead work with the maximal ideal spectrum of R. Namely, let R be a finitely generated, commutative, graded k-algebra with k algebraically closed and X = Proj(Spec(R)), and X_{max} = Proj(MaxSpec(R)). For any ideal I in R, set v(I) = {P ∈ X | P ⊇ I} be the closed set in X defined by I and let Z(I) be the zero locus of I in X_{max}. Let W be a closed set in X and consider W_{max} = W ∩X_{max} which is a closed set in X_{max}. Let V : K → X be a support data as in Section 2.4. We claim that if there exists M in K such that V_{max}(M) := V (M) ∩ X_{max} = W_{max} then V (M) = W. Let I and J be ideals of R such that v(I) = W and v(J) = V (M). Since V_{max}(M) = W_{max} it follows that Z(I) = Z(J) and, hence, the radicals of I and J coincide. This then implies v(I) = v(J), as claimed. Therefore, when X is the spectrum of a finitely generated commutative k-algebra we may verify (2.4.8) for X by instead verifying (2.4.8) for X_{max}. In the case when a group G acts on R we can apply the same reasoning to see that to verify (2.4.8) for X_{G} it suffices to verify it for X_{G,max}. In particular, using ρ we see that it suffices to realize the G-invariant closed subsets of X_{max}.

2.5. Extending a Support Data. We also have the notion of extending a support data. Namely, let X be a Zariski space, let X be the closed sets of X, and let X_{sp} be the specialization closed sets of X. Assume K is a TTC and V : K^{c} → X is a support data.

Definition 2.5.1. We say that V : K → X_{sp} extends V : K^{c} → X if (i) V satisfies properties (2.4.1)–(2.4.5) for objects in K; (ii) V(M) = V (M) for all M ∈ K^{c}; and (iii) if V satisfies (2.4.7) then V satisfies (2.4.7).

3. Localization and the Classification Theorems

3.1. Localization functors. In this section we identify a key tool: the localization and colocalization functors as given in [BIK12, Proposition 2.16]. Bousfield localization has its origins in homotopy theory but, starting with the work of Neeman and Rickard, it has proven invaluable to the study of triangulated categories in representation theory and other settings. For details about these localization functors, we refer the reader to [BIK12, Section 2.2]. The following theorem is a restatement of [BIK08, Lemma 2.1] in our setting. It was originally proven for finite groups by Rickard [Ric97].

Theorem 3.1.1. Let K be a compactly generated triangulated category. Given a thick subcategory C of K^{c} and an object M in K, there exists a functorial triangle in K,

Γ_{C}(M) → M → L_{C}(M) →

which is unique up to isomorphism, such that Γ_{C}(M) is in Loc(C) and there are no non-zero maps in K from C or, equivalently, from Loc(C) to L_{C}(M).

We also record the following fact, which will be needed in the proof of Theorem 3.3.1.

Lemma 3.1.2. Let K be a compactly generated triangulated category, and C a thick subcategory of K^{c}. Given an object M ∈ K, we have

M ∈ Loc(C) if and only if Γ_{C}(M) ≅ M.

Proof. Suppose that M ∈ Loc(C). By Theorem 3.1.1, there are no non-zero maps from Loc(C) to L_{C}(M). In particular, the second map in the triangle

Γ_{C}(M) → M → L_{C}(M) →

must be zero, and hence the first map is an isomorphism.

Conversely, suppose that Γ_{C}(M) ≅ M. In the proof of [BIK12, Proposition 2.32], Loc(C) (resp. Γ_{C}) is playing the role of S (resp. Γ) in [BIK12, Proposition 2.16]. But that result asserts that Im(Γ) = S. So we can conclude that Im(Γ_{C}) = Loc(C). However,

Im(Γ_{C}) := {N ∈ K | N ≅ Γ_{C}(Q) for some Q ∈ K}.

Since M ≅ Γ_{C}(M), it follows that M ∈ Im(Γ_{C}) = Loc(C). □

3.2. Extending Support Data via Localization Functors. Let X be a Zariski space, X its closed subsets, and X_{sp} its specialization closed subsets. As a first application of localization and colocalization functors, given a support data on K^{c}, V : K^{c} → X, we can construct an assignment on K, V : K → X_{sp}, which is near to being an extension of V.

Definition 3.2.1. Let K be a compactly generated TTC and let V : K^{c} → X be a support data.

(a) Given W ∈ X_{sp}, let I_{W} denote the thick tensor ideal of K^{c} given by all M ∈ K^{c} such that V (M) ⊆ W.

(b) Using Theorem 3.1.1 set

Γ_{W} = Γ_{I_{W}} and L_{W} = L_{I_{W}} .

Given ∅ ≠ W ∈ X_{irr}, let Z be the union of all the elements of X properly contained in W. Define

∇_{W} = Γ_{W} L_{Z} = L_{Z}Γ_{W};

cf. [BIK12, Definition 2.34].

Definition 3.2.2. Let K be a compactly generated TTC, and V : K^{c} → X be a support data. Define V : K → X_{sp} as follows. For M ∈ K set

V(M) = ⋃_{{W ∈X_{irr}|∇_{W} (M)≠0}} W;

cf. [BIK12, Definition 2.35].

The following theorem demonstrates that V : K → X_{sp} has properties (2.4.1)–(2.4.4) for a support data whenever (2.4.8) holds for V.

Theorem 3.2.3. Let M, N, Q, M_{α} ∈ K for α in some index set.

(i) V(0) = ∅.

(ii) V(M ⊕ N) = V(M) ∪ V(N), and more generally V(⊕_{α}M_{α}) = ⋃_{α} V(M_{α}).

(iii) V(ΣM) = V(M).

(iv) for any distinguished triangle M → N → Q → ΣM,

V(N) ⊆ V(M) ∪ V(Q).

Finally, assuming that (2.4.8) holds for V, then (v) V(1) = X.

Proof. (i) If M = 0 then ∇_{W} (M) = 0 for all W ∈ X_{irr}, implying that V(M) = ∅. (ii) ∇_{W} (M⊕N) = ∇_{W} (M)⊕∇_{W} (N) since ∇_{W} is an exact functor and, in particular, is additive. Thus ∇_{W} (M ⊕ N) ≠ 0 if and only if ∇_{W} (M) ≠ 0 or ∇_{W} (N) ≠ 0. The same argument works for arbitrary direct sums, and the result follows. (iii) Being exact, ∇_{W} commutes with Σ, which implies the result (cf. [BIK08, Proposition 5.1] and [BIK12, Section 1.3.6]). (iv) By exactness of ∇_{W}, ∇_{W} (M) → ∇_{W} (N) → ∇_{W} (Q) → ∇_{W} (ΣM) is a distinguished triangle. If ∇_{W} (N) ≠ 0, then at least one of ∇_{W} (M) ≠ 0 or ∇_{W} (Q) ≠ 0.

Finally assume that (2.4.8) holds for V, so for W in X_{irr} there exists M ∈ K^{c} with V (M) = W. Since M ∈ Loc(I_{W}), we have Γ_{W} (M) ≅ M (cf. Definition 3.2.1 and Lemma 3.1.2). Letting Z be the union of all elements of X properly contained in W, as in Definition 3.2.1, we have M ∉ I_{Z}. It follows from Lemma 3.1.2 again that Γ_{I_{Z}}(M) ≅ M̸ and hence (from the exact triangle) that L_{Z}(M) ≠ 0. Thus ∇_{W} (M) = L_{Z}Γ_{W} (M) ≅ L_{Z}(M) ≠ 0. But ∇_{W} (M) ≅ ∇_{W} (1 ⊗ M) ≅ ∇_{W} (1) ⊗ M by [BIK08, Corollary 8.3]. Thus ∇_{W} (1) ≠ 0 and hence W ⊆ V(1). This proves that V(1) = X. □

Thanks to Theorem 3.2.3, in order to show that V extends V it suffices to prove that (3.2.1) V(M ⊗ N) = V(M) ∩ V(N) for M, N ∈ K;

(3.2.2) V(M) = V (M) for all M ∈ K^{c}; and,

(3.2.3) if V satisfies (2.4.7), then V satisfies (2.4.7).

3.3. Hopkins’ Theorem. Given an object M ∈ K^{c}, let Tensor(M) ⊆ K^{c} be the thick tensor ideal in K^{c} generated by M. We can also use localization to provide a general version of a theorem of Hopkins [Hop87] and Neeman [Nee92a] in the context of our setting. Note that the following theorem holds in general for an arbitrary extension V of a support data V (cf. Definition 2.5.1).

Theorem 3.3.1. Let K be a compactly generated TTC, X be a Zariski space, and X be the closed subsets of X. Moreover, let V : K^{c} → X be a support data satisfying (2.4.7), and assume that V : K → X_{sp} extends V. Fix an object M ∈ K^{c}, and set

W = V (M) = ⋃_{{Z∈X |Z⊆V (M)}} Z.

Then I_{W} = Tensor(M), where I_{W} is as in Definition 3.2.1.

Proof. We apply the strategy presented in [BIK12, Theorem 2.39] which in turn is based upon the analogous result in [BCR97]. For brevity we set I = I_{W} and I′ = Tensor(M).

(⊇) Since Tensor(M) is the smallest thick tensor ideal of K^{c} containing M, from the definition of I_{W} it follows that I ⊇ I′.

(⊆) Let N ∈ K. Apply the exact triangle of functors Γ_{I′} → Id → L_{I′} → to Γ_{I}(N):

Γ_{I′}Γ_{I}(N) → Γ_{I}(N) → L_{I′}Γ_{I}(N) →  (3.3.1)

Since I′ ⊆ I, the first term belongs to Loc(I′) ⊆ Loc(I). The second term also belongs to Loc(I), a triangulated subcategory, and hence so does the third term: L_{I′}Γ_{I}(N) ∈ Loc(I). Using Lemma 2.4.1, we deduce that V(L_{I′}Γ_{I}(N)) ⊆ W.

By Theorem 3.1.1 (applied to I′), there are no non-zero maps from I′ to L_{I′}Γ_{I}(N). Thus, for any object S ∈ K^{c}, the duality property implies that

0 = Hom_{K}(S ⊗ M, L_{I′}Γ_{I}(N)) ≅ Hom_{K}(S, M^{∗} ⊗ L_{I′}Γ_{I}(N)).  (3.3.2)

Since K is compactly generated it follows that M^{∗} ⊗ L_{I′}Γ_{I}(N) = 0 in K. Hence,

∅ = V(M^{∗} ⊗ L_{I′}Γ_{I}(N))
= V(M) ∩ V(L_{I′}Γ_{I}(N))
= W ∩ V(L_{I′}Γ_{I}(N))
= V(L_{I′}Γ_{I}(N)),

by Definition 2.5.1, (2.4.6), and the observation at the end of the previous paragraph. Thus, by Definition 2.5.1(iii), L_{I′}Γ_{I}(N) = 0 in K. By (3.3.1) it follows that Γ_{I}(N) ≅ Γ_{I′}Γ_{I}(N).

Now specialize to N ∈ I. Then, using Lemma 3.1.2 twice, we have Γ_{I}(N) ≅ N, so Γ_{I′}(N) ≅ N, whence N ∈ Loc (I′). Applying [Nee92b, Lemma 2.2] we see that in fact N ∈ I′. This shows I ⊆ I′ and completes the proof. □

3.4. Classifying Thick Tensor Ideals in a TTC. The following theorem provides a classification of the thick tensor ideals of compact objects in our setting. Dell’Ambrogio proved a somewhat more general version of this result [Del10, Theorem 1.5].

Theorem 3.4.1. Let K be a compactly generated TTC. Let X be a Zariski space and let V : K^{c} → X be a support data defined on K^{c} satisfying the additional conditions (2.4.7) and (2.4.8). Moreover, assume V : K → X_{sp} extends V.

Given the above setup there is a pair of mutually inverse maps

\{thick tensor ideals of K^{c}\} \underrightarrow{Γ} ; \overleftarrow{Θ} X_{sp},

given by

Γ(I) = ⋃_{M∈I} V (M),
Θ(W) = I_{W} ,

where I_{W} = {M ∈ K^{c} | V (M) ⊆ W } as in Definition 3.2.1.

Proof. Observe that by (2.4.2) and (2.4.5), I_{W} is a thick tensor ideal of K^{c}.

(i) We first show that Γ ∘ Θ is the identity. Observe that

Γ(Θ(W)) = Γ(I_{W}) = ⋃_{M∈I_{W}} V (M) ⊆ W,

where the final inclusion follows from definition of I_{W}.

For the reverse inclusion, write W = ⋃_{j∈J} W_{j} for some index set J and closed subsets W_{j} ∈ X. By (2.4.8), there exist objects N_{j} ∈ K^{c} such that V (N_{j}) = W_{j} for j ∈ J. Then N_{j} ∈ I_{W} so W ⊆ ⋃_{M∈I_{W}} V (M). Hence, Γ(Θ(W)) = W.

(ii) Now we show that Θ ∘ Γ is the identity. Given a thick tensor ideal I, set W = Γ(I) = ⋃_{M∈I} V (M). Then

Θ(Γ(I)) = Θ(W) = I_{W} ⊇ I.

For the reverse inclusion, let N ∈ I_{W}, so V (N) ⊆ W. Since X is Noetherian, V (N) = W_{1} ∪⋯∪W_{n}, where the W_{i} are the irreducible components of V (N). Each W_{i} has a generic point x_{i} with \overline{\{x_{i}\}} = W_{i}, and since W_{i} ⊆ W, x_{i} ∈ W. By definition of W, there exists M_{i} ∈ I such that x_{i} ∈ V (M_{i}). Since V (M_{i}) is closed, W_{i} ⊆ V (M_{i}). Set M := ⨁_{i=1}^{n} M_{i} ∈ I. Then

V (N) ⊆ ⋃_{i=1}^{n} V (M_{i}) = V (M) ⊆ W.

We claim that N ∈ Tensor(M). Since I is a thick tensor ideal containing M, clearly Tensor(M) ⊆ I, so this assertion will complete the proof of the inclusion I_{W} ⊆ I, and thus of the theorem.

To prove the claim, we use Theorem 3.3.1. Namely, we have the equality Tensor(M) = I_{Z}, where

Z = V (M) = ⋃_{{Y ∈X |Y ⊆V (M) }} Y.

But V (N) ⊆ Z since V (N) ⊆ V (M), so N ∈ I_{Z} = Tensor(M) as claimed. □

3.5. Computing the Balmer Spectrum. We can now prove that the Balmer spectrum of K^{c} is homeomorphic to X in our setup.

Theorem 3.5.1. Let K be a compactly generated TTC and let X be a Zariski space. Assume that V : K^{c} → X is a support data defined on K^{c} satisfying the additional conditions (2.4.7) and (2.4.8). Further assume that we have a support data V : K → X_{sp} which extends V. Then there is a homeomorphism

f : X → Spc(K^{c}).

Proof. Since V : K^{c} → X is a support data the universal property of the Balmer spectrum implies that there is a continuous map f : X → Spc(K^{c}) as defined in [Bal05, Theorem 3.2]. By assumption X is Zariski and by Theorem 3.4.1 we know that V induces a bijection between the thick tensor ideals of K^{c} and the specialization closed subsets of X (noting that since K^{c} is assumed to have a duality we have that the radical thick tensor ideals are exactly the thick tensor ideals by [Bal05, Remark 4.3 and Proposition 4.4]). Thus the pair (X , V) is a classifying support data as defined in [Bal05, Definition 5.1]. The results in [Bal05, Theorem 5.2] then implies that f is a homeomorphism. □

4. Classical Lie Superalgebras

4.1. We now apply the technology of the previous sections to the representations of classical Lie superalgebras. We start with the notation and conventions in [BKN10, BKN09, BKN11, LNZ11]. Let 𝔞 = 𝔞¯0⊕𝔞¯1 be a Lie superalgebra over the complex numbers, ℂ, with Lie bracket [ , ] : 𝔞 ⊗ 𝔞 → 𝔞. A finite dimensional Lie superalgebra 𝔞 is called classical if there is a connected reductive algebraic group A¯0 such that Lie(A¯0) = 𝔞¯0 and an action of A¯0 on 𝔞¯1 which differentiates to the adjoint action of 𝔞¯0 on 𝔞¯1.

