STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

arXiv:1505.06628v2 [math.RT] 6 Jan 2017

DAVE BENSON, SRIKANTH B. IYENGAR, HENNING KRAUSE
AND JULIA PEVTSOVA
Abstract. We introduce the notion of π-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group,
we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding
classification for finite group schemes.

Contents
Introduction
1. Finite group schemes
2. π-points
3. π-cosupport and π-support
4. Finite groups
5. Cohomological support and cosupport
6. Stratification
References

1
2
4
6
10
13
15
16

Introduction
The theory of support varieties for finitely generated modules over the group
algebra of a finite group began back in the nineteen eighties with the work of Alperin
and Evens [1], Avrunin and Scott [2], Carlson [14, 15], among others. An essential
ingredient in its development was Carlson’s anticipation that for elementary abelian
p-groups the cohomological definition of support, which takes its roots in Quillen’s
fundamental work on mod p group cohomology [22], gave the same answer as the
“rank” definition through restriction to cyclic shifted subgroups.
To deal with infinitely generated modules, Benson, Carlson and Rickard [6] found
that they had to introduce cyclic shifted subgroups for generic points of subvarieties,
defined over transcendental extensions of the field of coefficients. Correspondingly,
all the homogeneous prime ideals in the cohomology ring are involved, not just the
maximal ones. This enabled them to classify the tensor ideal thick subcategories
of the stable category stmod(kG) of finitely generated kG-modules [7].
It seemed plausible that one should be able to use modifications of the same
techniques to classify the tensor ideal localising subcategories of the stable category
StMod(kG) of all kG-modules, but there were formidable technical obstructions to
realising this, and it was not until more than a decade later that this was achieved by
Date: 4th January 2016.
2010 Mathematics Subject Classification. 16G10 (primary); 20C20, 20G10 20J06 (secondary).
Key words and phrases. cosupport, stable module category, finite group scheme, localising
subcategory, support, thick subcategory.
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BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

the first three authors of this paper [10], using a rather different set of ideas than the
ones in [7]. The basic strategy was a series of reductions, via changes of categories,
that reduced the problem to that of classifying the localising subcategories of the
derived category of differential graded modules over a polynomial ring, where it was
solved using methods from commutative algebra. A series of papers [8, 9, 10, 11],
established machinery required to execute this strategy.
In this paper we give an entirely new, and conceptually different, proof of the
classification of the tensor ideal localising subcategories of StMod(kG) from [10]. It
is closer in spirit to the proof of the classification of the tensor ideal thick subcategories of stmod(kG) from [7], and rooted essentially in linear algebra. The crucial
new idea is to introduce and study π-cosupports for representations.
The inspiration for this comes from two sources. The first is the theory of
π-points developed by Friedlander and the fourth author [18] as a suitable generalisation of cyclic shifted subgroups. Whereas Carlson’s original construction only
applied to elementary abelian p-groups, and required an explicit choice of generators of the group algebra, π-points allow for a “rank variety” description of the
cohomological support for any finite group scheme; see [18, Theorem 3.6] and Section 2 of this paper. Based on this, in [18] the π-support of a module over a finite
group scheme G defined over a field k is introduced and used to classify the tensor
ideal thick subcategories of stmod(kG) (with an error corrected in a sequel [12] to
this paper). But the tensor ideal localising subcategories of StMod(kG) remained
inaccessible by these techniques alone, even for finite groups.
What is required is a π-version of the notion of cosupports introduced in [11].
The relevance of π-cosupport is through the following formula for the module of
homomorphisms, proved in Section 3. For any finite group scheme G over k, and
kG-modules M and N , there is an equality
π- cosuppG (Homk (M, N )) = π- suppG (M ) ∩ π- cosuppG (N ).
To be able to apply this formula to cohomological cosupport developed in [9] one
needs to identify the two notions of cosupport. Our strategy for making this identification is to prove that π-cosupport detects projectivity: a kG-module is projective
if and only if its π-cosupport is empty. The desired classification result would then
follow from general techniques developed in [12].
In Section 4 we prove such a detection theorem for projectivity for finite groups.
Besides yielding the desired classification theorem for StMod(kG) from [10], it implies that cohomological support and cosupport coincide with π-support and πcosupport, respectively. This remarkable fact is a vast generalisation of Carlson’s
original anticipation. The different origins of the two notions are reflected in the
fact that phenomena that are transparent, or at least easy to detect, for one may
be rather opaque and difficult to verify for the other. See Section 5 for illustrations.
The corresponding detection theorem for arbitrary finite group schemes has
turned out to be more challenging, and is dealt with in the sequel to this paper [12],
using a different approach, where again π-support and π-cosupport play a crucial
role. This brings us to the second purpose of this paper: To lay the groundwork for
the proof in [12]. For this reason parts of this paper are written in the language of
finite group schemes. However, we have attempted to present it in such a way that
the reader only interested in finite groups can easily ignore the extra generality.
1. Finite group schemes
This section summarises basic property of modules over affine group schemes;
for details we refer the reader to Jantzen [20] and Waterhouse [24].

STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

3

Let k be a field. An affine group scheme G over k is a functor from commutative
k-algebras to groups, with the property that, considered as a functor to sets, it is
representable as Homk-alg (R, −). The commutative k-algebra R has a comultiplication coming from the multiplicative structure of G, and an antipode coming from
the inverse. This makes R into a commutative Hopf algebra called the coordinate
algebra k[G] of G. Conversely, if k[G] is a commutative Hopf algebra over k then
Homk-alg (k[G], −) is an affine group scheme. This work concerns only affine group
schemes so henceforth we drop the qualifier “affine”.
A group scheme G over k is finite if k[G] is finite dimensional as a k-vector space.
The k-linear dual of k[G] is then a cocommutative Hopf algebra, called the group
algebra of G, and denoted kG.
We identify modules over a finite group scheme G with modules over its group
algebra kG; this is justified by [20, I.8.6]. Thus, we will speak of kG-modules (rather
than G-modules), and write Mod kG for the category of kG-modules.
Extending the base field. Let G be a finite group scheme over a field k. If K is
a field extension of k, we write K[G] for K ⊗k k[G], which is a commutative Hopf
algebra over K. This defines a group scheme over K denoted GK , and we have a
natural isomorphism KGK ∼
= K ⊗k kG.
For each kG-module M , we set
MK := K ⊗k M

and M K := Homk (K, M ),

viewed as KGK -modules. When K or M is finite dimensional over k, these are
related as follows.
Remark 1.1. For any kG-module M , there is a natural map of KGK -modules
Homk (K, k) ⊗k M −→ Homk (K, M ) .
This is a bijection when K or M is finite dimensional over k. Then M K is a direct
sum of copies of MK as a KGK -module, for Homk (K, k) is a direct sum of copies
of K, as a K-vector space.
The assignments M 7→ MK and M 7→ M K define functors from Mod kG to
Mod KGK that are left and right adjoint, respectively, to restriction of scalars along
the homomorphism of rings kG → KGK . The result below collects some basic facts
concerning how these functors interact with tensor products and homomorphisms.
In what follows, the submodule of G-invariants of a kG-module M is denoted M G ;
see [20, I.2.10] for the construction.
Lemma 1.2. Let G be a finite group scheme over k and K an extension of the
field k. Let M and N be kG-modules.
There are natural isomorphisms of KGK -modules:
(i) (M ⊗k N )K ∼
= M K ⊗ K NK .
(ii) (M ⊗k N )K ∼
= MK ⊗K N K when M is finite dimensional over k.
(iii) Homk (M, N )K ∼
= HomK (MK , N K ).
There are also natural isomorphisms of K-vector spaces:
(iv) HomkG (M, N )K ∼
= HomKGK (MK , N K ).
K GK
G K ∼
(v) (M ) = (M ) .
Proof. The isomorphisms in (i), (iii) and (iv) are standard whilst (v) is the special
case M = k and N = M of (iv). The isomorphism in (ii) can be realised as the
composition of natural maps
∼

∼

(M ⊗k K) ⊗K Homk (K, N ) −−→ M ⊗k Homk (K, N ) −−→ Homk (K, M ⊗k N )

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BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

where the last map is an isomorphism as M is finite dimensional over k.



Examples of finite group schemes. We recall some important classes of finite
group schemes relevant to this work.
Example 1.3 (Finite groups). A finite group G defines a finite group scheme
over any field k. More precisely, the group algebra kG is a finite dimensional
cocommutative Hopf algebra, and hence its dual is a commutative Hopf algebra
which defines a group scheme over k; it is also denoted G. A finite group E is an
elementary abelian p-group if it is isomorphic to (Z/p)r , for some prime number p.
The integer r is then the rank of E. Over a field k of characteristic p, there are
isomorphisms of k-algebras
p
k[E] ∼
= k ×r and kE ∼
= k[z1 , . . . , zr ]/(z , . . . , z p ).
1

r

The comultiplication on kE is determined by the map zi 7→ zi ⊗ 1 + zi ⊗ zi + 1 ⊗ zi
and the antipode is determined by the map zi 7→ (zi + 1)p−1 − 1.
Example 1.4 (Additive groups). Fix a positive integer r and let Ga(r) denote the
finite group scheme whose coordinate algebra is
r

k[Ga(r) ] = k[t]/(tp )
with comultiplication defined by t 7→ t ⊗ 1 + 1 ⊗ t and antipode t 7→ −t. There is
an isomorphism of k-algebras
p
p
kGa(r) ∼
= k[u0 , . . . , ur−1 ]/(u , . . . , u ).
0

r−1

We note that Ga(r) is the rth Frobenius kernel of the additive group scheme Ga
over k; see, for instance, [20, I.9.4]
Example 1.5 (Quasi-elementary group schemes). Following Bendel [3], a group
scheme over a field k of positive characteristic p is said to be quasi-elementary if it
is isomorphic to Ga(r) × (Z/p)s . Its group algebra structure is the same as that of
an elementary abelian p-group.
A finite group scheme G over a field k is unipotent if its group algebra kG is
local. Quasi-elementary group schemes are unipotent. Also, the group scheme over
a field of positive characteristic p defined by a finite p-group is unipotent.
2. π-points
In the rest of this paper G denotes a finite group scheme defined over a field k of
positive characteristic p. We recall the notion of π-points and basic results about
them. The primary references are the papers of Friedlander and Pevtsova [17, 18].
π-points. A π-point of G, defined over a field extension K of k, is a morphism of
K-algebras
α : K[t]/(tp ) −→ KGK
that factors through the group algebra of a unipotent abelian subgroup scheme C
of GK , and such that KGK is flat when viewed as a left (equivalently, as a right)
module over K[t]/(tp ) via α. It should be emphasised that C need not be defined
over k; see Examples 2.6. Restriction along α defines a functor
α∗ : Mod KGK −→ Mod K[t]/(tp ) .
The result below extends [18, Theorem 4.6], that dealt with MK .
Theorem 2.1. Let α : K[t]/(tp ) → KGK and β : L[t]/(tp ) → LGL be π-points of
G. Then the following conditions are equivalent.

STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

5

(i) For any finite dimensional kG-module M , the module α∗ (MK ) is projective
if and only if β ∗ (ML ) is projective.
(ii) For any kG-module M , the module α∗ (MK ) is projective if and only if
β ∗ (ML ) is projective.
(iii) For any finite dimensional kG-module M , the module α∗ (M K ) is projective
if and only if β ∗ (M L ) is projective.
(iv) For any kG-module M , the module α∗ (M K ) is projective if and only if
β ∗ (M L ) is projective.
Proof. The equivalence of (i) and (ii) is proved in [18, Theorem 4.6]. The equivalence of (iii) and (iv) can be proved in exactly the same way.
(i) ⇐⇒ (iii) Since M is finite dimensional, M K is a direct sum of copies of MK ,
by Remark 1.1. Hence α∗ (M K ) is projective if and only if α∗ (MK ) is projective.
The same is true of β ∗ (M L ) and β ∗ (ML ).

