# Rank varieties and for elementary supergroup schemes -points

To Jon F. Carlson on his 80th birthday

## Abstract

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic starting with a definition of a , generalising cyclic shifted subgroups of Carlson for elementary abelian groups and -point of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra -points where , has even degree and has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.

## Introduction

Carlson Reference 17 introduced two notions of variety for a finitely generated module over an elementary abelian One, the rank variety, is based on restrictions to cyclic shifted subgroups, while the other is a cohomological support variety. This theory was generalised to infinitely generated modules by Benson, Carlson and Rickard -group.Reference 6 by using cyclic shifted subgroups defined over extension fields where enough generic points exist.

The notion of rank variety was put in the more general context of a finite group scheme over a field by Friedlander and Pevtsova Reference 23, through the theory of A -points. is a flat algebra homomorphism from -point to where , is an extension field of There is an equivalence relation on . and in the case of an elementary abelian group, once we fix a minimal generating set for the radical ideal of the group algebra, there is exactly one shifted subgroup up to scalar multiple in each equivalence class over a large enough field. -points,

In a parallel development a theory of support varieties based on cohomology, and applicable in a rather broad context, was developed in Reference 7Reference 8. Combining those ideas with the theory of eventually led to a classification of the localising tensor ideal, and Hom closed colocalising, subcategories of the stable module category of a finite group scheme by the current authors -pointsReference 12.

In Reference 13 we began a program to extend all these results to the world of supergroup schemes. In that work we identified a family of elementary supergroup schemes and proved that the projectivity of modules over a unipotent supergroup scheme can be detected by its restrictions to the elementary ones, possibly defined over extensions fields. This is in analogy with ChouinardтАЩs theorem for finite groups.

In this paper we develop a theory of for the elementary supergroup schemes -points introduced in Reference 13, and classify the localising subcategories of its stable module category. This feeds into the proof of a similar classification for finite unipotent supergroup schemes, presented in Reference 14. It transpires that rather than flat maps from we have to consider the , -algebra

where is in even degree and is in odd degree, and maps of finite flat dimension

The basic new result in our work is that detect projectivity: a -points -module is projective if, and only if, for each -point as above the restriction of the -module to has finite flat dimension; a corresponding statement involving rather than , also holds. ,

From the point of view of commutative algebra, the group algebra is a complete intersection, and for such rings Avramov Reference 1 has developed a theory of support sets for modules, as yet another extension of CarlsonтАЩs work. Our proof of the detection theorem for goes by relating to support sets over -points However, we need a version of the theory that applies also to infinite dimensional modules. This is presented in Appendix .A.

With the detection theorem in hand, we put an equivalence relation on analogous to the one in -points,Reference 23, and exhibit an explicit set of representatives of these equivalence classes, up to linear multiples, analogous to the cyclic shifted subgroup approach in Reference 6Reference 17. These form a projective space over and give rise to various notions of support for modules over , This allows us to classify the localising subcategories of the stable category of . For a finite dimensional -modules. -module one also has a rank variety, in exact analogy with CarlsonтАЩs theory of rank varieties for elementary abelian groups. This variety is determined by a rank condition on an explicitly defined matrix, and in particular it is a closed subset.

### Outline of the paper

The basic definitions concerning supergroup schemes, including the structure of the elementary ones, are recalled in Section 1. In Section 2 we record the desired statements concerning support sets of modules over by specialising results for general complete intersections established in Appendix ,A. The link to and a proof of the detection theorem discussed above, is presented in Section -points,4. The proofs again require quite substantial input from the homological theory of complete intersection rings, and also basic facts about the representation theory of the algebra recalled in Section 3. From this point on, the narrative unfolds in the expected way: Rank varieties for finite dimensional modules are introduced in Section 5, leading to an explicit method for computing them. The equivalence relation on is discussed in Section -points6. Section 7 brings in the cohomological notions of support and cosupport, culminating with the classification results.

## 1. Elementary supergroup schemes

Throughout will be a field of positive characteristic A .*superalgebra* will mean a algebra, and a -graded*graded* module over such an algebra will be assumed to be a left module. The category of graded modules over a superalgebra -graded is denoted these are allowed to be infinitely generated. Naturally the morphisms in this category are ; maps that preserve the grading. The full subcategory of finitely generated ones is denoted -linear When . is a graded -module, denotes the module with the zero and one components swapped. For the action of an element , on is given by

A superalgebra is *commutative* if for all in .