Let U(𝔞) be the universal enveloping superalgebra of 𝔞. If M and N are 𝔞-supermodules (equivalently U(𝔞)-supermodules) one can use the coproduct and antipode of U(𝔞) to define an 𝔞-supermodule structure on the tensor product M ⊗N and, when M is finite dimensional, the dual M^{∗}. In this context we will use the term 𝔞-module to mean 𝔞-supermodule. Furthermore, in what follows we will only consider the underlying even category in which the Hom sets consist of morphisms of 𝔞-modules which preserve the ℤ_{2}-grading.

Let F_{(𝔞,𝔞¯0)} be the full subcategory of finite dimensional 𝔞-modules which have a compatible action by A¯0 and are completely reducible over A¯0. The category F := F_{(𝔞,𝔞¯0)} has enough injective (and projective) modules and is a Frobenius category (cf. [BKN11, Proposition 2.2.2]). Given M, N in F, let Ext_{F}^{d}(M, N) be the degree d extensions between N and M. These extension groups can be realized via relative Lie superalgebra cohomology for the pair (𝔞, 𝔞¯0):

Ext_{F}^{d}(M, N) ≅ Ext_{(𝔞,𝔞¯0)}^{d}(M, N) ≅ H^{d}(𝔞, 𝔞¯0; M^{∗} ⊗ N).

There exists a Koszul type resolution which can be used to calculate H^{d}(𝔞, 𝔞¯0; M^{∗} ⊗ N).

4.2. Let C_{(𝔞,𝔞¯0)} be the category of all 𝔞-modules which are finitely semisimple over A¯0 (i.e., the objects in C_{(𝔞,𝔞¯0)} have a compatible A¯0 action and as A¯0-modules they decompose into a direct sum of finite dimensional simple modules). The category F_{(𝔞,𝔞¯0)} is a full subcategory of C_{(𝔞,𝔞¯0)}. Since the projectives, injectives, and simple modules in C_{(𝔞,𝔞¯0)} are the same as in F_{(𝔞,𝔞¯0)}, it follows that C_{(𝔞,𝔞¯0)} is also a Frobenius category.

Let K = Stab(C_{(𝔞,𝔞¯0)}) be the stable module category of C_{(𝔞,𝔞¯0)}. Then K is a triangulated category (e.g. [BIK12, Section 1.3.5]), and using the coproduct and augmentation maps on U(𝔤) it follows that K is a tensor triangulated category. We also note that Stab(F_{(𝔞,𝔞¯0)}) is a tensor triangulated subcategory of K. We now verify that K is compactly generated and K^{c} = Stab(F_{(𝔞,𝔞¯0)}).

Proposition 4.2.1. Let K = Stab(C_{(𝔞,𝔞¯0)}).

(a) K is generated by Stab(F_{(𝔞,𝔞¯0)}).

(b) K^{c} = Stab(F_{(𝔞,𝔞¯0)}). In particular, K is compactly generated.

Proof. We first observe that since every object in F_{(𝔞,𝔞¯0)} is finite dimensional they give compact objects in K.

We now prove K is compactly generated. Let M ∈ K be non-zero. As in [Ric97, Lemma 3.1] we may assume M is stably isomorphic to an 𝔞-module, which we also call M, with no projective direct summands. By the PBW theorem for Lie superalgebras the module M is locally finite. Because of this the U(𝔞)-module generated by a fixed non-zero element of M is finite dimensional and, hence, is an object of F_{(𝔞,𝔞¯0)}. Call it N. Furthermore, N cannot have a projective direct summand for, if it did, it would provide a projective submodule of M and, hence (since projectives are injective), a projective summand of M. The inclusion map defines an isomorphism between N and its image in M in C_{(𝔞,𝔞¯0)}. But since M and N have no projective summands [Ric97, Lemma 3.1(b)] implies this map also defines a stable isomorphism between N and its image. In particular, it defines a non-zero homomorphism between N and M in K. Therefore K is generated by Stab(F_{(𝔞,𝔞¯0)}) and, in particular, K is compactly generated. To prove (b) we note that since K = Loc_{K} (Stab(F_{(𝔞,𝔞¯0)}) it follows from [Nee92b, Lemma 2.2] that K^{c} is precisely the thick subcategory generated by Stab(F_{(𝔞,𝔞¯0)}). That is, K^{c} is Stab(F_{(𝔞,𝔞¯0)}) itself. □

The antipode map on U(𝔞) gives a duality on F_{(𝔞,𝔞¯0)} and from this it follows that the compact objects of K are rigid.

4.3. Relative Cohomology and the Cohomological Support. The relative cohomology ring for (𝔞, 𝔞¯0) can be computed using an explicit complex (cf. [BKN10, Section 2.3]). Since [𝔞¯1, 𝔞¯1] ⊆ 𝔞¯0 the differentials in the complex are zero and from this it follows that

R := H^{•}(𝔞, 𝔞¯0; ℂ) = S^{•}(𝔞∗ ; ¯1)^{A¯0}.

Since A¯0 is assumed to be reductive it follows that R is a finitely generated commutative ℂ-algebra. Moreover, it was shown in [BKN10] that if M_{1}, M_{2} are in F_{(𝔞,𝔞¯0)} then Ext_{(𝔞,𝔞¯0)}^{•}(M_{1}, M_{2}) is a finitely generated R-module.

If X = Proj(Spec(R)), then X is a Zariski space. Recall that K^{c} = Stab(F_{(𝔞,𝔞¯0)}), and that X denotes the collection of closed subsets of X. We will now define an assignment V : K^{c} → X as follows. Let

V_{(𝔞,𝔞¯0)}(M) := Proj(Spec(R/J_{𝔞,M}))

where J_{𝔞,M} = Ann_{R}(Ext_{F}^{•}(M, M)) (i.e., the annihilator ideal of this module). An equivalent formulation of this assignment is

V_{(𝔞,𝔞¯0)}(M) = Proj({P ∈ Spec(R) | Ext_{(𝔞,𝔞¯0)}^{•}(M, M)_{P} ≠ 0})

(i.e., the primes ideals P for which the cohomology is non-zero when localized at P). More generally, for a pair of modules M, N ∈ F_{(𝔞,𝔞¯0)}, let

V_{(𝔞,𝔞¯0)}(M, N) = Proj({P ∈ Spec(R) | Ext_{(𝔞,𝔞¯0)}^{•}(M, N)_{P} ≠ 0}).

Note that V_{(𝔞,𝔞¯0)}(M) = V_{(𝔞,𝔞¯0)}(M, M).

As in the case with group algebras, one can verify that properties (2.4.1)–(2.4.4), (2.4.6) hold for V_{(𝔞,𝔞¯0)} (cf. [Eve91, Chapters 8, 10]). We call V_{(𝔞,𝔞¯0)} a pre-support data on K^{c} as we will need additional conditions on 𝔞 to have an actual support data which satisfies property (2.4.5).

4.4. Detecting Subalgebras. We say that a Lie superalgebra is of Type I if it admits a ℤ-grading 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} which is compatible with the Lie bracket and with 𝔤¯0 = 𝔤_{0} and 𝔤¯1 = 𝔤_{−1} ⊕𝔤_{1}. Otherwise, 𝔤 is of Type II. The prototypical example of a Type I classical Lie superalgebra is 𝔤𝔩(m|n). Simple Type I classical Lie superalgebras in the Kac classification [Kac77] include those of types A(m, n), C(n) and P (n).

Throughout this section we will assume that 𝔤 is a Type I classical Lie superalgebra admitting both a stable and polar action of G¯0 on 𝔤¯1. This is the same setup used in [LNZ11]. We quickly review the construction of the two families of detecting subalgebras for classical Lie superalgebras using the invariant theory of G¯0 on 𝔤¯1. For further details we refer the reader to [BKN10, Sections 3–4].

If the action of G¯0 on 𝔤¯1 is a stable action, then we can construct the detecting Lie subsuperalgebra 𝔣 as follows. The stable action insures that there is a generic point x_{0} in 𝔤¯1 (i.e., the G¯0-orbit of x_{0} is closed and has maximal dimension). Set H = Stab_{G¯0}(x_{0}) and N = Norm_{G¯0}(H). Now let 𝔣¯1 = 𝔤H ; ¯1 and set 𝔣¯0 = [𝔣¯1, 𝔣¯1]. We then put 𝔣 = 𝔣¯0 ⊕ 𝔣¯1. We note that the 𝔣 used here differs slightly from the one in [LNZ11]. However, we can follow the argument used in the proof of Theorem 4.1.1 in loc. cite to show that we may transfer their results to our 𝔣.

When the action of G¯0 on 𝔤¯1 is a polar action (as defined by Dadok and Kac [DK85]) we can construct a second detecting subalgebra 𝔢 as follows. We fix a Cartan subspace 𝔢¯1 ⊆ 𝔤¯1 and then obtain a classical Lie subalgebra 𝔢 = 𝔢¯0 ⊕ 𝔢¯1 with 𝔢¯0 := [𝔢¯1, 𝔢¯1]. From [BKN10, Section 8.9] it follows for 𝔤𝔩(m|n) and the simple classical Type I Lie superalgebras that 𝔢¯0 and 𝔣¯0 are tori and [𝔢¯0, 𝔢¯1] = [𝔣¯0, 𝔣¯1] = 0.

Since 𝔤 is a classical Lie algebra such that G¯0 admits a stable and polar action on 𝔤¯1 we can make an appropriate choice of generic element x_{0} to ensure that 𝔢 ≤ 𝔣 ≤ 𝔤. Furthermore, these inclusions induce restriction homomorphisms S(𝔤∗ ; ¯1) → S(𝔣∗ ; ¯1) → S(𝔢∗ ; ¯1) which provide isomorphisms

H^{•}(𝔤, 𝔤¯0; ℂ) ⟶^{res} H^{•}(𝔣, 𝔣¯0; ℂ)^{N} ⟶^{res} H^{•}(𝔢, 𝔢¯0; ℂ)^{W} .  (4.4.1)

where W ⊆ G¯0 is a pseudoreflection group. This implies that R = H^{•}(𝔤, 𝔤¯0; ℂ) is a polynomial algebra. See [BKN10] for further details.

4.5. Superalgebras of the form 𝔷 = 𝔱 ⊕ 𝔷¯1. In this section we will focus on Lie superalgebras of the form 𝔷 = 𝔷¯0 ⊕ 𝔷¯1 where 𝔷¯0 = 𝔱 is a torus and [𝔷¯0, 𝔷¯1] = 0. This is a classical Lie superalgebra. As an example, 𝔷 could be one of the detecting subalgebras introduced in the previous section. The main goal of this section will be to classify thick tensor ideals in Stab(K^{c}) and to compute Spc(K^{c}) where K^{c} = F_{(𝔷,𝔷¯0)}. We first verify that (3.2.1), (3.2.2) and (3.2.3) hold for 𝔷 by following the arguments in [BCR96] and indicating the places where modifications are necessary.

Let K denote an algebraically closed extension field of ℂ having sufficiently large transcendence degree; that is, larger than dim(𝔷¯1). For M ∈ C_{(𝔷,𝔷¯0)}, set

V_{𝔷¯1}^{r}(K ⊗ M) = { x ∈ Proj(K ⊗ 𝔷¯1) | K ⊗ M is not projective as a U(⟨x⟩)-module }. (4.5.1)

Here ⟨x⟩ is the Lie superalgebra generated by x and here as elsewhere ⊗ denotes the tensor product over ℂ.

We first verify the tensor product property for these generalized rank varieties. This is an analogue of [BCR96, Theorem 7.4].

Proposition 4.5.1. Let M and N be in C_{(𝔷,𝔷¯0)}. Then

V_{𝔷¯1}^{r}(K ⊗ (M ⊗ N)) = V_{𝔷¯1}^{r}(K ⊗ M) ∩ V_{𝔷¯1}^{r}(K ⊗ N).

Proof. Let 𝔷, 𝔷′ be Lie superalgebras which satisfy the assumptions at the beginning of this section, and M and N be 𝔷- and 𝔷′-modules, respectively. We see that

V_{𝔷¯1×𝔷¯1′}^{r}(K ⊗ (M ⊠ N)) = V_{𝔷¯1}^{r}(K ⊗ M) × V_{𝔷′ ; ¯1}^{r}(K ⊗ N)

as subvarieties of Proj(K ⊗ (𝔷¯1 × 𝔷′ ; ¯1)) = Proj(K ⊗ 𝔷¯1) × Proj(K ⊗ 𝔷′ ; ¯1).

The verification of this statement uses the same line of reasoning as given in [BCR96, Theorem 7.2]. The only modification necessary is to note that for any fixed x ∈ 𝔷¯1, the subalgebra ⟨x⟩ generated by x is either isomorphic to 𝔮(1) or a 1-dimensional abelian Lie superalgebra concentrated in degree ¯1. As a consequence, any (possibly infinite dimensional)

⟨x⟩-module M in C_{(⟨x⟩,⟨x⟩¯0)} decomposes as a direct sum M = ⨁ M_{λ} of weight spaces for ⟨x⟩¯0 and that this is a decomposition as ⟨x⟩-modules. Writing t for a vector which spans ⟨x⟩¯0, we may view M_{λ} as a module for the Clifford algebra A_{λ} = U(𝔮(1))/(t − λ(t)). Applying [Aus74, Corollary 4.8] we have that M_{λ} is a direct sum of finite dimensional indecomposable modules. If λ ≠ 0 then the only indecomposable modules are two dimensional projective simple modules. When λ = 0 the indecomposables consist of the one dimensional trivial module and its two dimensional projective cover (see [BKN09, Proposition 5.2]). One can then analyze the variety of the external product of such indecomposables as in [BCR96, Theorem 7.2].

In order to finish the proof for a given 𝔷 one can use the coproduct map Δ : U(𝔷) → U(𝔷) ⊗ U(𝔷) and then proceed as in the proof of [BCR96, Theorem 7.4]. □

We can now prove a version of Dade’s Lemma for modules in C_{(𝔷,𝔷¯0)}.

Proposition 4.5.2. Let M ∈ C_{(𝔷,𝔷¯0)}. Then M is projective if and only if V_{𝔷¯1}^{r}(K ⊗ M) = ∅.

Proof. If M is projective in C_{(𝔷,𝔷¯0)} then K ⊗ M is projective when restricted to ⟨x⟩ for any x ∈ Proj(K ⊗ 𝔷¯1) and so V_{𝔷¯1}^{r}(K ⊗ M) = ∅.

For the converse, note that by our assumptions on M and 𝔷 we have M = ⨁_{λ∈𝔱^{∗}} M_{λ} where M_{λ} is the λ-weight space for 𝔷¯0 = 𝔱. Furthermore, this decomposition is as 𝔷-modules. Since the generalized rank variety of a direct sum is the union of the varieties of the direct summands it suffices to prove the result for M_{λ}. Now M_{λ} is a module for the Clifford algebra U(𝔷)/(t − λ(t) | t ∈ 𝔱). By [BKN09, Section 5.2] the associated block B_{λ} containing the module M_{λ} of C_{(𝔷,𝔷¯0)} has only a single simple module up to parity shift. Call it L(λ).

Now assume V_{𝔷¯1}^{r}(K ⊗ M_{λ}) = ∅. Since

V_{𝔷¯1}^{r}(K ⊗ M_{λ} ⊗ L(−λ)) ⊆ V_{𝔷¯1}^{r}(K ⊗ M_{λ}) ∩ V_{𝔷¯1}^{r}(K ⊗ L(−λ))

it follows that V_{𝔷¯1}^{r}(K ⊗ M_{λ} ⊗ L(−λ)) = ∅. But, M_{λ} ⊗ L(−λ) has trivial 𝔱-action, so it is a module for U(𝔷/𝔷¯0) ≅ Λ^{•}(𝔷¯1), where 𝔷¯1 is regarded as an abelian Lie superalgebra. The proof of Dade’s Lemma (Theorem 5.2) in [BCR96] directly carries over to the algebra Λ^{•}(𝔷¯1) (which behaves like the group algebra of an elementary abelian 2-group). Hence, M_{λ} ⊗ L(−λ) is projective as a module for Λ^{•}(𝔷¯1).