Definition 2.2. When π-points α and β satisfy the conditions of Theorem 2.1,
they are said to be equivalent, and denoted α ∼ β.
For ease of reference, we list some basic properties of π-points.
Remark 2.3. (1) Let α : K[t]/(tp ) → KGK be a π-point and L a field extension of
K. Then L ⊗K α : L[t]/(tp ) → LGL is a π-point and it is easy to verify, say from
condition (i) of Theorem 2.1, that α ∼ L ⊗K α.
(2) Every π-point of a subgroup scheme H of G is naturally a π-point of G. This
follows from the fact that an embedding of group schemes always induces a flat
map of group algebras.
(3) Every π-point is equivalent to one that factors through a quasi-elementary
subgroup scheme over the same field extension; see [17, Proposition 4.2].
π-points and cohomology. The cohomology of G with coefficients in a kGmodule M is denoted H ∗ (G, M ). It can be identified with Ext∗kG (k, M ). Recall that
H ∗ (G, k) is a k-algebra that is graded-commutative (because kG is a Hopf algebra)
and finitely generated, as was proved by Friedlander and Suslin [19, Theorem 1.1].
Let Proj H ∗ (G, k) denote the set of homogeneous prime ideals H ∗ (G, k) that are
properly contained in the maximal ideal of positive degree elements.
Given a π-point α : K[t]/(tp ) → KGK we write H ∗ (α) for the composition of
homomorphisms of k-algebras.
K⊗ −

H ∗ (G, k) = Ext∗kG (k, k) −−−−k−→ Ext∗KGK (K, K) −→ Ext∗K[t]/(tp ) (K, K),
where the second map is induced by restriction. By Frobenius reciprocity and
the theorem of Friedlander and Suslin recalled above, Ext∗K[t]/(tp ) (K, K) is finitely
generated as a module over Ext∗KGK (K, K). Since the former is nonzero, it follows
that the radical of the kernel of the map Ext∗KGK (K, K) → Ext∗K[t]/(tp ) (K, K) is a
1
∗
prime ideal different from Ext>
KGK (K, K) and hence that the radical of Ker H (α)
∗
>1
is a prime ideal in H (G, k), different from H (G, k).
Remark 2.4. Fix a point p in Proj H ∗ (G, k). There exists a field K and a π-point
αp : K[t]/(tp ) −→ KGK

p
such that Ker H ∗ (αp ) = p. In fact, there is such a K that is a finite extension of
the degree zero part of the homogenous residue field at p; see [18, Theorem 4.2].
It is shown in [18, Corollary 2.11] that α ∼ β if and only if there is an equality
p
p
Ker H ∗ (α) = Ker H ∗ (β) .

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BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

In this way, the equivalence classes of π-points are in bijection with Proj H ∗ (G, k).
Theorem 2.5 ([18, Theorem 3.6]). Let G be a finite group scheme over a field k.
Taking a π-point α to the radical of Ker H ∗ (α) induces a bijection between the set
of equivalence classes of π-points of G and Proj H ∗ (G, k).

We illustrate these ideas on the Klein four group that will be the running example
in this work.
Example 2.6. Let V = Z/2 × Z/2 and k a field of characteristic two. The group
algebra kV is isomorphic to k[x, y]/(x2 , y 2 ), where x+ 1 and y + 1 correspond to the
generators of V . Let J = (x, y) denote the Jacobson radical of kV . It is well-known
that H ∗ (V, k) is the symmetric algebra on the k-vector space Homk (J/J 2 , k); see,
for example, [4, Corollary 3.5.7]. Thus H ∗ (V, k) is a polynomial ring over k in two
variables in degree one and Proj H ∗ (V, k) ∼
= P1k .
The π-point corresponding to a rational point [a, b] ∈ P1k (using homogeneous
coordinates) is represented by the map of k-algebras
k[t]/(tp ) −→ k[x, y]/(x2 , y 2 ) where t 7→ ax + by.
More generally, for each closed point p ∈ P1k there is some finite field extension
K of k such that P1K contains a rational point [a′ , b′ ] over p (with Aut(K/k) acting
transitively on the finite set of points over p). Then the π-point corresponding to
p is represented by the map of K-algebras
K[t]/(tp ) −→ K[x, y]/(x2 , y 2 ) where t 7→ a′ x + b′ y.
Now let K denote the field of rational functions in a variable s. The generic
point of P1k then corresponds to the map of K-algebras
K[t]/(tp ) −→ K[x, y]/(x2 , y 2 ) where t 7→ x + sy.

3. π-cosupport and π-support
As before, G is a finite group scheme over a field k of positive characteristic p.
In this section, we introduce a notion of π-cosupport of a kG-module, by analogy
with the notion of π-support introduced in [18, §5]. The main result, Theorem 3.4,
is a formula that computes the π-cosupport of a function object, in terms of the
π-support and π-cosupport of its component modules.
Definition 3.1. The π-cosupport of a kG-module M is the subset of Proj H ∗ (G, k)
defined by
π- cosuppG (M ) := {p ∈ Proj H ∗ (G, k) | α∗p (Homk (K, M )) is not projective}.
Here αp : K[t]/(tp ) → KGK denotes a representative of the equivalence class of
π-points corresponding to p; see Remark 2.4. The definition is modelled on that of
the π-support of M , introduced in [18] as the subset
π- suppG (M ) := {p ∈ Proj H ∗ (G, k) | α∗p (K ⊗k M ) is not projective}.
This is denoted Π(G)M in [18]; our notation is closer to the one used in [8] for
cohomological support.

STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

7

Projectivity. For later use, we record the well-known property that the module
of homomorphisms preserves and detects projectivity.
Lemma 3.2. Let M and N be kG-modules.
(i) If M or N is projective, then so is Homk (M, N ).
(ii) M is projective if and only if Endk (M ) is projective.
Proof. We repeatedly use the fact that a kG-module is projective if and only if it
is injective; see, for example, [20, Lemma I.3.18].
(i) The functor Homk (M, −) takes injective kG-modules to injective kG-modules
because it is right adjoint to an exact functor. Thus, when N injective, so is
Homk (M, N ). The same conclusion follows also from the projectivity of M because
Homk (−, N ) takes projective modules to injective modules, as follows from the
natural isomorphisms:
HomkG (−, Homk (M, N )) ∼
= HomkG (− ⊗k M, N )
∼ HomkG (M ⊗k −, N )
=
∼
= HomkG (M, Homk (−, N )) .
The first and the last isomorphisms are by adjunction, and the one in the middle
holds because kG is cocommutative.
(ii) When M is projective, so is Endk (M ), by (i). For the converse, observe
that when Endk (M ) is projective, so is Endk (M ) ⊗k M , since − ⊗k M preserves
projectivity being the left adjoint of an exact functor. It remains to note that M is a
direct summand of Endk (M )⊗k M , because the composition of the homomorphisms
ν

ε

M −−→ Endk (M ) ⊗k M −−→ M,
ν(m) = idM ⊗m

where

and ε(f ⊗ m) = f (m),

of kG-modules equals the identity on M .



We now work towards Theorem 3.4 that gives a formula for the cosupport of a
function object, and the support of a tensor product. These are useful for studying
modules over finite group schemes, as will become clear in Section 5; see also [12].
Function objects and tensor products. The proof of Theorem 3.4 is complicated by the fact that, in general, a π-point α : K[t]/(tp ) → KGK does not
preserve Hopf structures, so restriction along α does not commute with taking tensor products, or the module of homomorphisms. To deal with this situation, we
adapt an idea from the proof of [17, Lemma 3.9]—see also [15, Lemma 6.4] and [23,
Lemma 6.4]—where the equivalence of (i) and (iii) in the following result is proved.
The hypothesis on the algebra A is motivated by Remark 2.3(3) and Example 1.5.
Lemma 3.3. Let K be a field of positive characteristic p and A a cocommutative
Hopf K-algebra that is isomorphic as an algebra to K[t1 , . . . , tr ]/(tp1 , . . . tpr ). Let
α : K[t]/(tp ) −→ A
be a flat homomorphism of K-algebras. For any A-modules M and N , the following
conditions are equivalent:
(i) α∗ (M ⊗K N ) is projective.
(ii) α∗ (HomK (M, N )) is projective.
(iii) α∗ (M ) or α∗ (N ) is projective.

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BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

Proof. As noted before, (i) ⇐⇒ (iii) is [17, Lemma 3.9]; the hypotheses of op. cit.
includes that M and N are finite dimensional, but that is not used in the proof.
We employ a similar argument to verify that (ii) and (iii) are equivalent.
Let σ : A → A be the antipode of A, ∆ : A → A ⊗K A its comultiplication, and
set I = Ker(A → K), the augmentation ideal of A. Identifying t with its image in
A, one has
(1 ⊗ σ)∆(t) = t ⊗ 1 − 1 ⊗ t + w with w ∈ I ⊗K I;
see [20, I.2.4]. Recall that the action of a ∈ A on HomK (M, N ) is given by multiplication with (1 ⊗ σ)∆(a), so that for f ∈ HomK (M, N ) and m ∈ M one has
X
X
(a · f )(m) =
a′ f (σ(a′′ )m) where ∆(a) =
a′ ⊗ a′′ .
Given a module over A ⊗K A, we consider two K[t]/(tp )-structures on it: One
where t acts via multiplication with (1 ⊗ σ)∆(t) and another where it acts via
multiplication with t ⊗ 1 − 1 ⊗ t. We claim that these two K[t]/(tp )-modules are
both projective or both not projective. This follows from a repeated use of [17,
Proposition 2.2] because w can be represented as a sum of products of nilpotent
elements of A ⊗K A, and each nilpotent element x of A ⊗K A satisfies xp = 0.
We may thus assume that t acts on HomK (M, N ) via t ⊗ 1 − 1 ⊗ t. There is then
an isomorphism of K[t]/(tp )-modules
α∗ (HomK (M, N )) ∼
= HomK (α∗ (M ), α∗ (N )) ,
where the action of K[t]/(tp ) on the right hand side is the one obtained by viewing
it as a Hopf K-algebra with comultiplication defined by t 7→ t ⊗ 1 + 1 ⊗ t and
antipode t 7→ −t. It remains to observe that for any K[t]/(tp )-modules U, V , the
module HomK (U, V ) is projective if and only if one of U or V is projective.
Indeed, if U or V is projective, so is HomK (U, V ), by Lemma 3.2. As to the
converse, every K[t]/(tp )-module is a direct sum of cyclic modules, so when U and
V are not projective, they must have direct summands isomorphic to K[t]/(tu ) and
K[t]/(tv ), respectively, for some 1 ≤ u, v < p. Then HomK (K[t]/(tu ), K[t]/(tv )) is
a direct summand of HomK (U, V ). The former cannot be projective as a K[t]/(tp )module because its dimension as a K-vector space is uv, while the dimension of any
projective module must be divisible by p: projective K[t]/(tp )-modules are free. 
The first part of the result below is [18, Proposition 5.2].
Theorem 3.4. Let M and N be kG-modules. Then there are equalities
(i) π- suppG (M ⊗k N ) = π- suppG (M ) ∩ π- suppG (N ),
(ii) π- cosuppG (Homk (M, N )) = π- suppG (M ) ∩ π- cosuppG (N ).
Proof. We prove part (ii). Part (i) can be proved in the same fashion; see [18,
Proposition 5.2].
Fix a π-point α : K[t]/(tp ) → KGK . By Remark 2.3(3), we can assume α factors
as K[t]/(tp ) → KC → KGK , where C is a quasi-elementary subgroup scheme of
GK . As noted in Lemma 1.2(iii), there is an isomorphism of KGK -modules
Homk (M, N )K ∼
= HomK (MK , N K ) .
We may restrict a KGK module to K[t]/(tp ) by first restricting to KC, and
Homk (−, −) commutes with this operation. Thus the desired result follows from
the equivalence of (ii) and (iii) in Lemma 3.3, applied to the map K[t]/(tp ) → KC,
keeping in mind the structure of KC; see Example 1.5.