An *affine supergroup scheme* over is a covariant functor from commutative superalgebras to groups, whose underlying functor to sets is representable. If is a supergroup scheme its *coordinate ring* is the representing object. By applying YonedaтАЩs lemma to the group multiplication and inverse maps, it is a commutative Hopf superalgebra. This gives a contravariant equivalence of categories between affine supergroup schemes and commutative Hopf superalgebras.

A *finite* supergroup scheme is an affine supergroup scheme whose coordinate ring is finite dimensional. In this case, the dual is a finite dimensional cocommutative Hopf superalgebra called the *group ring* of This gives a covariant equivalence of categories between finite supergroup schemes and finite dimensional cocommutative Hopf superalgebras. We denote by . the *stable module category* which is obtained from by annihilating all projective modules. Note that this carries the structure of a triangulated category since is a self-injective algebra. We write for the full subcategory of finite dimensional modules.

Following Reference 13 we say that a finite supergroup scheme over is *elementary* if it is isomorphic to a quotient of where , is the Witt elementary, explicitly described in Reference 13, Definition 3.3. The main theorem of that paper states that if is a unipotent finite supergroup scheme over then a -module is projective if and only if, for all extension fields and all elementary sub-supergroup schemes , of the module , is a projective It also gives a similar condition for the nilpotence of an element of cohomology. -module.

We will be concerned *only* with the algebra structure of and the existence of a comultiplicaton; the explicit formula for the latter plays no role. Nevertheless, here is a brief description of the supergroup schemes and their finite quotients.

Let be the Frobenius kernel of the additive group scheme th We denote by . the supergroup scheme with the group algebra with generator in odd degree and primitive. Let be the affine group scheme of Witt vectors of length and let , be the Frobenius kernel of th Hence, . is a finite connected group scheme of height Then the finite group scheme . is defined as a quotient

The *Witt elementary* super group is a finite supergroup scheme determined uniquely by the following two properties:

- (1)
It fits into an extension ,

- (2)
.

See Reference 13, ┬з8 for details.

The quotients of can be completely classified using the theory of Dieudonn├й modules and, up to isomorphism, fall into one of the following classes:

- (i)
with ,

- (ii)
with ,

- (iii)
with , , or ,

- (iv)
with , and .

The last family involves supergroups which are quotients of but have the same group algebra structure as In the present work we are only concerned with the algebra structure of . so we do not distinguish between cases (iii) and (iv). Therefore, ignoring the comultiplication, the group algebra of interest is one of the following: ,

where the have degree and has degree The first case occurs when the supergroup scheme is a group scheme. For these the representation theory has been analysed in detail in .Reference 6Reference 9Reference 17Reference 23.

Our goal is to write down suitable analogues of the main theorems of these papers for elementary supergroup schemes with group algebras of the second and third form. Among these, the third one presents the most challenges and is the focus of the bulk of this work. The second one is discussed in Section 8.

Henceforth will be, as before, a field of positive characteristic and the supergroup scheme with group algebra

with and , and , .

## 2. Support sets

In this section we describe supports sets of following the general theory for complete intersections developed in Appendix -modules,A.

We write for the of elements, -tuples of , and , for the non-zero elements of modulo scalar multiplication. The image of in is denoted .

For a singly or doubly graded ring we write for the set of homogeneous prime ideals other than the maximal ideal of non-zero degree elements, topologised with the Zariski topology.

Next we discuss cohomology rings, starting with a brief remark about graded-commutativity of bigraded rings.

### Cohomology of

The cohomology ring of was recorded in Reference 13, Theorems 9.9 and 9.10. It takes the form

where the degrees are and , , Here, the first degree is cohomological and the second comes from the . on -grading The numbering is chosen so that . and restrict to zero on the subalgebra generated by if and , has non-zero restriction to the subalgebra generated by .

The nil radical of is generated by The quotient modulo this ideal is .

This ring has the property that the internal degree of a non-zero element is congruent to its cohomological degree modulo two.

It follows that if we take the of this ring as a doubly graded ring or as a singly graded ring by ignoring the internal degree, we get the same homogeneous prime ideals, that is to say, the natural map is a homeomorphism:

Moreover the inclusion

Thus

### Cohomology of

Fix

where the

In the notation of Equation 2.11 and Equation 2.12, and for

Let

So one gets a map

When

## 3. The algebra

In this section we discuss modules over the superalgebra

where

### Cohomology

The ring

is well-knownтАФsee Reference 28, Theorem 5тАФand easy to compute using the minimal resolution of

with

We are interested in the representation theory of *G-projective modules*, where тАЬGтАЭ stands for тАЬGorensteinтАЭ. In what follows, we speak of тАЬG-projectivesтАЭ rather than MCM modules, for consistency.

### G-projective modules

Recall that the ring

Let