It remains to show that M_{λ} is projective in C_{(𝔷,𝔷¯0)}. Recalling that L(λ) is the only simple module in the block B_{λ} up to parity shift and that L(λ) is finite dimensional, it suffices to show that

Ext_{(𝔷,𝔷¯0)}^{n}(L(λ), M_{λ}) = Ext_{(𝔷,𝔷¯0)}^{n}(ℂ, M_{λ} ⊗ L(−λ)) = 0

for all n > 0. Apply the Lyndon-Hochschild-Serre spectral sequence for the pair (𝔷¯0, 𝔷¯0) ◁ (𝔷, 𝔷¯0):

E_{2}^{i,j} = Ext_{(𝔷¯1,{0})}^{i}(ℂ, Ext_{(𝔷¯0,𝔷¯0)}^{j}(ℂ, M_{λ} ⊗ L(−λ))) ⇒ Ext_{(𝔷,𝔷¯0)}^{i+j}(ℂ, M_{λ} ⊗ L(−λ)).

The spectral sequence collapses yielding

Ext_{(𝔷,𝔷¯0)}^{n}(ℂ, M_{λ} ⊗ L(−λ)) ≅ Ext_{𝔷¯1}^{n}(ℂ, M_{λ} ⊗ L(−λ)) = 0

for all n > 0 by the last sentence of the preceding paragraph. □

We let V_{(𝔷,𝔷¯0)}(−) be the cohomological variety theory for 𝔷 as defined in Section 4.3. As explained in the proof of Theorem 4.5.4 (below), this is a support data and we can define the variety theory V_{(𝔷,𝔷¯0)}(−) on K following Definition 3.2.2. Assuming that the transcendence degree of K over ℂ is at least dim(𝔷¯1) we have a bijective correspondence between V_{𝔷¯1}^{r}(K) and V_{(𝔷,𝔷¯0)}(ℂ), which equals X by a direct calculation. Namely, a point in V_{𝔷¯1}^{r}(K) corresponds to a generic point for a uniquely determined irreducible subvariety of X (see [BCR96, Propositions 2.1, 3.1]). Let β^{∗} : V_{(𝔷,𝔷¯0)}(ℂ) → V_{𝔷¯1}^{r}(K) denote this correspondence. Under this correspondence one can use the same strategy as outlined in [BCR96, Sections 9–10] to show that for all M ∈ C_{(𝔷,𝔷¯0)}

β^{∗} : V_{(𝔷,𝔷¯0)}(M) → V_{𝔷¯1}^{r}(K ⊗ M)

is a bijection. For the analogous statement, see [BCR96, Theorem 10.5]. In order to make this translation, for W ∈ X_{sp}, E(W) corresponds to our Γ_{W} (1) and F (W) corresponds to L_{W} (1). Moreover, if W ∈ X_{irr}, κ(W) corresponds to ∇_{W} (1) (cf. [BIK08, p. 609, Table]). Hence, we obtain the following result.

Proposition 4.5.3. Let M ∈ C_{(𝔷,𝔷¯0)}. Then V_{(𝔷,𝔷¯0)}(M) ≅ V_{𝔷¯1}^{r}(K ⊗ M).

We can now classify the thick tensor ideals of Stab(F_{(𝔷,𝔷¯0)}).

Theorem 4.5.4. Let 𝔷 = 𝔷¯0 ⊕ 𝔷¯1 where 𝔷¯0 = 𝔱 is a torus and [𝔷¯0, 𝔷¯1] = 0. Set R = S^{•}(𝔷∗ ; ¯1). Then the thick tensor ideals of K^{c} = Stab(F_{(𝔷,𝔷¯0)}) are in bijective correspondence with specialization closed subsets of Proj(Spec(R)). Furthermore, Spc(K^{c}) homeomorphic to Proj(Spec(R)).

Proof. Since 𝔷¯0 acts trivially on 𝔷¯1 via the adjoint action, R = S^{•}(𝔷∗ ; ¯1) ≅ H^{•}(𝔷, 𝔷¯0; ℂ). Set K = Stab(C_{(𝔷,𝔷¯0)}) and K^{c} = Stab(F_{(𝔷,𝔷¯0)}). In this case the cohomological variety V_{(𝔷,𝔷_{0})}(−) of Section 4.3 has a concrete “rank variety” type description so (2.4.5) holds and V_{(𝔷,𝔷¯0)} is a support data. Moreover, by using (2.4.5) and Carlson’s L_{ζ} modules (see Section 7.2 for another use of the same technique) one can show that (2.4.8) is satisfied.

Since 𝔷 has only finitely many simple modules in each block property (2.4.7) holds by [BKN11, Theorem 2.9.1]. Given V_{(𝔷,𝔷¯0)} let V_{(𝔷,𝔷¯0)} be defined as in Definition 3.2.2. The earlier results in this section verify that (3.2.1), (3.2.2) and (3.2.3) hold for 𝔷, thus one can conclude that V_{(𝔷,𝔷¯0)} extends V_{(𝔷,𝔷¯0)}. The results now follow from Theorems 3.4.1 and 3.5.1. □

Since R is the cohomology ring for C_{(𝔷,𝔷¯0)} it follows that K is an R-linear triangulated category in the sense of [BIK12]. By [BIK12, Corollary 4.7] the local global principle holds and by Section 4.1, one has that K is Noetherian in the sense of [BIK12, Definition 4.21]. We expect that R stratifies K and that the previous theorem can be deduced from [BIK12, Theorem 4.23].

4.6. Support Data for Type I superalgebras. We will now investigate the construction of support data via cohomological support varieties for the classical Lie superalgebra 𝔤, its detecting subalgebras 𝔢 and 𝔣, and the relation between them.

Example 4.6.1. Let 𝔷 = 𝔷¯1 (i.e., 𝔷¯0 = 0) be an abelian Lie superalgebra consisting of odd degree elements. Here R = H^{•}(𝔷, 𝔷¯0; ℂ) ≅ S^{•}(𝔷∗ ; ¯1). From Theorem 4.5.4 the thick tensor ideals of K^{c} are in bijective correspondence with specialization closed subsets of Proj(Spec(R)) and Spc(K^{c}) identifies with Proj(Spec(R)). Note that this is analogous to the case of the group algebra of an elementary abelian p-group.

Example 4.6.2. Let 𝔷 denote the detecting subalgebra 𝔢 or 𝔣 of Section 4.4. In either case 𝔷¯0 acts trivially on R ≅ S^{•}(𝔷∗ ; ¯1) and arguing as in the preceding example, we can classify thick tensor ideals in K^{c} by specialization closed subsets of Proj(Spec(R)), and Spc(K^{c}) ≅ Proj(Spec(R)).

Example 4.6.3. Let 𝔤 be a classical Lie superalgebra of Type I admitting both a stable and polar action of G¯0 on 𝔤¯1. Let R = H^{•}(𝔤, 𝔤¯0; ℂ). The ring homomorphisms given in (4.4.1) induce morphisms of topological spaces,

V_{(𝔢,𝔢¯0)}(ℂ) ⟶^{res^{∗}} V_{(𝔣,𝔣¯0)}(ℂ) ⟶^{res^{∗}} V_{(𝔤,𝔤¯0)}(ℂ),

and isomorphisms (by passing to quotients),

V_{(𝔢,𝔢¯0)}(ℂ)//W ⟶^{res^{∗}} V_{(𝔣,𝔣¯0)}(ℂ)//N ⟶^{res^{∗}} V_{(𝔤,𝔤¯0)}(ℂ).

Let M be in K^{c} = Stab(F_{(𝔤,𝔤¯0)}). Then res^{∗} induces maps between the cohomologically defined pre-support data:

V_{(𝔢,𝔢¯0)}(M) → V_{(𝔣,𝔣¯0)}(M) → V_{(𝔤,𝔤¯0)}(M).

Since M is a G¯0-module it follows that V_{(𝔢,𝔢¯0)}(M) (resp. V_{(𝔣,𝔣¯0)}(M)) is invariant under the action of W (resp. N). Thus we obtain induced embeddings between the geometric quotients (cf. [BKN10, (6.1.3)]):

V_{(𝔢,𝔢¯0)}(M)//W ↪ V_{(𝔣,𝔣¯0)}(M)//N ↪ V_{(𝔤,𝔤¯0)}(M).  (4.6.1)

By [LNZ11, Theorem 5.1.1(b)] the maps in (4.6.1) are in fact homeomorphisms. Therefore, we can use the fact that (2.4.1)–(2.4.6) hold for V_{(𝔢,𝔢¯0)} to see that they also hold for V_{(𝔤,𝔤¯0)} (cf. [LNZ11, Section 5.2]). By [LNZ11, Section 5.2] property (2.4.8) holds as well.

The preceding arguments demonstrate that V_{(𝔤,𝔤¯0)} is indeed a support data for K^{c}. However, we cannot use V_{(𝔤,𝔤¯0)} to classify thick tensor ideals or compute the Balmer spectrum because property (2.4.7) does not hold in general. For example, for any Kac module K^{+}(λ) for 𝔤 = 𝔤𝔩(m|n), V_{(𝔤,𝔤¯0)}(K^{+}(λ)) = ∅ by [BKN09, Corollary 3.3.1], but it is well known that Kac modules need not be projective in F_{(𝔤,𝔤¯0)}.

We remark that K is an R-linear triangulated category and the local global principal holds by [BIK12, Corollary 4.7] but in general the minimality condition of loc. cit. fails and, hence, R fails to stratify K. This can already be seen in the case of 𝔤 = 𝔤𝔩(1|1).

Example 4.6.4. Consider the associated varieties for classical Lie superalgebras defined by Duflo and Serganova [DS05]. Let X = Proj({x ∈ 𝔞¯1 | [x, x] = 0}), a projective variety over ℂ, and let X be the set of closed subsets of X. Define V : K^{c} → X in the following way. For each M in K^{c}, let

V (M) := X_{M} = Proj ({x ∈ X | M is not free as a U(⟨x⟩)-module}) .

The assignment M ↦ V (M) satisfies properties (2.4.1)–(2.4.6). In many instances, for example when 𝔞 = 𝔤𝔩(m|n), V satisfies (2.4.7). Moreover, one can check that (2.4.8) holds for 𝔤𝔩(1|1) and 𝔤𝔩(2|2). These low rank examples also show that V does not classify the thick tensor ideals of K^{c}. From the point of view of our setup what fails is that X is not a Zariski space.

Example 4.6.5. Let 𝔤 = 𝔤𝔩(m|n). The algebra 𝔤 is Type I with 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} where 𝔤_{0} ≅ 𝔤𝔩(m) ⊕ 𝔤𝔩(n). Set 𝔮^{+} = 𝔤_{0} ⊕ 𝔤_{1} and note that it is a classical Lie superalgebra. We will investigate the cohomologically defined support data for this parabolic subalgebra, K = Stab (C_{(𝔮^{+},𝔤_{0})}), and the classification of thick tensor ideals in K^{c} = Stab (F_{(𝔮^{+},𝔤_{0})}).

We first consider the Lie subsuperalgebra 𝔤_{1}. Let R = H^{•}(𝔤_{1}, ℂ) = H^{•}(𝔤_{1}, {0}; ℂ) ≅ S^{•}(𝔤_{1}^{∗}) with Y = Proj(Spec(R)). Since 𝔤_{1} is an abelian Lie superalgebra consisting of odd degree elements one has a support data V_{𝔤_{1}} : L^{c} → Y where L is the stable module category of C_{(𝔤_{1},{0})} as in Example 4.6.1 and Y is the collection of all closed subsets of Y. Moreover, let V_{𝔤_{1}} : L → Y_{sp} be the extension of V_{𝔤_{1}} as used in Theorem 4.5.4.

Using the support data V_{𝔤_{1}} define a support data V̂ : K^{c} → Y by V̂(M) = V_{𝔤_{1}}(M). Properties (2.4.1)–(2.4.6) follow because V_{𝔤¯1} is a support data for L^{c} and restriction is an exact functor. For (2.4.7), one can use the fact that a module in C_{(𝔮^{+},𝔤_{0})} is projective if and only if it is projective over U(𝔤_{1}) [BKN12, Theorem 3.3.1]; note that there only finite dimensional modules are considered but the relevant part of the proof does not use this fact. Thus one can define an extension V̂ : K → Y_{sp} of V̂ by setting V̂(M) = V_{𝔤_{1}}(M).

Now observe that the group G_{0} = GL(m) × GL(n) acts on R and so we may consider X = Y_{G_{0}} = Proj(G_{0}- Spec(R)). If M ∈ K^{c} then V̂(M) is a G_{0}-invariant closed set in Y and thus the support data V̂ can be viewed as having image in X via the map ρ : Y → X given in Section 2.3. In order to show that V̂ : K^{c} → X classifies thick tensor ideals we need to verify that (2.4.8) holds. The number of G_{0}-orbits on Y_{max} is finite with orbit representatives given by matrices I_{t} in 𝔤_{1} of rank t with 0 ≤ t ≤ r and r = min(m, n). Furthermore, \overline{G_{0} · I_{s}} ⊆ \overline{G_{0} · I_{t}} if and only if s ≤ t. The closed G_{0}-invariant sets in Y_{max} have the form W_{t} = \overline{G_{0} · I_{t}} for some 0 ≤ t ≤ r.

There certainly exists a module N ∈ F_{(𝔭^{+},𝔤_{0})} such that V_{𝔤_{1}}(N) = W_{t}; for example, one may take N to be a simple module of atypicality t (see [BKN11, (3.8.1)]). We can now conclude that Γ and Θ yield a bijective correspondence between thick tensor ideals of Stab(F_{(𝔮^{+},𝔤_{0})}) and specialization closed G_{0}-invariant subsets of Proj(Spec(S^{•}(𝔤_{1}^{∗}))) (i.e., specialization closed subsets of X).

Let M_{t} be the maximal ideal in R corresponding to the point I_{t}. According to [Lor09, Proposition 13], X = Y_{G_{0}} is finite and X = {∩_{g}gM_{t} : 1 ≤ t ≤ r}. Furthermore, irreducible sets of X are ordered by inclusion: \overline{\{∩_{g} gM_{s}\}} ⊆ \overline{\{∩_{g} gM_{t}\}} if and only if s ≤ t. Consequently, every specialization closed set in X is irreducible, thus all thick tensor ideals in Stab(F_{(𝔮^{+},𝔤_{0})}) are prime.

An identical analysis applies also to 𝔮^{−} = 𝔤_{0} ⊕ 𝔤_{−1} using the support data V̂(M) = V_{𝔤_{−1}}(M).

5. Classification of Thick Tensor Ideals for 𝔤𝔩(m|n)

5.1. Let 𝔤 = 𝔤𝔩(m|n) and let K = Stab (C_{(𝔤,𝔤¯0)}) with K^{c} = Stab (F_{(𝔤,𝔤¯0)}). Our goal is to construct a support data V̂ : K^{c} → X.

First, let L (resp. L^{c}) be the stable module category for C_{(𝔣,𝔣¯0)} (resp. F_{(𝔣,𝔣¯0)}). Set

R = H^{•}(𝔣, 𝔣¯0; ℂ) ≅ S^{•}(𝔣∗ ; ¯1),

Y = Proj(Spec(R)), Y to be the closed subsets of Y, and Y_{sp} to be the specialization closed subsets of Y. From Example 4.6.2 there exists a cohomologically defined support data V : L^{c} → Y with an extension V : L → Y_{sp}.