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9

Basic computations. Next we record, for later use, some elementary observations
concerning support and cosupport. The converse of (i) in the result below also holds.
For finite groups this is proved in Theorem 4.4 below, and for finite group schemes
this is one of the main results of [12].
Lemma 3.5. Let M be a kG-module.
(i) π- suppG (M ) = ∅ = π- cosuppG (M ) when M is projective.
(ii) π- cosuppG (M ) = π- suppG (M ) when M is finite dimensional.
(iii) π- suppG (k) = Proj H ∗ (G, k) = π- cosuppG (k).
Proof. Part (ii) is immediate from definitions, given Remark 1.1. For the rest, fix
a π-point α : K[t]/(tp ) → KGK .
(i) When M is projective, so are the KGK -modules K ⊗k M and Homk (K, M ),
and restriction along α preserves projectivity, as α is a flat map. This justifies (i).
(iii) Evidently, kK equals K and α∗ (K) is non-projective. Since α was arbitrary,
one deduces that π- suppG (k) is all of Proj H ∗ (G, k). That this also equals the
π-cosupport of k now follows from (ii).

Corollary 3.6. If M is a kG-module, then there is an equality
π- cosuppG (Homk (M, k)) = π- suppG (M ).
Proof. This follows from Theorem 3.4 (ii) and Lemma 3.5 (iii).



The equality of π-support and π-cosupport for finite dimensional modules, which
holds by Lemma 3.5(ii), may fail for infinite dimensional modules. We describe an
example over the Klein four group; see Example 5.5 for a general construction.
Example 3.7. Let V be the Klein four group Z/2 × Z/2 and k a field of characteristic two. The π-points of kV were described in Example 2.6. We keep the notation
from there. Let M be the infinite dimensional kG-module with basis elements ui
and vi for i ≥ 0 and kG-action defined by
xui = vi ,

yui = vi−1 ,

xvi = 0,

yvi = 0

where v−1 is interpreted as the zero vector. A diagram for this module is as follows:
u2

u1

u0

◦

◦

◦

x

...

y

◦

v0

◦

v1

◦

v2

...

Claim. The π-support of M is the closed point {[0, 1]} of P1k whilst its π-cosupport
contains also the generic point.
Given a finite field extension K of k, it is not hard to verify that for any rational
point [a, b] ∈ P1K the image of multiplication by ax + by on MK is its socle, that
is spanned by the elements {vi }i>0 . For [a, b] 6= [0, 1], this is also the kernel of
ax + by, whilst for [a, b] = [0, 1] it contains, in addition, the element u0 . In view of
Remark 1.1, this justifies the assertions about the closed points of P1k .
For the generic point let K denote the field of rational functions in a variable s.
It is again easy to check that the kernel of x + sy and the image of multiplication
by x + sy on MK are equal to its socle. So the generic point is not in π- suppV (M ).
For cosupport, consider the k-linear map f : K → M defined as follows. Given
a rational function φ(s), its partial fraction expansion consists of a polynomial
ψ(s) plus the negative parts of the Laurent expansions at the poles. If ψ(s) =

10

BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

α0 +α1 s+· · · we define f (φ) to be α0 u0 +α1 u1 +· · · . By definition, (x+sy)(f )(a) =
xf (a) + yf (sa); using this it is easy to calculate that f is in the kernel of x + sy.
On the other hand, any function in the image of x + sy lands in the socle of M . It
follows that the kernel of x + sy is strictly larger than the image, and so the generic
point is in the π-cosupport of M .
4. Finite groups
The focus of the rest of the paper is on finite groups. In this section we prove that
a module over a finite group is projective if and only if it has empty π-cosupport.
The key ingredient is a version of Dade’s lemma for elementary abelian groups. This
in turn is based on an analogue of the Kronecker quiver lemma [6, Lemma 4.1].
Lemma 4.1. Let k be an algebraically closed field, ℓ a non-trivial extension field of
k, and let V, W be k-vector spaces. If there exist k-linear maps f, g : V → W with
the property that for every pair of scalars λ and µ in ℓ, not both zero, the linear
map λf + µg : Homk (ℓ, V ) → Homk (ℓ, W ) is an isomorphism, then V = 0 = W .
Remark 4.2. Note that λf + µg here really means Homk (λ, f ) + Homk (µ, g) since
this is the way ℓ acts on homomorphisms.
Proof. Since k is algebraically closed, we may as well assume that ℓ = k(s), a simple
transcendental extension of k, since further extending the field only strengthens the
hypothesis without changing the conclusion.
Use g to identify V and W . Then f is a k-endomorphism of V with the property
that for all µ ∈ ℓ the endomorphism Homk (1, f ) + Homk (µ, id) of Homk (ℓ, V ) is
invertible. The action of f makes V into a k[t]-module, with t acting as f does.
Since f + µ. id is invertible for all µ ∈ k, and k is algebraically closed, V is a
k(t)-module.
Consider the homomorphism k(t) ⊗k k(s) → k(t) of rings that is induced by the
assignment p(t) ⊗ q(s) 7→ p(t)q(t). It is not hard to verify that its kernel is the ideal
generated by t − s, so there is an exact sequence of k(t) ⊗k k(s)-modules
t−s

0 −→ k(t) ⊗k k(s) −−−→ k(t) ⊗k k(s) −→ k(t) −→ 0
Applying Homk(t) (−, k(t)) and using adjunction yields an exact sequence
t−s

0 −→ k(t) −→ Homk (k(s), k(t)) −−−→ Homk (k(s), k(t)) −→ 0 .
Thus t− s is not invertible on Homk (k(s), k(t)). If V is nonzero then it has k(t) as a
summand as a k(t)-module, so that t − s, that is to say, Homk (1, t) + Homk (−s, id),
is not invertible. This contradicts the hypothesis.

Dade’s lemma for cosupport. The result below, and its proof, are modifications
of [6, Theorem 5.2]. Given an k-algebra R and an extension field K of k, we write
RK for the K-algebra K ⊗k R and M K for the RK -module Homk (K, M ).
Theorem 4.3. Let k be a field of positive characteristic p and set
R = k[t1 , . . . , tr ]/(tp1 , . . . , tpr ).
Let K be an algebraically closed field of transcendence degree at least r − 1 over k.
Then an R-module M is projective if and only if for all flat maps α : K[t]/(tr ) → RK
the module α∗ (M K ) is projective.

STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

11

Proof. If M is projective as an R-module, then M K is projective as an RK -module,
and because α is flat, it follows that α∗ (M K ) is projective.
Assume that α∗ (M K ) is projective for all α as in the statement of the theorem.
We verify that M is projective as an R-module by induction on r, the case r = 1
being trivial. Assume r ≥ 2 and that the theorem is true with r replaced by r − 1.
It is easy to verify that the hypothesis and the conclusion of the result are
unchanged if we pass from k to any extension field in K. In particular, replacing
k by its algebraic closure in K we may assume that k is itself algebraically closed.
The plan is to use Lemma 4.1 to prove that Ext1R (k, M ) = 0. Since R is Artinian
it would then follow that M is injective, and hence also projective, because R is a
self-injective algebra.
We first note that for any extension field ℓ of k, tensoring with ℓ gives a one-toone map Ext∗R (k, k) → Ext∗Rℓ (ℓ, ℓ), which we view as an inclusion, and a natural
isomorphism of ℓ-vector spaces
ExtiR (k, M )ℓ ∼
= ExtiRℓ (ℓ, M ℓ ) for i ∈ Z.
These remarks will be used repeatedly in the argument below. They imply, for
example, that ExtiR (k, M ) = 0 if and only if ExtiRℓ (ℓ, M ℓ ) = 0.
Let β : Ext1R (k, k) → Ext2R (k, k) be the Bockstein map; see [5, §4.3] or, for a
slightly different approach, [20, I.4.22]. Recall that this map is semilinear through
the Frobenius map, in the sense that
β(λε) = λp β(ε)

for ε ∈ Ext1R (k, k) and λ ∈ k.

Fix an extension field ℓ of k in K that is algebraically closed and of transcendence
degree 1. Choose linearly independent elements ε and γ of Ext1R (k, k). The elements
β(ε) and β(γ) of Ext2R (k, k) induce k-linear maps
f, g : Ext1R (k, M ) −→ Ext3R (k, M ) .
Let λ and µ be elements in ℓ, not both zero, and consider the element
(4.1)

λ1/p ε + µ1/p γ ∈ Ext1Rℓ (ℓ, ℓ) ∼
= Homℓ (J/J 2 , ℓ) ,

where J is the radical of the ring Rℓ . It defines a linear subspace of codimension
one in the ℓ-linear span of t1 , . . . , tr in Rℓ . Let S be the subalgebra of Rℓ generated
by this subspace and view M ℓ as an S-module, by restriction of scalars.
Claim. As an S-module, M ℓ is projective.
p
Indeed, S is isomorphic to ℓ[z1 , . . . , zr−1 ]/(z1p , . . . , zr−1
), as an ℓ-algebra. It is
not hard to verify that the hypotheses of the theorem apply to the S-module M ℓ
and the extension field ℓ ⊂ K. Since the transcendence degree of K over ℓ is r − 2,
the induction hypothesis yields that the S-module M ℓ is projective, as claimed.

We give Rℓ the structure of a Hopf algebra by making the generators ti primitive:
that is, comultiplication is determined by the map ti 7→ ti ⊗1+1⊗ti and the antipode
is determined by the map ti 7→ −ti . Note that Rℓ ⊗S ℓ has a natural structure of
an Rℓ -module, with action induced from the left hand factor.
Claim. The Rℓ -module (Rℓ ⊗S ℓ) ⊗ℓ M ℓ , with the diagonal action, is projective.
Indeed, it is not hard to see the comultiplication on Rℓ induces one on S, so the
latter is a sub Hopf-algebra of the former. As M ℓ is projective as an S-module, by
the previous claim, so is Homℓ (M ℓ , N ) for any S-module N ; see Lemma 3.2. Since

12

BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

projective S-modules are injective, the desired claim is then a consequence of the
following standard isomorphisms of functors
HomRℓ ((Rℓ ⊗S ℓ) ⊗ℓ M ℓ , −) ∼
= HomRℓ (Rℓ ⊗S ℓ, Homℓ (M ℓ , −))
∼ HomS (ℓ, Homℓ (M ℓ , −))
=
on the category of Rℓ -modules.
The Bockstein of the element (4.1) is
λβ(ε) + µβ(γ) ∈ Ext2Rℓ (ℓ, ℓ),
and is represented by an exact sequence of the form
(4.2)

0 −→ ℓ −→ Rℓ ⊗S ℓ −→ Rℓ ⊗S ℓ −→ ℓ −→ 0 .