Now define V̂ : K^{c} → Y and V̂ : K → Y_{sp} by setting V̂(M) = V (M) and V̂(M) = V(M), respectively. Let N = Norm_{G¯0}(𝔣¯1). Then N acts on R and so set X = Proj(N-Spec(R)) which is a Zariski space. Let X be the closed sets in X. Any M in F_{(𝔤,𝔤¯0)} is a G¯0-module, and hence V (M) will be an N-invariant closed set. As a consequence we can and will view V̂(M) as lying in X under the map ρ : Proj(Spec(R)) → Proj(N-Spec(R)) from Section 2.3. That is, V̂ : K^{c} → X. Similarly, one can restrict the extension V to K and compose with the map ρ to obtain a map

V̂ : K → X_{sp}.

The fact that V is a support data shows that (2.4.1)–(2.4.6) hold for V̂. The fact that V extends V immediately implies V̂ also satisfies (2.4.1), (2.4.2), (2.4.4), (2.4.5), and Definition 2.5.1(ii). Since the restriction functor is exact, the fact that V satisfies (2.4.3) implies that V̂ does as well. Using [LNZ11, Theorem 3.3.1] we see that a module M ∈ C_{(𝔤,𝔤¯0)} is projective if and only if its restriction is projective in C_{(𝔣,𝔣¯0)}; note that in loc. cit. they assume M lies in F_{(𝔤,𝔤¯0)} but the proof of the needed results does not use this fact. This along with the fact that V satisfies (2.4.7) implies it also holds for V̂. In short, V̂ extends V.

To apply the results of Section 5 it only remains to verify that V̂ satisfies (2.4.8). This verification involves results on geometric induction which will be obtained in the subsequent sections. Using the discussion in Section 2.3 and the rank variety description of V, we can reduce this problem to showing that for any N-invariant closed subvariety W in Proj(𝔣¯1) there exists a module M in F_{(𝔤,𝔤¯0)} such that

W = V̂_{max}(M) ≅ V_{𝔣¯1}^{r}(M) := Proj({x ∈ 𝔣¯1 | M is not projective as a U(⟨x⟩)-module}).

This will be proved in Section 7 and the notation V_{𝔣¯1}^{r}(M) will be used for the remainder of the paper.

5.2. Assuming (2.4.8) is verified for V̂ : K^{c} → X, we can now state the main result. Namely, V̂ provides a classification of the thick tensor ideals for F_{(𝔤𝔩(m|n),𝔤𝔩(m|n)¯0)} and X provides a realization of the Balmer spectrum for this category. These results are consequences of our general setup in Sections 3.4 and 3.5.

Theorem 5.2.1. Let 𝔤 = 𝔤𝔩(m|n), let 𝔣 be the detecting subalgebra of 𝔤, and let N = Norm_{G¯0}(𝔣¯1). Then there is a bijection between the set of thick tensor ideals of Stab(F_{(𝔤,𝔤¯0)}) and the set of specialization closed subsets of Proj(N- Spec(S^{•}(𝔣∗ ; ¯1))).

Theorem 5.2.2. Let 𝔤 = 𝔤𝔩(m|n), let 𝔣 be the detecting subalgebra of 𝔤, and let N = Norm_{G¯0}(𝔣¯1). Then there is a homeomorphism between Spc(K^{c}) and Proj(N- Spec(S^{•}(𝔣∗ ; ¯1))). In particular, there is a bijection between the set of prime thick tensor ideals of Stab(F_{(𝔤,𝔤¯0)}) and the collection of irreducible N-stable closed subsets of Proj(Spec(S^{•}(𝔣∗ ; ¯1))).

Let us also mention that the analogues of the above theorems are also true for the type C Lie superalgebras 𝔬𝔰𝔭(2|2n). In this case realization can be achieved using Carlson’s L_{ζ} modules, Kac modules, and dual Kac modules.

6. Geometric Induction

6.1. In order to verify (2.4.8) for 𝔤 = 𝔤𝔩(m|n) we introduce modules which are constructed via geometric induction from parabolic subgroups. Let G denote the supergroup GL(m|n) and let P be a parabolic subgroup of G. By parabolic subgroup we will always mean the stabilizer of a flag of subsuperspaces in the natural representation of G. In particular, define the standard parabolic Q = Q^{+} to be the subgroup of block upper triangular matrices with Levi subgroup isomorphic to G¯0 = GL(m) × GL(n). Set Q^{−} to be the opposite standard parabolic of block lower triangular matrices with Levi subgroup isomorphic to G¯0.

We define induction from P -modules to G-modules using Lie superalgebras as in [Ser12]. See also [GS10] and references therein for further details on parabolic induction. Let 𝔭 be the Lie subsuperalgebra of 𝔤 corresponding to P, let 𝔲^{+} denote the nilpotent radical of 𝔭, 𝔩 the Levi subalgebra of 𝔭, and 𝔲^{−} the opposite nilpotent radical. Thus 𝔭 = 𝔩 ⊕ 𝔲^{+} and 𝔤 = 𝔲^{−} ⊕ 𝔭. Note that the ℤ-grading on 𝔤 is compatible with our choice of standard parabolic in the sense that for 𝔭 = Lie(Q^{+}) we have 𝔤_{0} = 𝔩, 𝔤_{1} = 𝔲^{+}, and 𝔤_{−1} = 𝔲^{−}.

Given an arbitrary 𝔤-module M let

Γ(M) = { m ∈ M | U(𝔤¯0)m is finite dimensional } .

It is straightforward to see that Γ is an endofunctor on the category of 𝔤-modules and that it coincides with the analogous functor defined by replacing 𝔤¯0 with 𝔤. The functor Γ is left exact and we let Γ^{i} denote its ith right derived functor. In particular, we define a functor from P -modules to G-modules by

H^{i}(G/P, N) = Γ^{i} (Hom_{U(𝔭)}(U(𝔤), N)) .

As 𝔤¯0-modules these modules have the following useful alternative description. For any finite dimensional 𝔭-module N we have

Hom_{U(𝔭)} (U(𝔤), N^{∗}) ≅ Hom_{U(𝔭¯0)} (U(𝔤¯0), [Λ^{•}(𝔤¯1/𝔭¯1) ⊗ N]^{∗}) ,

where the isomorphism is as 𝔤¯0-modules [Ser12, Section 4]. Applying Γ^{i} to both sides yields the following result.

Proposition 6.1.1. Let N be a finite dimensional P -module. Then for all i ≥ 0 there is an isomorphism of 𝔤¯0-modules

H^{i}(G/P, N^{∗}) ≅ H^{i}(G¯0/P¯0, [Λ^{•}(𝔤¯1/𝔭¯1) ⊗ N]^{∗}).

6.2. These functors can be used to construct the simple modules for 𝔤 as follows. Let T be the subgroup of G consisting of diagonal matrices and let 𝔱 = Lie(T) be the corresponding Cartan subalgebra of 𝔤¯0 consisting of all diagonal matrices. Let X_{0}^{+} ⊆ 𝔱^{∗} be the set of dominant integral highest weights with respect to the Borel subalgebra 𝔟¯0 (of all upper triangular matrices) of 𝔤¯0. For λ ∈ X_{0}^{+}, let L_{0}(λ) be the simple finite dimensional 𝔤¯0module of highest weight λ.

Recall that Q^{±} is our notation for the standard parabolic subgroups. Set 𝔮^{±} = 𝔤_{0}⊕𝔤_{±1} = Lie(Q^{±}). Since the Lie superalgebras 𝔤_{±1} are abelian ideals of 𝔮^{±} L_{0}(λ) can be viewed as a simple 𝔮^{±}-module via inflation. For each λ ∈ X_{0}^{+}, the Kac modules are constructed in the following way:

K^{±}(λ) = U(𝔤) ⊗_{U(𝔮^{±})} L_{0}(λ).

From the PBW theorem for Lie superalgebras it follows that

H^{0}(G/Q^{±}, L_{0}(λ)^{∗})^{∗} ≅ K^{±}(λ)

for any simple finite dimensional 𝔮^{±}-module L_{0}(λ).

The module K^{+}(λ) has a unique maximal submodule. The head of K^{+}(λ) is the simple finite dimensional 𝔤-module L(λ). Up to parity shift, the set {L(λ) | λ ∈ X_{0}^{+}} constitutes a complete set of non-isomorphic simple modules in F_{(𝔤,𝔤¯0)}. We disregard the parity shift as it will not play a role in what follows.

6.3. Let G = GL(m|n), let G_{±1} be the closed subgroup of G with Lie(G_{±1}) = 𝔤_{±1}, and let P denote an arbitrary parabolic subgroup of G with P = L⋉U, where L is a Levi subgroup and U is a unipotent subgroup. From this point on we always assume that G_{1} ⊆ P. Let Z be the closed subgroup P ∩ G_{−1} and 𝔷 be the Lie superalgebra associated to Z.

For a supergroup scheme D with closed subgroup C we have an induction functor from C-modules to D-modules, ind_{C}^{D} −, which can be defined as in the non-super setting (e.g. see [Zub06]). We use the notation and results of [Jan03] extended to the supergroup setting. This induction functor is left exact and admits higher right derived functors R^{i} ind_{C}^{D} − for i > 0. When D = G and C = P these functors coincide with the functors H^{i}(G/P, −) introduced in Section 6.1. Our first result relates the induction for the pairs (G, P) and (Q^{−}, P_{0} ⋉ Z).

Proposition 6.3.1. Let N be a module for P. Then for each j > 0, R^{j} ind_{P}^{G} N = 0 if and only if R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N = 0.

Proof. We first employ the spectral sequence:

E_{2}^{i,j} = R^{i} ind_{Q^{−}}^{G} R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N ⇒ R^{i+j} ind_{P_{0}⋉Z}^{G} N.

The functor ind_{Q^{−}}^{G}(−) coincides with the coinduction functor Hom_{U(𝔮^{−})}(U(𝔤), −) and so it is exact and the spectral sequence collapses. This yields the isomorphism:

ind_{Q^{−}}^{G}[R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N] ≅ R^{j} ind_{P_{0}⋉Z}^{G} N

for j > 0. Since ind_{Q^{−}}^{G}(−) ≅ Hom_{U(𝔮^{−})}(U(𝔤), −) we have that ind_{Q^{−}}^{G}(−) takes non-zero

modules to non-zero modules. We can now conclude that for j > 0, R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N = 0 if and only if R^{j} ind_{P_{0}⋉Z}^{G} N = 0. Next we involve the spectral sequence:

E_{2}^{i,j} = R^{i} ind_{P}^{G} R^{j} ind_{P_{0}⋉Z}^{P} N ⇒ R^{i+j} ind_{P_{0}⋉Z}^{G} N.

Since ind_{P_{0}⋉Z}^{P}(−) is exact, it follows that for j ≥ 0,

R^{j} ind_{P_{0}⋉Z}^{G} N ≅ R^{j} ind_{P}^{G}[ind_{P_{0}⋉Z}^{P} N].

Since G_{1} ⊆ P we see that K^{−}(0) is isomorphic to ind_{P_{0}⋉Z}^{P} ℂ when restricted to P. Using this and the tensor identity twice yields

R^{j} ind_{P}^{G}[ind_{P_{0}⋉Z}^{P} N] ≅ R^{j} ind_{P}^{G}[(ind_{P_{0}⋉Z}^{P} ℂ) ⊗ N] ≅ K^{−}(0) ⊗ R^{j} ind_{P}^{G} N.

These isomorphisms show for j > 0 that R^{j} ind_{P}^{G} N = 0 if and only if R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N = 0. □

Proposition 6.3.2. Let R = H^{•}(𝔤_{−1}, ℂ), N be a module for P, and M = ind_{P}^{G} N. If R^{j} ind_{P}^{G} N = 0 for all j > 0 then

Ext_{𝔤_{−1}}^{•}(M, M) ≅ Ext_{𝔤_{−1}}^{•}(M, ind_{P_{0}⋉Z}^{Q^{−}} N)

as R-modules.

Proof. Set M = ind_{P}^{G} N. For each λ ∈ X_{0}^{+} let m_{λ} = dim L_{0}(λ). Then as R-modules

Ext_{𝔤_{−1}}^{•}(M, M) ≅ Hom_{𝔤_{0}}(⊕_{λ∈X_{0}^{+}}L_{0}(λ)^{m_{λ}}, Ext_{𝔤_{−1}}^{•}(M, M))
≅ Ext_{(𝔮^{−},𝔤_{0})}^{•}(⊕_{λ∈X_{0}^{+}}L_{0}(λ)^{m_{λ}} ⊗ M, M)
≅ Ext_{(𝔤,𝔤_{0})}^{•}(⊕_{λ∈X_{0}^{+}}K^{−}(λ)^{m_{λ}} ⊗ M, M),

by the proof of [BKN12, Theorem 3.3.1] and Frobenius reciprocity. We have a spectral sequence:

E_{2}^{i,j} = Ext_{(𝔤,𝔤_{0})}^{i}(⊕_{λ∈X_{0}^{+}}K^{−}(λ)^{m_{λ}} ⊗ M, R^{j} ind_{P}^{G} N) ⇒ Ext_{(𝔭,𝔭_{0})}^{i+j}(⊕_{λ∈X_{0}^{+}}K^{−}(λ)^{m_{λ}} ⊗ M, N).

Since R^{j} ind_{P}^{G} N = 0 for j > 0 this spectral sequence collapses and yields the isomorphism:

Ext_{(𝔤,𝔤_{0})}^{•}(⊕_{λ∈X_{0}^{+}}K^{−}(λ)^{m_{λ}} ⊗ M, M) ≅ Ext_{(𝔭,𝔭_{0})}^{•}(⊕_{λ∈X_{0}^{+}}K^{−}(λ)^{m_{λ}} ⊗ M, N).

Since G_{1} ⊆ P we have that K^{−}(λ) ≅ U(𝔭) ⊗_{U(𝔭_{0}⊕𝔷)} L_{0}(λ) as 𝔭-modules and

Ext_{𝔤_{−1}}^{•}(M, M) ≅ Ext_{(𝔭,𝔭_{0})}^{•}(⊕_{λ∈X_{0}^{+}}K^{−}(λ)^{m_{λ}} ⊗ M, N)
≅ Ext_{(𝔭_{0}⊕𝔷,𝔭_{0})}^{•}(⊕_{λ∈X_{0}^{+}}L_{0}(λ)^{m_{λ}} ⊗ M, N)
≅ Ext_{(𝔮^{−},𝔤_{0})}^{•}(⊕_{λ∈X_{0}^{+}}L_{0}(λ)^{m_{λ}} ⊗ M, ind_{P_{0}⋉Z}^{Q^{−}} N)
≅ Ext_{𝔤_{−1}}^{•}(M, ind_{P_{0}⋉Z}^{Q^{−}} N).

The second to last line holds by a spectral sequence argument similar to the one above after replacing (G, P) with (Q^{−}, P_{0} ⋉Z) and using Proposition 6.3.1. The last line uses the proof of [BKN12, Theorem 3.3.1]. □

6.4. Consider the pairs (Q^{−}, G_{−1}) and (P_{0} ⋉ Z, Z) with quotient groups Q^{−}/G_{−1} ≅ G_{0} and (P_{0} ⋉ Z)/Z ≅ P_{0}. Applying [Jan03, I, (6.12)] when M is a Q^{−}-module and N is a P_{0} ⋉ Z-module we have two spectral sequences which converge to the same abutment:

Ê_{2}^{i,j} = Ext_{𝔤_{−1}}^{i}(M, R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N) ⇒ (R^{i+j}G_{1})(M, N),  (6.4.1)
E_{2}^{i,j} = R^{i} ind_{P_{0}}^{G_{0}} Ext_{𝔷}^{j}(M, N) ⇒ (R^{i+j}G_{2})(M, N).  (6.4.2)

Here G_{i}, i = 1, 2 are the composition of the functors used to construct these spectral sequences with G_{1} isomorphic to G_{2} (cf. [Jan03, I, (6.12)]).

In the case when R^{j} ind_{P_{0}⋉Z}^{Q^{−}} N = 0 for all j > 0, the first spectral sequence collapses and one can identify the abutment in the second spectral sequence. This yields:

E_{2}^{i,j} = R^{i} ind_{P_{0}}^{G_{0}} Ext_{𝔷}^{j}(M, N) ⇒ Ext_{𝔤_{−1}}^{i+j}(M, ind_{P_{0}⋉Z}^{Q^{−}} N).