For any Rℓ -module N the Hopf algebra structure on Rℓ gives a map of ℓ-algebras
Ext∗Rℓ (ℓ, ℓ) → Ext∗Rℓ (N, N ) such that the two actions of Ext∗Rℓ (ℓ, ℓ) on Ext∗Rℓ (ℓ, N )
coincide, up to the usual sign [4, Corollary 3.2.2]. What this entails is that the map
λf + µg : Ext1Rℓ (ℓ, M ℓ ) −→ Ext3Rℓ (ℓ, M ℓ )
may be described as splicing with the extension
0 −→ M ℓ −→ (Rℓ ⊗S ℓ) ⊗ℓ M ℓ −→ (Rℓ ⊗S ℓ) ⊗ℓ M ℓ −→ M ℓ −→ 0 ,
which is obtained from the exact sequence (4.2) by applying − ⊗ℓ M ℓ . By the
preceding claim, the modules in the middle are projective, and so the element
λβ(ε) + µβ(γ) induces a stable isomorphism
∼

Ω2 (M ℓ ) −−→ M ℓ .
It follows that λf + µg is an isomorphism for all λ, µ in l not both zero. Thus
Lemma 4.1 applies and yields Ext1R (k, M ) = 0 as desired.

Support and cosupport detect projectivity. The theorem below is the main
result of this work. Several consequences are discussed in the subsequent sections.
Theorem 4.4. Let k be a field and G a finite group. For any kG-module M , the
following conditions are equivalent.
(i) M is projective.
(ii) π- suppG (M ) = ∅
(iii) π- cosuppG (M ) = ∅.
Proof. We may assume that the characteristic of k, say p, divides the order of G.
The implications (i) =⇒ (ii) and (i) =⇒ (iii) are by Lemma 3.5.
(iii) =⇒ (i) Let E be an elementary abelian p-subgroup of G and let M ↓E
denote M viewed as a kE-module. The hypothesis implies π- cosuppE (M ↓E ) = ∅,
by Remark 2.3(2), and then it follows from Theorem 4.3 that M ↓E is projective.
Chouinard’s theorem [16, Theorem 1] thus implies that M is projective.
(ii) =⇒ (i) When π- suppG (M ) = ∅, it follows from Theorem 3.4 that
π- cosuppG Endk (M ) = ∅ .
The already settled implication (iii) =⇒ (i) now yields that Endk (M ) is projective.
Hence M is projective, by Lemma 3.2.


STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

13

5. Cohomological support and cosupport
The final part of this paper is devoted to applications of Theorem 4.4. We proceed in several steps and derive global results about the module category of a finite
group from local properties, including a comparison of π-support and π-cosupport
with cohomological support and cosupport. In the next section we consider the
classification of thick and localising subcategories of the stable module category.
From now on G denotes a finite group, k a field of positive characteristic dividing
the order of G. Let StMod(kG) be the stable module category of all (meaning,
also infinite dimensional) kG-modules modulo the projectives; see, for example, [4,
§2.1]. This is not an abelian category; rather, it has the structure of a compactly
generated tensor triangulated category and comes equipped with a natural action
of the cohomology ring H ∗ (G, k); see [8, Section 10]. This yields the notion of
cohomological support and cosupport developed in [8, 11]. More precisely, for each
homogeneous prime ideal p of H ∗ (G, k) that is different from the maximal ideal of
positive degree elements, there is a distinguished object Γp k. Using this one defines
for each kG-module M its cohomological support
suppG (M ) = {p ∈ Proj H ∗ (G, k) | Γp k ⊗k M is not projective}
and its cohomological cosupport
cosuppG (M ) = {p ∈ Proj H ∗ (G, k) | Homk (Γp k, M ) is not projective}.
The result below reconciles these notions with the corresponding notions defined
in terms of π-points.
Theorem 5.1. Let G be a finite group and M a kG-module. Then
cosuppG (M ) = π- cosuppG (M )

and

suppG (M ) = π- suppG (M ) ,

∗

regarded as subsets of Proj H (G, k).
Proof. We use the fact that π- suppG (Γp k) = {p}; see [18, Proposition 6.6]. Then
using Theorems 4.4 and 3.4 (ii) one gets
def

p ∈ cosuppG (M ) ⇐⇒ Homk (Γp k, M ) is not projective
4.4
⇐⇒ π- cosuppG (Homk (Γp k, M )) 6= ∅
3.4
⇐⇒ π- suppG (Γp k) ∩ π- cosuppG (M ) 6= ∅
⇐⇒ p ∈ π- cosuppG (M ).
This gives the equality involving cosupports.
In the same vein, using Theorems 4.4 and 3.4 (i) one gets
def

p ∈ suppG (M ) ⇐⇒ Γp k ⊗k M is not projective
4.4
⇐⇒ π- suppG (Γp k ⊗k M ) 6= ∅
3.4
⇐⇒ π- suppG (Γp k) ∩ π- suppG (M ) 6= ∅
⇐⇒ p ∈ π- suppG (M ).
This gives the equality involving supports.



Here is a first consequence of this result; we are unable to verify it directly,
except for closed points in the π-support and π-cosupport.

14

BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

Corollary 5.2. For any kG-module M the maximal elements, with respect to inclusion, in π- cosuppG (M ) and π- suppG (M ) coincide.
Proof. Given Theorem 5.1, this follows from [11, Theorem 4.13].



We continue with two useful formulas for computing cohomological supports and
cosupports; they are known from previous work [6, 11] and are now accessible from
the perspective of π-points.
Corollary 5.3. For all kG-modules M and N there are equalities
(i) suppG (M ⊗k N ) = suppG (M ) ∩ suppG (N ).
(ii) cosuppG (Homk (M, N )) = suppG (M ) ∩ cosuppG (N ).
Proof. This follows from Theorems 3.4 and 5.1.