Now one can apply Proposition 6.3.2 when M = ind_{P}^{G} N to rewrite the aforementioned spectral sequence as

E_{2}^{i,j} = R^{i} ind_{P_{0}}^{G_{0}} Ext_{𝔷}^{j}(M, N) ⇒ Ext_{𝔤_{−1}}^{i+j}(M, M).  (6.4.3)

6.5. A Version of the Borel-Weil-Bott Theorem. We now specialize to the case when the parabolic subgroup is block upper triangular. Namely, for the remainder of the paper we will always have in mind a fixed integer 0 ≤ k ≤ n and the fixed parabolic subgroup P = P_{k} which is block upper triangular and has Levi subgroup isomorphic to GL(m|n − k) × GL(k). Note that the assumption from the previous section that G_{1} ⊆ P is satisfied by this choice of parabolic subgroup.

Let 𝔭 = Lie(P) be the parabolic subalgebra of 𝔤 = 𝔤𝔩(m|n) corresponding to P and let 𝔩 be the corresponding Levi subalgebra.

Lemma 6.5.1. Let D > m(n − k) be fixed. Let L_{𝔭}(λ) be a simple 𝔭-module with highest weight λ satisfying λ_{i} − λ_{i+1} > D for i = 1, …, m − 1, m + 1, …, m + n − 1. Then as an 𝔩¯0-module we have

L_{𝔭}(λ) ≅ ⨁_{γ∈Γ_{λ}} L_{𝔩¯0}(γ)

with Γ_{λ} ⊆ 𝔱^{∗} a finite set such that for all γ ∈ Γ_{λ} we have γ_{i} − γ_{i+1} > D − m(n − k) for i = 1, …, m − 1, m + 1, …, m + n − 1.

Proof. As L_{𝔭}(λ) = L_{𝔩}(λ) = L_{𝔤𝔩(m|n−k)}(λ_{1}, …, λ_{m+n−k}) ⊠ L_{𝔤𝔩(k)}(λ_{m+n−k+1}, …, λ_{m+n}) and a simple 𝔤𝔩(m|n − k)-module is a quotient of the Kac module of the same highest weight, it suffices to prove that as 𝔩¯0-modules we have

K_{𝔤𝔩(m|n−k)}^{+}(λ_{1}, …, λ_{m+n−k}) ⊠ L_{𝔤𝔩(k)}(λ_{m+n−k+1}, …, λ_{m+n}) ≅ ⨁_{γ∈Γ̃_{λ}} L_{𝔩¯0}(γ)  (6.5.1)

with Γ̃_{λ} satisfying the conditions stated in the theorem. This implies the desired result since Γ_{λ} ⊆ Γ̃_{λ}.

From the PBW theorem for Lie superalgebras we have that

K_{𝔤𝔩(m|n−k)}^{+}(λ_{1}, …, λ_{m+n−k}) ≅ Λ^{•}(𝔩_{−1}) ⊗ L_{𝔤𝔩(m|n−k)¯0}(λ_{1}, …, λ_{m+n−k}),

where the isomorphism is as 𝔤𝔩(m|n − k)¯0 ≅ 𝔤𝔩(m) ⊕ 𝔤𝔩(n − k)-modules. From this we see that as a 𝔤𝔩(m) ⊕ 𝔤𝔩(n − k)-module the Kac module K_{𝔤𝔩(m|n−k)}^{+}(λ_{1}, …, λ_{m+n−k}) appears as a direct summand of

⨁_{t=0}^{dim(𝔩_{−1})=m(n−k)} (𝔩_{−1})^{⊗t} ⊗ L_{𝔤𝔩(m)⊕𝔤𝔩(n−k)}(λ_{1}, …, λ_{m+n−k}) ≅ ⨁_{σ} L_{𝔤𝔩(m|n−k)¯0}(σ).

To describe the σ’s which appear in the direct sum, we observe that as a 𝔤𝔩(m) ⊕ 𝔤𝔩(n − k)- module we have 𝔩_{−1} ≅ V_{m}^{∗} ⊠ V_{n−k}, where V_{m} (resp. V_{n−k}) denotes the natural module for 𝔤𝔩(m) (resp. 𝔤𝔩(n − k)). A calculation using Pieri’s formula [Mac95, (5.16)] shows that the σ’s which appear in the above sum satisfy σ_{i} − σ_{i+1} > D − m(n − k) for i = 1, …, m − 1, m + 1, …, m + n − k − 1.

Inserting this description of the Kac module into the left hand side of (6.5.1) yields the right hand side and, hence, the desired result. □

Lemma 6.5.2. Fix D ≥ km and let L_{𝔩¯0}(γ) be a simple 𝔩¯0-module with γ_{i} − γ_{i+1} > D for i = 1, …, m−1, m+1, …, m+n−1. Then as 𝔩¯0-modules we have the following isomorphism:

Λ^{•} (u− ; ¯1) ⊗ L_{𝔩¯0}(γ) ≅ ⨁_{σ∈Σ_{γ}} L_{𝔩¯0}(σ)

for a finite set Σ_{γ} ⊂ 𝔱^{∗} such that for every σ ∈ Σ_{γ} we have σ_{i} − σ_{i+1} > D − km for i = 1, …, m − 1, m + 1, …, m + n − 1.

Proof. As 𝔩¯0 ≅ 𝔤𝔩(m)⊕𝔤𝔩(n−k)⊕𝔤𝔩(k)-modules we have that 𝔲− ; ¯1 is isomorphic to V_{m}^{∗}⊠ℂ⊠V_{k}, where, as before, V_{q} denotes the natural module of 𝔤𝔩(q). And

L_{𝔩¯0}(γ) = L_{𝔤𝔩(m)}(γ_{1}, …, γ_{m}) ⊠ L_{𝔤𝔩(n−k)}(γ_{m+1}, …, γ_{m+n−k}) ⊠ L_{𝔤𝔩(k)}(γ_{m+n−k+1}, …, γ_{m+n}).

The claim follows as in the proof of the previous lemma using Pieri’s formula [Mac95, (5.16)]. □

We can now prove a coarse version of the parabolic Borel-Weil-Bott Theorem in the super setting for sufficiently dominant weights. Note we write it using the bound d as it will be needed in what follows.

Theorem 6.5.3. Fix d ≥ 0. Let λ ∈ X_{0}^{+} with λ_{i} − λ_{i+1} > d + mn for i = 1, …, m − 1, m + 1, …, m + n − 1. Then

H^{j}(G/P, L_{𝔭}(λ)^{∗}) = 0

for j > 0. Furthermore, as 𝔤¯0-modules we have

H^{0}(G/P, L_{𝔭}(λ)^{∗}) ≅ ⨁_{γ∈Γ_{λ}} ⨁_{σ∈Σ_{γ}} L_{𝔤¯0}(σ)^{∗}

where Γ_{λ} and Σ_{γ} are as in the previous lemmas and, in particular, every σ which appears in the direct sum satisfies σ_{i} − σ_{i+1} > d for i = 1, …, m − 1, m + 1, …, m + n − 1.

Proof. For the statement of the theorem it suffices to compute the induced module as a 𝔤¯0-module. By Proposition 6.1.1, we have

H^{j}(G/P, L_{𝔭}(λ)^{∗}) ≅ H^{j}(G¯0/P¯0, [Λ^{•}(𝔤¯1/𝔭¯1) ⊗ L_{𝔭}(λ)]^{∗}).

From our previous two lemmas we have an isomorphism as 𝔩¯0-modules:

[Λ^{•}(𝔤¯1/𝔭¯1) ⊗ L_{𝔭}(λ)]^{∗} ≅ ⨁_{γ∈Γ_{λ}} ⨁_{σ∈Σ_{γ}} L_{𝔩¯0}(σ)^{∗}

where each σ satisfies σ_{i} − σ_{i+1} > d for i = 1, …, m − 1, m + 1, …, m + n − 1. In particular, each σ is dominant regular. That is, as a 𝔭¯0-module [Λ^{•}(𝔤¯1/𝔭¯1) ⊗ L_{𝔭}(λ)]^{∗} has a composition series consisting of simple modules with dominant regular highest weights. The theorem then follows by using the non-super parabolic Borel-Weil-Bott Theorem and an induction on the length of the composition series of modules with such a composition series. □

6.6. The Parabolic Grading. Fix positive real numbers a, b and let h ∈ T be the diagonal matrix

h = diag(a, …, a, b, …, b)

where there are m + n − k a’s and k b’s. The adjoint action of h on 𝔤 defines a ℤ-grading compatible with our choice of parabolic. That is, h acts by (ab^{−1})^{−1}, 1, and ab^{−1}, respectively, on 𝔲^{−}, 𝔩, and 𝔲^{+}. This element can be used to give a compatible ℤ-grading to any weight module where the degree of a weight vector is determined by its weight. In particular, for any integral weight μ we write deg(μ) for the corresponding degree. We call this the parabolic grading to distinguish it from the ℤ_{2}-grading and the ℤ-grading introduced in Section 6.1 (although the two ℤ-gradings coincide when k = n).

Given a weight module M we write M_{t} for the t-th parabolic graded summand of M. For example, the parabolic Verma module U(𝔤) ⊗_{U(𝔭)} L_{𝔭}(λ) has a parabolic grading and by the PBW theorem it is non-zero only in degrees less than or equal to deg(λ). As this module surjects onto the simple module L_{𝔤}(λ), it follows that the non-zero degrees of L_{𝔤}(λ) are also bounded above by deg(λ). Similarly, since there exists a surjective 𝔤-module homomorphism by [GS10, Lemma 2],

U(𝔤) ⊗_{U(𝔭)} L_{𝔭}(λ) → H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗},  (6.6.1)

it follows that the non-zero degrees of H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗} are bounded above by deg(λ).

The parabolic grading, (6.6.1) and Theorem 6.5.3 will be used to obtain a partial description of H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗} when λ is sufficiently dominant. To do so we will need the following lemma.

Lemma 6.6.1. Fix D > 0. Let L_{𝔤¯0}(τ) be a finite dimensional 𝔤¯0-module of highest weight τ satisfying τ_{i} − τ_{i+1} > D for i = 1, …, m − 1, m + 1, …, m + n − 1. Then as 𝔩¯0-modules we have

S^{t}(𝔤¯0/𝔭¯0) ⊗ L_{𝔭¯0}(τ) ≅ (U(𝔤¯0) ⊗_{U(𝔭¯0)} L_{𝔭¯0}(τ))_{deg(τ)−t} ≅ L_{𝔤¯0}(τ)_{deg(τ)−t}

for t = 0, …, D.

Proof. Let M = U(𝔤¯0) ⊗_{U(𝔭¯0)} L_{𝔭¯0}(τ) be the parabolic Verma 𝔤¯0-module of highest weight τ. Then one has a surjective map M → L_{𝔤¯0}(τ) which preserves the parabolic grading. Using the PBW theorem for Lie algebras and the description of the left ideal which is the annihilator of the highest weight vector of L_{𝔤¯0}(τ) (see [Hum78, Section 21.4]), we see that the graded surjective map is actually an isomorphism for degrees deg(τ), …, deg(τ) − D. This implies the second isomorphism and the first is from the PBW theorem. □

We can now prove that for sufficiently dominant weights the map (6.6.1) is an isomorphism for the degrees which are sufficiently close to deg(λ). In the statement of the theorem we use the notation S_{super}^{t}(𝔤/𝔭) to denote the t-th supersymmetric power of 𝔤/𝔭.

Theorem 6.6.2. Fix d ≥ 0 and let L_{𝔭}(λ) be a finite dimensional simple 𝔭-module with λ_{i} − λ_{i+1} > D = d + mn for i = 1, …, m − 1, m + 1, …, m + n − 1. Then

S_{super}^{t}(𝔤/𝔭) ⊗ L_{𝔭}(λ) ≅ [U(𝔤) ⊗_{U(𝔭)} L_{𝔭}(λ)]_{deg(λ)−t} ≅ [H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}]_{deg(λ)−t}

for t = 0, …, d.

Proof. Fix t ∈ {0, …, d}. Combining Theorem 6.5.3 and Lemma 6.6.1 we have that

[H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}]_{deg(λ)−t} ≅ ⨁_{γ∈Γ_{λ}} ⨁_{σ∈Σ_{γ}} L_{𝔤¯0}(σ)_{deg(λ)−t}

≅ ⨁_{γ∈Γ_{λ}} ⨁_{σ∈Σ_{γ}} [S^{•}(𝔤¯0/𝔭¯0) ⊗ L_{𝔭¯0}(σ)]_{deg(λ)−t} .

Let us briefly explain the second line. We first observe that in the parabolic grading L_{𝔤¯0}(μ) is non-zero only in degrees less than or equal to deg(μ) for any dominant μ. Furthermore, (6.6.1) implies that if L_{𝔤¯0}(σ) is a composition factor of H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}, then it is a composition factor of U(𝔤) ⊗_{U(𝔭)} L_{𝔭}(λ). Taken together this implies that if L_{𝔤¯0}(σ) appears in the first line then we must have that deg(σ) ≤ deg(λ) and it can contribute only to degrees less than or equal to deg(σ).

Now since σ_{i} −σ_{i+1} > d for i = 1, …, m−1, m+1, …, m+n−1 for all σ which appear in the first line it follows that Lemma 6.6.1 can be applied to the degrees deg(σ), …, deg(σ)−d. That is, it holds for d degrees down from deg(σ) for each σ. Taken together with the previous paragraph we see that the above isomorphism holds for d degrees down from deg(λ).

Now, to prove the stated theorem we merely need to unwind the above isomorphisms using Lemmas 6.5.1 and 6.5.2:

[H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}]_{deg(λ)−t} ≅ ⨁_{γ∈Γ_{λ}} ⨁_{σ∈Σ_{γ}} [S^{•}(𝔤¯0/𝔭¯0) ⊗ L_{𝔭¯0}(σ)]_{deg(λ)−t}

≅ ⨁_{γ∈Γ_{λ}} [S^{•}(𝔤¯0/𝔭¯0) ⊗ ( ⨁_{σ∈Σ_{γ}} L_{𝔭¯0}(σ))]_{deg(λ)−t}

≅ ⨁_{γ∈Γ_{λ}} [S^{•}(𝔤¯0/𝔭¯0) ⊗ Λ^{•} (u− ; ¯1) ⊗ L_{𝔩¯0}(γ)]_{deg(λ)−t}

≅ [S^{•}(𝔤¯0/𝔭¯0) ⊗ Λ^{•} (u− ; ¯1) ⊗ ( ⨁_{γ∈Γ_{λ}} L_{𝔩¯0}(γ))]_{deg(λ)−t}

≅ [S^{•}(𝔲− ; ¯0) ⊗ Λ^{•} (u− ; ¯1) ⊗ L_{𝔭}(λ)]_{deg(λ)−t}

≅ [U(𝔤) ⊗_{U(𝔭)} L_{𝔭}(λ)]_{deg(λ)−t} .

□

7. Support Varieties

7.1. We are now prepared to compute the support varieties of various 𝔤-modules. This will enable us to verify that in this setting (2.4.8) holds for the support data V̂ of Section 5.

To proceed we need to set notation. Fix 𝔤 = 𝔤𝔩(m|n) and write 𝔣 for the detecting subalgebra of 𝔤. Since 𝔤𝔩(m|n) ≅ 𝔤𝔩(n|m) we can and will assume that m ≤ n from this point on. In particular 𝔣¯1 has a basis given by the matrix units

e_{1,2m}, e_{2,2m−1}, …, e_{m,m+1}, e_{m+1,m}, …, e_{2m,1}.

Let T denote the torus of G¯0 consisting of diagonal matrices. Then T acts on 𝔤 by the adjoint action and, in particular, the above basis for 𝔣¯1 consists of weight vectors.

As mentioned Section 5, the underlying geometry in this setting is given by the spectrum of the cohomology ring of 𝔣; that is, the ring

R := H^{•}(𝔣, 𝔣¯0; ℂ) ≅ S^{•}(𝔣∗ ; ¯1) = ℂ[𝔣¯1] ≅ ℂ[X_{1}, X_{2}, …, X_{m}, Y_{1}, Y_{2}, …, Y_{m}],

where X_{j} and Y_{j} are defined below. In particular,

V_{(𝔣,𝔣¯0)}(ℂ) ≅ V_{𝔣¯1}^{r}(ℂ) = Proj (MaxSpec(R)) = Proj(𝔣¯1).

We set coordinates by identifying R with the polynomial ring on 2m variables. This is done by letting X_{j} : 𝔣¯1 → ℂ to be the linear functional given on our basis of matrix units for 𝔣¯1 by X_{j} (e_{m−j+1,m+j}) = 1 and otherwise zero. Similarly, we define Y_{j} : 𝔣¯1 → ℂ to be the linear functional given on our basis by Y_{j} (e_{m+j,m−j+1}) = 1 and otherwise zero.

Let Σ_{m} denote the symmetric group on m letters embedded diagonally in G¯0 ≅ GL(m)× GL(n) as permutation matrices so that the action of Σ_{m} on R is via simultaneous permutation of X_{1}, …, X_{m} and Y_{1}, …, Y_{m}. Let N = Norm_{G¯0}(H) as in Section 4.4 or, equivalently, N = Norm_{G¯0}(𝔣¯1). A direct calculation (see [BKN10, Section 8.11] for the analogous calculation for 𝔰𝔩(n|n)) verifies that N = Σ_{m}T K where K ≅ GL(n − m) is the subgroup of G consisting of (n − m) × (n − m) invertible matrices in the lower right corner of G. It is convenient to redefine

N = Σ_{m}T

and instead consider the action of this group on 𝔣¯1. Since K fixes R pointwise there is no harm in doing this.

As explained in Section 5.1, to verify that (2.4.8) holds for the support data V̂ it suffices to prove that we can realize every N-invariant closed subvariety of V_{𝔣¯1}^{r}(ℂ) ≅ Proj(𝔣¯1) as V_{𝔣¯1}^{r}(M) for some M ∈ F_{(𝔤,𝔤¯0)}. Let W be such a variety. Since Σ_{m} is a finite group we may write

W = Σ_{m}V

for some T -invariant closed subvariety V of Proj(𝔣¯1). Furthermore, since the action of Σ_{m} distributes across unions we may use this along with (2.4.2) to assume without loss that V is irreducible among the T -invariant subvarieties. That is, if we write Z(I) for the closed subvariety of Proj(𝔣¯1) determined by a homogeneous ideal I of R, it suffices to consider the case when V = Z(I) for a homogeneous T -prime ideal I. But since T is connected it follows from [Lor08, Proposition 19] that I is a homogeneous T -invariant prime ideal in R. Therefore, V must be of the form

V = Z(X_{a_{1}}, …, X_{a_{s}}, Y_{b_{1}}, …, Y_{b_{t}}, g_{1}, …, g_{r})  (7.1.1)

where g_{1}, …, g_{r} are homogeneous polynomials of weight zero for T. In short, to verify (2.4.8) in this setting we must prove that for any V as in (7.1.1) there exists an M ∈ F_{(𝔤,𝔤¯0)} such that V_{𝔣¯1}^{r}(M) = Σ_{m}V. This will be accomplished in the following sections.

Before continuing it is convenient to introduce the following notation. Let s, t, p be nonnegative integers with m ≥ s, t ≥ p ≥ 0. Set

V (s, t, p) = Z(X_{1}, …, X_{s}, Y_{s−p+1}, …, Y_{s−p+t}).  (7.1.2)

This is the variety given by the vanishing of s X coordinates, t Y coordinates, with p “overlapping pairs” (that is, there are precisely p indices i such that X_{i} and Y_{i} are both required to vanish). It is useful to note that since we may replace V by its conjugate under the action of Σ_{m}the case r = 0 in (7.1.1) will be complete once we realize Σ_{m}V (s, t, p).

7.2. Weight Zero Polynomials. We first consider the case when s = t = 0 in (7.1.1). That is, we show how to realize

Σ_{m}Z(g_{1}, g_{2}, …, g_{r}),

where g_{1}, …, g_{r} are homogeneous weight zero polynomials on 𝔣¯1, as the support of a module in F_{(𝔤,𝔤¯0)}. This will be accomplished using Carlson’s L_{ζ} modules. These modules are the standard tool used to prove realization in the setting of finite groups and are all that is needed there to prove realization in that setting.

Proposition 7.2.1. Let g_{1}, g_{2}, …, g_{r} be homogenous polynomials in R of weight zero with respect to T. Then there exists a module M in F such that V_{𝔣¯1}^{r}(M) = Σ_{m}Z(g_{1}, g_{2}, …, g_{r}).

Proof. We show that we can reduce to the case when g_{1}, …, g_{r} are N = Σ_{m}T -invariant polynomials. In order to do so, we first show that Σ_{m}Z(g_{1}, g_{2}, …, g_{r}) is the zero set for a finite Σ_{m}-invariant collection of weight zero polynomials.

Set q = m!. Let

Γ = {σ = (σ_{1}, …, σ_{q}) ∈ Σ_{m} × ⋯ × Σ_{m} | σ_{i} ≠ σ_{j} for 1 ≤ i ≠ j ≤ q} .

Then Γ acts diagonally on the set of monomials of degree q in g_{1}, …, g_{r} by

σ.(g_{1}^{m_{1}}g_{2}^{m_{2}} … g_{r}^{m_{r}}) = (σ_{1}g_{1}) … (σ_{m_{1}}g_{1})(σ_{m_{1}+1}g_{2}) … (σ_{m_{1}+···+m_{r−1}+1}g_{r}) … (σ_{q}g_{r}).

We set

Y = {σ.(g_{1}^{m_{1}}g_{2}^{m_{2}} … g_{r}^{m_{r}}) | σ ∈ Γ, m_{1}, …, m_{r} ∈ ℤ_{≥0} such that _{∑i}m_{i} = q}

and note that Y is a finite set which is invariant under the action of Σ_{m}. Furthermore, we have

Σ_{m}Z(g_{1}, …, g_{r}) = Z (Y) .  (7.2.1)

To see this, we first let b ∈ 𝔣¯1 be an element of the left hand set. Then b = τa for some τ ∈ Σ_{m} and a ∈ Z(g_{1}, …, g_{r}). Computing

(σ.(g_{1}^{m_{1}}g_{2}^{m_{2}} … g_{r}^{m_{r}})) (b) = σ.(g_{1}^{m_{1}}g_{2}^{m_{2}} … g_{r}^{m_{r}})(τa)
= (σ_{1}g_{1}) … (σ_{q}g_{r})(τa)
= g_{1}(σ_{1}^{−1}(τa)) ⋯ g_{r}(σ_{q}^{−1}(τa)).

From the definition of Γ we see that there is an index 1 ≤ i ≤ q such that σ_{i} = τ and, hence, this product contains the factor g_{j}(a) for some 1 ≤ j ≤ r. From this we see that (σ.(g_{1}^{m_{1}}g_{2}^{m_{2}} … g_{r}^{m_{r}})) (b) = 0 and so b is contained in the right hand side.

On the other hand, let b be an element of the right hand set. If b is not an element of the left hand set, then σ_{i}b is not an element of V (g_{1}, …, g_{r}) for any σ_{i} ∈ Σ_{m} and, hence, there exists a polynomial g_{σ_{i}} ∈ {g_{1}, …, g_{r}} such that g_{σ_{i}}(σ_{i}b) ≠ 0. If we set σ = (σ_{1}^{−1}, …, σ_{q}^{−1}) and consider the monomial g_{σ_{1}} ⋯ g_{σ_{q}}, then

(σ.(g_{σ_{1}} ⋯ g_{σ_{q}}))(b) ≠ 0.

This then contradicts our choice of b since, up to reordering factors, σ.(g_{σ_{1}} ⋯ g_{σ_{q}}) ∈ Y.

We can now reduce to a finite set of Σ_{m}T -invariant polynomials as follows. For each Σ_{m}-orbit in Y, say {y_{1}, …, y_{s}}, let ỹ_{1}, …, ỹ_{s} be the elementary symmetric polynomials in y_{1}, …, y_{s}. One can then verify that

Z(y_{1}, …, y_{s}) = Z(ỹ_{1}, …, ỹ_{s}).

If we let Ỹ be the set of all such elementary symmetric polynomials obtained by ranging over the Σ_{m}-orbits in Y, we then have

Σ_{m}Z(g_{1}, …, g_{r}) = Z(Y) = Z(Ỹ),

where Ỹ is a finite set of Σ_{m}T -invariant polynomials.

We now show that we can realize this variety using Carlson’s L_{ζ}-modules. Say Ỹ = {q_{1}, …, q_{u}}. From [BKN10, Theorem 4.1.1] we have a ring isomorphism induced by the restriction map:

H^{•}(𝔤, 𝔤¯0; ℂ) ≅ H^{•}(𝔣, 𝔣¯0; ℂ)^{N} ≅ S^{•}(𝔣¯1)^{N}.

Therefore, for each i = 1, 2, …, u there exists ζ_{i} ∈ H^{n_{i}}(𝔤, 𝔤¯0; ℂ) corresponding to q_{i} under this isomorphism. Corresponding to this cohomology class is a 𝔤-module homomorphism Ω^{n_{i}}(ℂ) → ℂ. Let L_{ζ_{i}} be the kernel of this map; that is, the Carlson module for ζ_{i}.

When we restrict L_{ζ_{i}} to 𝔣 it will decompose as an 𝔣-module into the direct sum of the Carlson module associated with the cohomology class q_{i} in H^{•}(𝔣, 𝔣¯0; ℂ)^{N} ⊆ H^{•}(𝔣, 𝔣¯0; ℂ) and a projective 𝔣-module. Standard arguments (e.g. see [BKN08, Lemma 8.5.1] for the Lie superalgebra version) show that

V_{𝔣¯1}^{r}(L_{ζ_{i}}) = Z(q_{i}).

Now applying the tensor product property we obtain

Σ_{m}Z(g_{1}, …, g_{r}) = Z (Y) = Z(Ỹ) = Z (q_{1}, …, q_{u}) = V_{𝔣¯1}^{r} (L_{ζ_{1}} ⊗ ⋯ ⊗ L_{ζ_{u}}) .

This proves the desired result. □

7.3. We now calculate the 𝔣-variety for the simple modules of 𝔤𝔩(m|n) and thereby achieve realization for Σ_{m}V (p, p, p) for any p = 0, …, m. Before stating the theorem, recall that given a simple 𝔤𝔩(m|n)-module L one can assign an integer to L from among 0, …, m called the atypicality of L. See, for example, [BKN12, Section 2.4] for the precise definition. All we need to know here is that there are simple modules of every possible atypicality.

Theorem 7.3.1. If L is a simple 𝔤 = 𝔤𝔩(m|n) module of atypicality ℓ, then

V_{𝔣¯1}^{r} (L) = Σ_{m}V (m − ℓ, m − ℓ, m − ℓ).

Proof. From the proof of [BKN12, Theorem 9.2.1] we have that V (m − ℓ, m − ℓ, m − ℓ) is a subset of V_{𝔣¯1}^{r} (L). Since V_{𝔣¯1}^{r}(L) is stable under the action of Σ_{m} it follows that Σ_{m}V (m − ℓ, m − ℓ, m − ℓ) ⊆ V_{𝔣¯1}^{r}(L). If ℓ = 0 then L is projective and both sides are the zero variety. If ℓ = m, then both sides are equal to 𝔣¯1. In either case the theorem holds and so for the reverse containment it suffices to consider the case when 0 < ℓ < m.

We now consider the reverse containment. Let

z = ∑_{j=1}^{m} x_{j}e_{m−j+1,m+j} + y_{j}e_{m+j,m−j+1} ∈ V_{𝔣¯1}^{r} (L) .

We claim that z ∈ Σ_{m}V (m − ℓ, m − ℓ, m − ℓ). Using the action of Σ_{m} we may assume without loss that there is 1 ≤ u ≤ v ≤ w ≤ m such that x_{i} ≠ 0 and y_{i} ≠ 0 for i = 1, …, u, x_{i} = 0 and y_{i} ≠ 0 for i = u + 1, …, v, x_{i} ≠ 0 and y_{i} = 0 for i = v + 1, …, w, and x_{i} = 0 and y_{i} = 0 for i = w + 1, …, m. In particular, z has m − w matching pairs of coordinates which are zero.

For any fixed non-zero a ∈ ℝ let g_{a} ∈ G¯0 be the group element given by the diagonal matrix

g_{a} = Diag (d_{1}, …, d_{m+n})

with

d_{m−i+1} = \begin{cases}a^{−1}, & i = 1, …, u; \\ a, & i = u + 1, …, v; \\ a^{−1}, & i = v + 1, …, w; \\ 1, & i = w + 1, …, m; \\ 1, & i = 1 − n, …, 0; \end{cases} .

Since V_{𝔣¯1}^{r} (L) is a conical N-stable variety we have ag_{a}zg_{a}^{−1} ∈ V_{𝔣¯1}^{r} (L). That is,

∑_{j=1}^{u} (x_{j}e_{m−j+1,m+j} + a^{2}y_{j}e_{m+j,m−j+1})+ ∑_{j=u+1}^{v} y_{j}e_{m+j,m−j+1}+ ∑_{j=v+1}^{w} x_{j}e_{m−j+1,m+j} ∈ V_{𝔣¯1}^{r} (L) .

Iterating, we have

∑_{j=1}^{u} (x_{j}e_{m−j+1,m+j} + a^{2t}y_{j}e_{m+j,m−j+1})+ ∑_{j=u+1}^{v} y_{j}e_{m+j,m−j+1}+ ∑_{j=v+1}^{w} x_{j}e_{m−j+1,m+j} ∈ V_{𝔣¯1}^{r} (L)

for all integers t > 0. By taking 0 < a < 1 and using that V_{𝔣¯1}^{r} (L) is a closed variety we see that in the limit we have

z′ := ∑_{j=1}^{u} x_{j}e_{m−j+1,m+j} + ∑_{j=u+1}^{v} y_{j}e_{m+j,m−j+1} + ∑_{j=v+1}^{w} x_{j}e_{m−j+1,m+j} ∈ V_{𝔣¯1}^{r} (L) .

However, [z′, z′] = 0 and so z′ lies in the associated variety for L defined by Duflo-Serganova in [DS05]. But [DS05, Theorem 5.3] then implies that w ≤ ℓ. That is, z has m − w ≥ m − ℓ matching pairs of coordinates which are zero and, hence, z ∈ Σ_{m}V (m − ℓ, m − ℓ, m − ℓ). □

7.4. Lower Bound for V_{𝔣¯1}^{r}(H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}). Let 𝔤 = 𝔤𝔩(m|n) with m ≤ n, and fix n − m ≤ k ≤ n. As in Section 6.5, let P = P_{k} be the parabolic subgroup of block upper triangular matrices which has Levi subgroup isomorphic to GL(m|n − k) × GL(k). In this section we determine a lower bound for the 𝔣¯1-rank variety of H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗} when λ ∈ X_{0}^{+} satisfies certain additional conditions.

It is convenient to define certain subspaces of 𝔤. Set

I = {1, …, m + k − n, m + n − k + 1, …, m + n} .

Let 𝔞 be the subspace spanned by the matrix units e_{i,j} with i, j ∈ I. Let 𝔟 be the subspace spanned by the matrix units e_{i,j} with at least one of i or j not in I. Then

𝔤 = 𝔞 ⊕ 𝔟  (7.4.1)

as vector spaces and 𝔞 is the Lie subsuperalgebra isomorphic to 𝔤𝔩(m + k − n|k) in the “corners” of 𝔤. Note that our assumption that n−m ≤ k ≤ n ensures that 0 ≤ m+k−n ≤ m.

Recall the parabolic ℤ-grading defined in Section 6.6 and that if γ is an integral weight, then we write deg(γ) for the degree of γ in the parabolic grading. For example, in the parabolic grading the module H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗} has deg(λ) as its highest non-zero degree. From this it follows that 𝔞 ∩ 𝔤_{1} ⊆ 𝔭_{1} acts trivially on [H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}]_{deg(λ)}.

Also, note that when x ∈ 𝔣¯1 ∩ 𝔭 is written as a sum according to the parabolic grading we have

x = x_{0} + x_{1},

with x_{0} and x_{1} have degrees 0 and 1, respectively.

Lemma 7.4.1. Let n − m ≤ k ≤ n and λ ∈ X_{0}^{+} such that λ_{i} = −λ_{2m+1−i} for i = 1, …, m. Let x ∈ 𝔣¯1 ∩ 𝔭. Then

[H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}]_{deg(λ)} ≅ Q ⊕ U

where the decomposition is as 𝔞¯0- and ⟨x_{0}⟩-modules.

Furthermore,

Q ≅ L_{𝔤𝔩(m+k−n)}(λ_{1}, …, λ_{m+k−n}) ⊠ L_{𝔤𝔩(k)}(λ_{m+n−k+1}, …, λ_{m+n})

as an 𝔞¯0 ≅ 𝔤𝔩(m + k − n) ⊕ 𝔤𝔩(k)-module and Q is a direct sum of trivial modules for ⟨x_{0}⟩.

Proof. We first introduce a refinement of the parabolic grading. Let a, b, and c be non-zero real numbers and set

h = diag(a, …, a, b, …, b, c, …, c) ∈ T

be the diagonal matrix with (m + k − n) a’s, (2n − 2k) b’s, and k c’s. The action of h on a weight module for T provides a ℤ × ℤ × ℤ-grading. As a matter of notation we write (deg_{a}(γ), deg_{b}(γ), deg_{c}(γ)) for the degree of an integral weight γ with respect to this grading. Note that by setting a = b = c^{−1} we recover the parabolic grading and that for any weight module M and integral weight γ we have M_{(deg_{a}(γ),deg_{b}(γ),deg_{c}(γ))} ⊆ M_{deg(γ)}.

Using this grading (for suitably generic a, b, c) we see that under the adjoint action we have

𝔤_{(0,0,0)} ≅ 𝔤𝔩(m + k − n) ⊕ 𝔤𝔩(n − k|n − k) ⊕ 𝔤𝔩(k).

Let M = H^{0}(G/P, L(λ)^{∗})^{∗}. Using the ℤ × ℤ × ℤ-grading we have that

M_{(deg_{a}(λ),deg_{b}(λ),deg_{c}(λ))} ≅ L_{𝔤𝔩(m+k−n)}(λ_{1}, …, λ_{m+k−n})⊠ L_{𝔤𝔩(n−k|n−k)}(λ_{m+k−n+1}, …, λ_{m+n−k}) ⊠ L_{𝔤𝔩(k)}(λ_{m+n−k+1}, …, λ_{m+n}) (7.4.2)

as 𝔤_{(0,0,0)}-modules.

Observe that x_{0} is an element of the detecting subalgebra of 𝔤𝔩(n−k|n−k) ⊆ 𝔤_{(0,0,0)}. Our assumption on λ implies that L_{𝔤𝔩(n−k|n−k)}(λ_{m+k−n+1}, …, λ_{m+n−k}) is a simple 𝔤𝔩(n−k|n−k)- module with atypicality n − k (the maximum possible). Theorem 7.3.1 then implies that L_{𝔤𝔩(n−k|n−k)}(λ_{m+k−n+1}, …, λ_{m+n−k}) is not projective as a U(⟨x_{0}⟩)-module. That is, we can find a non-zero vector

v_{0} ∈ L_{𝔤𝔩(n−k|n−k)}(λ_{m+k−n+1}, …, λ_{m+n−k})

such that v_{0} spans an ⟨x_{0}⟩-trivial direct summand and we have

L_{𝔤𝔩(n−k|n−k)}(λ_{m+k−n+1}, …, λ_{m+n−k}) = ℂv_{0} ⊕ J  (7.4.3)

as ⟨x_{0}⟩-modules.

Set

A = L_{𝔤𝔩(m+k−n)}(λ_{1}, …, λ_{m+k−n}) ⊠ ℂv_{0} ⊠ L_{𝔤𝔩(k)}(λ_{m+n−k+1}, …, λ_{m+n})
B = L_{𝔤𝔩(m+k−n)}(λ_{1}, …, λ_{m+k−n}) ⊠ J ⊠ L_{𝔤𝔩(k)}(λ_{m+n−k+1}, …, λ_{m+n})

Combining (7.4.3) with (7.4.2) we obtain

M_{(deg_{a}(λ),deg_{b}(λ),deg_{c}(λ))} ≅ A ⊕ B  (7.4.4)

as 𝔞¯0 and ⟨x_{0}⟩-modules. Now set Q equal to A and U equal to the sum of B and all trigraded components of M_{deg(λ)} excluding M_{(deg_{a}(λ),deg_{b}(λ),deg_{c}(λ))}. Then M_{deg(λ)} ≅ Q ⊕ U as vector spaces. The fact that x_{0} and 𝔞¯0 are of degree (0, 0, 0) along with our choice of A and B makes it straightforward to see that this decomposition has the asserted properties. □

Let 𝔲^{−} be the opposite nilradical for our fixed parabolic subalgebra 𝔭. Set 𝔞^{−} = 𝔞 ∩ 𝔲^{−} and 𝔟^{−} = 𝔟 ∩ 𝔲^{−}. In the PBW basis for U(𝔲^{−}) given by all monomials in the matrix units, set S to be the subspace of U(𝔲^{−}) spanned by all monomials containing at least one matrix unit from 𝔟^{−}. Since 𝔲^{−} = 𝔞^{−} ⊕ 𝔟^{−} the PBW theorem for Lie superalgebras implies that the following decomposition holds as vector spaces:

U(𝔲^{−}) ≅ U(𝔞^{−}) ⊕ S.  (7.4.5)

We can now provide a lower bound for the 𝔣¯1-rank variety of H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}.

Theorem 7.4.2. Let n − m ≤ k ≤ n and let λ ∈ X_{0}^{+} be a dominant integral weight such that λ_{i} = −λ_{2m+1−i} for i = 1, …, m. Furthermore, assume λ_{i} − λ_{i+1} > (m + k − n)k + mn for i = 1, …, m − 1, m + 1, …, m + n. Then

𝔣¯1 ∩ 𝔭 ⊆ V_{𝔣¯1}^{r} (H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}) .  (7.4.6)

Proof. Let x ∈ 𝔣¯1 ∩ 𝔭. From the rank variety description it suffices to show that M := H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗} is not projective as a U(⟨x⟩)-module. By the previous lemma we may write

M_{deg(λ)} ≅ Q ⊕ U.

Consider U(𝔞)Q ⊆ M. With respect to 𝔞 U(𝔞)Q is generated by a highest weight vector of weight γ := (λ_{1}, …, λ_{m+k−n}|λ_{m+n−k+1}, …, λ_{m+n}) and there exists a surjective 𝔞homomorphism from the Kac module K_{𝔤𝔩(m+k−n|k)}^{+}(γ) onto U(𝔞)Q. The key observation is that Theorem 6.6.2 (with d = (m + k − n)k) implies that this map is also injective. Thus

U(𝔞)Q ≅ K_{𝔤𝔩(m+k−n|k)}^{+}(γ)

as 𝔞-modules. Furthermore, our assumption on λ ensures that γ has atypicality m + k − n (the maximum possible) for 𝔞.

Since M = U(𝔲^{−})M_{deg(λ)}, we can apply Theorem 6.6.2, Lemma 7.4.1, and (7.4.5) to see that we have a vector space decomposition:

M = U(𝔞)Q ⊕ U(𝔞)U ⊕ (SM_{deg(λ)}) .  (7.4.7)

We now consider this decomposition under the action of x_{0} and x_{1}. A direct calculation shows that for i = 0, 1 we have [x_{i}, 𝔞] ⊆ 𝔞 and [x_{i}, 𝔟] ⊆ 𝔟. This plus the fact that Q, U, M_{deg(λ)} are ⟨x_{i}⟩-modules implies this is a decomposition into ⟨x_{i}⟩-modules and, hence, into ⟨x⟩-modules.

Since U(𝔞)Q is isomorphic to a Kac module of maximal atypicality for 𝔞, and x_{1} lies in the detecting subalgebra of 𝔞, it follows from the proof of [BKN12, Theorem 6.2.1] that U(𝔞)Q has a direct sum decomposition as an ⟨x_{1}⟩-module with one direct summand being trivial. Furthermore, since [x_{0}, 𝔞] = 0 and x_{0} acts as zero on Q, we have that ⟨x_{0}⟩ acts trivially on U(𝔞)Q. From this it follows that the decomposition of U(𝔞)Q is also as ⟨x_{0}⟩-modules and, hence, as ⟨x⟩-modules. That is, U(𝔞)Q contains a trivial direct summand as an ⟨x⟩-module and hence, by (7.4.7), so does M. Therefore, M is not projective as a U(⟨x⟩)-module and so x ∈ V_{𝔣¯1}^{r}(M) as desired. □

7.5. Recall from Example 4.6.5 that 𝔤 = 𝔤𝔩(m|n) admits a ℤ-grading 𝔤 = 𝔤_{−1} ⊕𝔤_{0} ⊕𝔤_{1}. We continue the setup of Section 6.5 with G = GL(m|n) and with G_{−1} as the closed subgroup of G with Lie(G_{−1}) = 𝔤_{−1}. In particular, the parabolic subgroup P ⊆ G consists of block upper triangular matrices with Levi subgroup L = GL(m|n−k)×GL(k) for some 0 ≤ k ≤ n. Then P = L ⋉ U, where L is the Levi subgroup and U is a unipotent subgroup. Let Z be the closed subgroup P ∩ G_{−1} and 𝔷 be the Lie superalgebra associated to Z.

The following theorem provides a relationship between supports for a P -module N and the G-module ind_{P}^{G} N (cf. [NPV02, (5.4.1) Theorem]). Note that 𝔷 and 𝔤_{−1} are abelian Lie superalgebras concentrated in odd degree and, hence, the support data V_{𝔷}(−) and V_{𝔤_{−1}}(−) are as in Example 4.6.1 and Example 4.6.5.

Theorem 7.5.1. Let N be a finite dimensional P -module and M = ind_{P}^{G} N. Suppose that H^{j}(G/P, N) = R^{j} ind_{P}^{G} N = 0 for all j > 0. Then

(a) V_{𝔤_{−1}}(ind_{P}^{G} N) = G_{0} · V_{𝔷}(M, N).

(b) V_{𝔤_{−1}}(ind_{P}^{G} N) ⊆ G_{0} · V_{𝔷}(N).

Proof. Part (b) follows from part (a) since V_{𝔷}(M, N) ⊆ V_{𝔷}(N). This inclusion also shows that

V_{𝔷}(M, N) ⊆ V_{𝔷}(M) ⊆ V_{𝔤_{−1}}(M).

Since V_{𝔤_{−1}}(M) is G_{0}-invariant, it follows that

G_{0} · V_{𝔷}(M, N) ⊆ V_{𝔤_{−1}}(M).

We now consider the spectral sequence (6.4.3). Let T = Ext_{𝔷}^{•}(M, N). One can apply [NPV02, (5.2.1) Proposition] to show that the ideal I in R = H^{•}(𝔤_{−1}, ℂ) which defines G_{0} · V_{𝔷}(M, N) annihilates E_{2}^{m,n} for all m ≥ 0.

Therefore, I ⊆ Ann_{R} E_{∞}^{•,•}. From the Grothendieck vanishing theorem, R^{j} ind_{P_{0}}^{G_{0}} N = 0 for j > s = dim G_{0}/P_{0}. Consequently, Ext_{𝔤_{−1}}^{•}(ind_{P}^{G} N, ind_{P}^{G} N) has an R-stable filtration 0 = F^{0} ⊆ F^{1} ⊆ ⋯ ⊆ F^{s} = Ext_{𝔤_{−1}}^{•}(ind_{P}^{G} N, ind_{P}^{G} N) where F^{i}/F^{i−1} = ⨁_{j} E_{∞}^{s−i+1,j}.

Observe that I · F^{i} ⊆ F^{i−1}, thus I^{N} annihilates Ext_{𝔤_{−1}}^{•}(ind_{P}^{G} N, ind_{P}^{G} N) which proves that V_{𝔤_{−1}}(ind_{P}^{G} N) ⊆ G_{0} · V_{𝔷}(ind_{P}^{G} N, N). □

7.6. The previous theorem allows us to give us a constraint on the 𝔣¯1 rank variety of H^{0}(G/P, N) for certain P -modules N.

Theorem 7.6.1. Fix n − m ≤ k ≤ n and let P be the parabolic defined above. Let N be a finite dimensional P -module such that R^{j} ind_{P}^{G} N = 0 for j > 0. Then,

V_{𝔣¯1}^{r}(H^{0}(G/P, N)) ⊆ Σ_{m}V (0, m − (n − k), 0).

Proof. Let

z ∈ V_{𝔣¯1}^{r}(H^{0}(G/P, N))

and write z = z_{−1} + z_{1} where z_{±1} ∈ 𝔤_{±1}.

For any fixed non-zero a ∈ ℝ let g_{a} ∈ G¯0 be the group element given by the diagonal matrix

g_{a} = Diag (d_{1}, …, d_{m+n})

with

d_{i} = \begin{cases}a, & i = 1, …, m; \\ a^{−1}, & i = m + 1, …, m + n; \end{cases} .

Since V_{𝔣¯1}^{r}(H^{0}(G/P, N)) is a conical N-stable variety we have a^{2}g_{a}zg_{a}^{−1} ∈ V_{𝔣¯1}^{r}(H^{0}(G/P, N)). That is,

a^{2}g_{a}zg_{a}^{−1} = z_{−1} + a^{4}z_{1} ∈ V_{𝔣¯1}^{r}(H^{0}(G/P, N))

Iterating, we have

a^{2t}g_{a}^{t}zg_{a}^{−t} = z_{−1} + a^{4t}z_{1} ∈ V_{𝔣¯1}^{r}(H^{0}(G/P, N))

for all integers t > 0. By taking 0 < a < 1 and using that V_{𝔣¯1}^{r}(H^{0}(G/P, N)) is a closed variety we see that in the limit we have

z_{−1} ∈ V_{𝔣¯1}^{r}(H^{0}(G/P, N)).

But then z_{−1} ∈ V_{𝔤_{−1}}(H^{0}(G/P, N)) and so it follows from Theorem 7.5.1(b) that

z_{−1} ∈ G_{0} · V_{𝔷}(N) ⊆ G_{0} · 𝔷.

However, since 𝔷 consists matrices of rank at most n − k, so too must the elements of G_{0} · 𝔷. For z_{−1} to have rank at most n − k it follows that z ∈ Σ_{m}V (0, m − (n − k), 0). □

7.7. The case when s = p = 0. We can now realize Σ_{m}V (0, t, 0) for any 0 ≤ t ≤ m.

Theorem 7.7.1. Let 0 ≤ t ≤ m and let λ ∈ X_{0}^{+} be a dominant integral weight with λ_{i} = −λ_{2m+1−i} for i = 1, …, m and λ_{i} − λ_{i+1} > t(n − m + t) + mn for i = 1, …, m − 1, m + 1, …, m + n − 1. Let P be the parabolic subgroup of GL(m|n) which is block upper triangular and has Levi subgroup isomorphic to GL(m|m − t) × GL(n − m + t). Then

V_{𝔣¯1}^{r} (H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}) = Σ_{m}V (0, t, 0).

Proof. Set M = H^{0}(G/P, L_{𝔭}(λ)^{∗}). Since the higher derived functors vanish by Theorem 6.5.3 and since V_{𝔣¯1}^{r}(M^{∗}) = V_{𝔣¯1}^{r}(M), we can use Theorem 7.6.1 to obtain the inclusion

V_{𝔣¯1} (H^{0}(G/P, L_{𝔭}(λ)^{∗})^{∗}) ⊆ Σ_{m}V (0, t, 0).

On the other hand, if we set k = n − m + t in Theorem 7.4.2, then we have the inclusion

𝔣¯1 ∩ 𝔭 ⊆ V_{𝔣¯1}^{r}(M).

But since V (0, t, 0) = 𝔣¯1 ∩ 𝔭 and V_{𝔣¯1}^{r}(M) is stable under the action of Σ_{m}, we then have

Σ_{m}V (0, t, 0) ⊆ V_{𝔣¯1}^{r}(M).

This proves the desired equality. □

7.8. The case when t = p = 0. If we set ī ∈ ℤ_{2} by the rule ī = ¯0 for i = 1, …, m and ī = ¯1 for i = m + 1, …, m + n, then we can define an automorphism of 𝔤𝔩(m|n) by e_{i,j} ↦ −(−1)^{ī(ī+j̄)}e_{j,i}. Given a finite dimensional 𝔤-module M let M^{τ} be the 𝔤-module M^{∗} with the action twisted by this automorphism.

Theorem 7.8.1. If 0 ≤ s ≤ m, then there exists a finite dimensional 𝔤-module M such that

V_{𝔣¯1}^{r}(M) = Σ_{m}V (s, 0, 0).

Proof. This follows from Theorem 7.7.1 and the observation that for any finite dimensional 𝔤-module M we have the identity

V_{𝔣¯1}^{r}(M^{τ}) = τ (V_{𝔣¯1}^{r}(M)) .
□

7.9. The case when s, t ≥ 0 and p = 0. In what follows we make extensive use of the fact that 𝔣-support varieties are a support data as defined in Section 2.4. We also use the fact that if G is a group acting on a set and A, B are subsets of that set then

GA ∩ GB = G(A ∩ GB) = G(GA ∩ B).

Proposition 7.9.1. Fix s, t ≥ 0. The variety Σ_{m}V (s, t, 0) is realizable as the support of a module in F.

Proof. The proof is by induction on t. The base case, t = 0, is Theorem 7.8.1. So let t > 0 and assume the result is true for Σ_{m}V (s, t − 1, 0). Consider first

U := Σ_{m}V (s, t − 1, 0) ∩ Σ_{m}V (0, t, 0).

This variety is realizable as the support of a tensor product, since the first term on the right is realizable by the induction hypothesis, and the second by Theorem 7.8.1. Viewing the intersection as imposing the condition of the second variety on the first, we need one more Y_{j} to vanish, and this can either overlap with an X_{i} that is already required to vanish, or not. Thus

U = Σ_{m}V (s, t, 0) ∪ Σ_{m}V (s, t, 1).

For clarity in what follows we will write Z_{i} = X_{i}Y_{i} for i = 1, …, m. Next consider

W := Σ_{m}V (s, t − 1, 0) ∩ Σ_{m}Z(Z_{s+t}Z_{s+t+1} … Z_{m}).

Again, W is realizable as the support of a tensor product, by the induction hypothesis and Section 7.2. If we write p_{m−s−(t−1)} for the degree m − s − (t − 1) elementary symmetric polynomial in the variables Z_{1}, …, Z_{m}, then Σ_{m}Z(Z_{s+t}Z_{s+t+1} … Z_{m}) = Z(p_{m−s−(t−1)}). We then have

W = Σ_{m}V (s, t − 1, 0) ∩ Z(p_{m−s−(t−1)})
= Σ_{m}V (s, t, 0) ∪ Σ_{m}V (s + 1, t − 1, 0),

where the second equality follows from the fact that for p_{m−s−(t−1)} to equal zero we must have strictly more than m − s − (t − 1) Z_{i}’s vanishing. Finally, U ∩ W is realizable as the support of a tensor product and a calculation of set intersections shows that

U ∩ W = V (s, t, 0).
□

7.10. The case when s, t ≥ p > 0.

Proposition 7.10.1. Fix s, t ≥ p > 0. The variety Σ_{m}V (s, t, p) is realizable as the support of a module in F.

Proof. First consider the variety V (p, p, p) where p > 0. Recall from Theorem 7.3.1 that V_{𝔣¯1}^{r}(L(λ)) = Σ_{m} · V (p, p, p) for a simple module L(λ) of highest weight λ having atypicality m − p.

Now consider the general case V (s, t, p) with s, t ≥ p > 0. Observe that

Σ_{m}V (s, t, p) = Σ_{m}V (s, t − p, 0) ∩ Σ_{m}V (0, t, 0) ∩ Σ_{m}V (p, p, p).

But Σ_{m}V (s, t−p, 0) is realizable as the support of a module in F by Proposition 7.9.1; likewise Σ_{m}V (0, t, 0) is realizable by Theorem 7.7.1; and, as we noted above, so is Σ_{m}V (p, p, p). Therefore, Σ_{m}V (s, t, p) is the support of the tensor product. □

We have now proven that all varieties of the form Σ_{m}V (s, t, p) are realizable. Any variety V as in (7.1.1) in which r = 0 is conjugate to some V (s, t, p) under the action of Σ_{m}. Therefore, we have in fact realized all varieties of the form Σ_{m}V with V as in (7.1.1) and with r = 0.

7.11. Mixed Coordinate and Weight Zero Functions. In this subsection, we will verify the realization property in the case where V is given by the vanishing of one or more coordinate functions and one or more weight zero polynomials on 𝔣¯1.

Proposition 7.11.1. Let V = V (s, t, 0) ∩ Z(g_{1}, …, g_{r}), where g_{1}, …, g_{r} are weight zero polynomials. Then the variety Σ_{m}V is realizable as the support of a module in F.

Proof. For i = 1, …, m, set Z_{i} = X_{i}Y_{i} and observe that, since they are weight zero, g_{1}, …, g_{r} are polynomials in Z_{1}, …, Z_{m}. However, in V (s, t, 0) we have X_{1} = ⋯ = X_{s} = Y_{s+1} = ⋯ = Y_{s+t} = 0, so we may drop any terms in the g_{k} that involve Z_{1}, …, Z_{s+t} without changing V, and thus we may assume that the g_{k} are polynomials in Z_{s+t+1}, …, Z_{m}.

Set

U = Σ_{m}Z(Z_{1}, …, Z_{s+t}, g_{1}, …, g_{r}).

Then U is realizable as the support of a module in F by Section 7.2, and Σ_{m}V ⊆ U. Set

W = Σ_{m}V (s, t, 0).

Then W is realizable by Proposition 7.9.1 and clearly Σ_{m}V ⊂ W. Consequently U ∩ W is realizable by the tensor product property and Σ_{m}V ⊆ U ∩ W. Therefore it suffices to prove

U ∩ W ⊆ Σ_{m}V.

We first observe that since Z_{i} = 0 if and only if X_{i} = 0 or Y_{i} = 0, and since we can permute the indices 1, …, s + t without affecting the polynomials g_{1}, …, g_{r}, we can write

U = Σ_{m} [(⋃_{j=0}^{s+t} V (j, s + t − j, 0)) ∩ Z(g_{1}, …, g_{r})]

= ⋃_{j=0}^{s+t} Σ_{m} (V (j, s + t − j, 0) ∩ Z(g_{1}, …, g_{r})) =: ⋃_{j=0}^{s+t} U_{j}.

From this we see that to prove the claim it suffices to show U_{j} ∩ W ⊂ Σ_{m}V for all j. Assume j ≤ s; the proof for j > s is similar and is left to the reader.

Write

U_{j} ∩ W = Σ_{m}(V (j, s + t − j, 0) ∩ Z(g_{1}, …, g_{r}) ∩ Σ_{m}V (s, t, 0)).

Viewing the intersection as imposing the conditions of W on V (j, s + t − j, 0), we are now required to have s coordinates X_{i} to be zero. As those in the range 1, …, j are already zero, we must have an additional number, say u, in the range j + 1, …, s + t and, say v, in the range s + t + 1, …, m. We note that then j + u + v = s. We also observe that those which are in the range 1 ≤ i ≤ s + t can have their positions permuted without changing g_{1}, …, g_{r}. Combining these observations we see that

U_{j} ∩ W ⊆ ⋃_{u=0}^{s−j} U_{j,u},

where

U_{j,u} := Σ_{m}(V (j + u, s + t − j, u) ∩ Z(X_{s+t+1}, …, X_{s+t+v}) ∩ Z(g_{1}, …, g_{r})).

Here we assume for simplicity of notation that the v coordinates which involve indices in the range s + t + 1, …, m are X_{s+t+1}, …, X_{s+t+v}. An identical argument applies in the general case.

Let π ∈ Σ_{m} be the product of the transpositions (j + u + i, s + t + i) for 1 ≤ i ≤ v. Recalling that g_{1}, …, g_{r} are polynomials in Z_{s+t+1}, …, Z_{m} we see that while πg_{1}, … πg_{r} are not the same polynomials as g_{1}, …, g_{r}, we do have

U_{j,u} = Σ_{m}(V (j + u, s + t − j, u) ∩ Z(X_{s+t+1}, …, X_{s+t+v}) ∩ Z(πg_{1}, …, πg_{r})). (7.11.1)

But now applying π to the entire intersection in (7.11.1) and using that j + u + v = s, we see that

U_{j,u} ⊆ Σ_{m}V.

This proves the claim and the theorem.

□

7.12. General Case. Finally we can prove realization in general. That is, we will have finally verified (2.4.8) holds for the support data V̂ introduced in Section 5.

Theorem 7.12.1. Let m ≥ s, t ≥ p ≥ 0 and let g_{1}, …, g_{r} be homogeneous weight zero polynomials. Set V = V (s, t, p) ∩ Z(g_{1}, …, g_{r}). Then there exists a finite dimensional 𝔤-module M such that

V_{𝔣¯1}^{r}(M) = Σ_{m}V.

Proof. Let U = Σ_{m} (V (s, t − p, 0) ∩ Z(g_{1}, …, g_{r})), W = Σ_{m}V (p, p, p), Y = Σ_{m}V (0, t, 0). By Proposition 7.11.1 and Theorems 7.3.1 and 7.7.1, respectively, these varieties can be realized as the support varieties of finite dimensional 𝔤-modules. Their tensor product then realizes U ∩ W ∩ Y. An analysis similar to the proof of the previous result shows that Σ_{m}V = U ∩ W ∩ Y. □

References

[Aus74] Maurice Auslander, Representation theory of Artin algebras. II, Comm. Algebra 1 (1974), 269–310. MR 0349747 (50 #2240)

[Bal05] Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168. MR MR2196732

[Bal10] , Tensor triangular geometry, Proceedings of the International Congress of Mathematicians. Volume II (New Delhi), Hindustan Book Agency, 2010, pp. 85–112. MR 2827786 (2012j:18016)

[BCR96] D. J. Benson, Jon F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules. II, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 4, 597–615. MR MR1401950 (97f:20008)

[BCR97] D. J. Benson, Jon F. Carlson, and Jeremy Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80. MR MR1450996 (98g:20021)

[BIK08] Dave Benson, Srikanth B. Iyengar, and Henning Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 4, 573–619. MR 2489634 (2009k:18012)

[BIK11] David J. Benson, Srikanth B. Iyengar, and Henning Krause, Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), no. 3, 1643–1684. MR 2846489

[BIK12] David J. Benson, Srikanth Iyengar, and Henning Krause, Representations of finite groups: local cohomology and support, Oberwolfach Seminars, vol. 43, Birkhäuser/Springer Basel AG, Basel, 2012. MR 2951763

[BKN08] Irfan Bagci, Jonathan R. Kujawa, and Daniel K. Nakano, Cohomology and support varieties for Lie superalgebras of type W (n), Int. Math. Res. Not. IMRN (2008), Art. ID rnn115, 42. MR 2448087 (2009i:17033)

[BKN09] Brian D. Boe, Jonathan R. Kujawa, and Daniel K. Nakano, Cohomology and Support Varieties for Lie Superalgebras II, Proc. London Math. Soc. 98 (2009), no. 1, 19–44.

[BKN10] , Cohomology and support varieties for Lie superalgebras, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6551–6590. MR 2678986

[BKN11] , Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN (2011), no. 3, 696–724. MR 2764876

[BKN12] , Complexity for modules over the classical Lie superalgebra 𝔤𝔩(m|n), Compos. Math. 148 (2012), no. 5, 1561–1592. MR 2982440

[Bru03] Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra 𝔤𝔩(m|n), J. Amer. Math. Soc. 16 (2003), no. 1, 185–231 (electronic). MR 2003k:17007

[Del10] Ivo Dell’Ambrogio, Tensor triangular geometry and KK-theory, J. Homotopy Relat. Struct. 5 (2010), no. 1, 319–358. MR 2812924 (2012i:19008)

[DK85] Jiri Dadok and Victor Kac, Polar representations, J. Algebra 92 (1985), no. 2, 504–524. MR MR778464 (86e:14023)

[DS05] Michel Duflo and Vera Serganova, On associated variety for Lie superalgebras, 2005.

[Eve91] Leonard Evens, The cohomology of groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications. MR 1144017 (93i:20059)

[FP07] Eric M. Friedlander and Julia Pevtsova, Π-supports for modules for finite group schemes, Duke Math. J. 139 (2007), no. 2, 317–368. MR 2352134 (2008g:14081)

[GQS07] Gerhard Götz, Thomas Quella, and Volker Schomerus, Representation theory of 𝔰𝔩(2|1), J. Algebra 312 (2007), no. 2, 829–848. MR MR2333186 (2008e:17004)

[GS10] Caroline Gruson and Vera Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852– 892. MR 2734963 (2012a:17010)

[Hop87] Michael J. Hopkins, Global methods in homotopy theory, Homotopy theory (Durham, 1985), London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 73–96. MR 932260 (89g:55022)

[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114. MR 1388895 (98a:55017)

[Hum78] James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York, 1978, Second printing, revised. MR 499562 (81b:17007)

[Jan03] Jens Carsten Jantzen, Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR MR2015057 (2004h:20061)

[Kac77] Victor G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 58 #5803

[LNZ11] Gustav I. Lehrer, Daniel K. Nakano, and Ruibin Zhang, Detecting cohomology for Lie superalgebras, Adv. Math. 228 (2011), no. 4, 2098–2115. MR 2836115 (2012i:17021)

[Lor08] Martin Lorenz, Group actions and rational ideals, Algebra Number Theory 2 (2008), no. 4, 467– 499. MR 2411408 (2009i:16058)

[Lor09] , Algebraic group actions on noncommutative spectra, Transform. Groups 14 (2009), no. 3, 649–675. MR 2534802 (2010m:14060)

[Mac95] I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144 (96h:05207)

[Nee92a] Amnon Neeman, The chromatic tower for D(R), Topology 31 (1992), no. 3, 519–532, With an appendix by Marcel Bökstedt. MR 1174255 (93h:18018)

[Nee92b] , The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566. MR 1191736 (93k:18015)

[Nee01] , Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507 (2001k:18010)

[NPV02] Daniel K. Nakano, Brian J. Parshall, and David C. Vella, Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002), 15–49. MR 2003b:20063

[PS08] Julia Pevtsova and S. Paul Smith, Spectra of tensor triangulated categories,

https://www.msri.org/workshops/405/schedules/2846, 2008, accessed February 1, 2014.

[Ric97] Jeremy Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), no. 1, 149–170. MR 1462832 (98d:20058)

[Ser12] Vera Serganova, Structure and representation theory of Kac-Moody superalgebras, Highlights in Lie algebraic methods, Progr. Math., vol. 295, Birkhäuser/Springer, New York, 2012, pp. 65–102. MR 2866847 (2012k:17034)

[Zub06] A. N. Zubkov, Some properties of general linear supergroups and of Schur superalgebras, Algebra Logika 45 (2006), no. 3, 257–299, 375. MR 2289090 (2007m:20070)

Department of Mathematics, University of Georgia, Athens, GA 30602

E-mail address: brian@math.uga.edu

Department of Mathematics, University of Oklahoma, Norman, OK 73019

E-mail address: kujawa@math.ou.edu

Department of Mathematics, University of Georgia, Athens, GA 30602

E-mail address: nakano@math.uga.edu