We wrap up this section with a couple of examples. The first one shows that the
π-support of a module M may be properly contained in that of Endk (M ).
Example 5.4. Let V be the Klein four group Z/2 × Z/2 and k a field of characteristic two. Thus, Proj H ∗ (V, k) = P1k , and a realisation of points of P1k as π-points
of kV was given in Example 2.6. Let M be the infinite dimensional kV -module
described in Example 3.7. As noted there, the π-support of M consists of a single
point, namely, the closed point [0, 1]. We claim
π- suppV (Endk (M )) = {[0, 1]} ∪ {generic point of P1k } .
Indeed, since the π-cosupport of M contains [0, 1], by Example 3.7, it follows from
Theorem 3.4 that the π-cosupport of Endk (M ) is exactly {[0, 1]}. Corollary 5.2
then implies that [0, 1] is the only closed point in the π-support of Endk (M ). It
remains to verify that the latter contains also the generic point.
Let K be the field of rational functions in a variable s. The π-point defined by
K[t]/(tp ) → K[x, y]/(x2 , y 2 ) with t 7→ x + sy corresponds to the generic point of
P1k ; see Example 2.6. The desired result follows once we verify that the element
1 ⊗ idM of Endk (M )K is in the kernel of x + sy but not in its image. It is in
the kernel because idM is a kV -module homomorphism. Suppose there exists an
f in Endk (M )K with (x + sy)f = 1 ⊗ idM . Then for each n ≥ 0, the identity
((x + sy)f )(un ) = un yields
f (vn ) + sf (vn−1 ) = un + (x + sy)f (un ) .
Noting that v−1 = 0, by convention, it follows that
f (vn ) ≡ un + sun−1 + · · · + sn u0
modulo the submodule K(v0 , v1 , · · · ) of MK . This cannot be, as f is in Endk (M )K .
In Example 3.7 it is proved that, for M as above, π- suppV (M ) 6= π- cosuppV (M ).
The remark below is a conceptual explanation of this phenomenon, since M is of
the form T (Ip ) for p = [0, 1] in P1k .
Example 5.5. Given p ∈ Proj H ∗ (G, k), there is a kG-module T (Ip ) which is
defined in terms of the following natural isomorphism
b ∗ (G, −), Ip ) ∼
HomH ∗ (G,k) (H
= HomkG (−, T (Ip ))

b ∗ (G, −) is Tate cohomolwhere Ip denotes the injective envelope of H ∗ (G, k)/p, H
ogy, and HomkG (−, −) is the set of homomorphisms in StMod kG; see [13, §3].
The cohomological support and cosupport of this module have been computed in

STRATIFICATION AND π-COSUPPORT: FINITE GROUPS

15

[10, Lemma 11.10] and [11, Proposition 5.4], respectively. Combining this with
Theorem 5.1 gives
π- suppG (T (Ip )) = {p} and π- cosuppG (T (Ip )) = {q ∈ Proj H ∗ (G, k) | q ⊆ p} .
6. Stratification
The results of this section concern the triangulated category structure of the
stable module category, StMod(kG). Recall that a full subcategory C of StMod(kG)
is localising if it is a triangulated subcategory and is closed under arbitrary direct
sums. In a different vein, C is tensor ideal if for all C in C and arbitrary M , the
kG-module C ⊗k M is in C.
Following [10, §3], we say that the stable module category StMod(kG) is stratified
by H ∗ (G, k) if for each homogeneous prime ideal p of H ∗ (G, k) that is different from
the maximal ideal of positive degree elements the localising subcategory
{M ∈ StMod(kG) | suppG (M ) ⊆ {p}}
admits no proper non-zero tensor ideal localising subcategory.
We are now in the position to give a simplified proof of [10, Theorem 10.3]. We
refer the reader to [10, Introduction] for a version of this result dealing entirely with
the (abelian) category of kG-modules.
Theorem 6.1. Let k be a field and G a finite group. Then the stable module
category StMod(kG) is stratified as a tensor triangulated category by the natural
action of the cohomology ring H ∗ (G, k). Therefore the assignment
[
(6.1)
C 7−→
suppG (M )
M∈C

induces a one to one correspondence between the tensor ideal localising subcategories
of StMod(kG) and the subsets of Proj H ∗ (G, k).
Proof. It suffices to show that HomkG (M ⊗k −, N ) 6= 0 whenever M, N are kGmodules with suppG (M ) = {p} = suppG (N ); see [10, Lemma 3.9]. By adjointness,
this is equivalent to Homk (M, N ) being non-projective. Thus the assertion follows
from Corollary 5.3, once we observe that suppG (N ) = {p} implies p ∈ cosuppG (N ).
But this is again a consequence of Corollary 5.3, since Endk (N ) is non-projective
by Lemma 3.2.
The second part of the assertion is a formal consequence of the first; see [10, Theorem 3.8]. The inverse map sends a subset V of Proj H ∗ (G, k) to the subcategory
consisting of all kG-modules M such that suppG (M ) ⊆ V.

The next results concern stmod(kG), the full subcategory of StMod G consisting
of finite dimensional modules. A tensor ideal thick subcategory C of stmod(kG) is a
triangulated subcategory that is closed under direct summands and has the property
that for any C in C and finite dimensional kG-module M , the kG-module C ⊗k M
is in C. The classification of the tensor ideal thick subcategories of stmod(kG) is
the main result of [7] and can be deduced from the classification of the tensor ideal
localising subcategories of StMod(kG). This is based on the following lemma.
Lemma 6.2. Let M be a finite dimensional kG-module. Then suppG (M ) is a
Zariski-closed subset of Proj H ∗ (G, k). Conversely, each Zariski-closed subset of
Proj H ∗ (G, k) is of this form.
Proof. The first statement follows from [8, Theorem 5.5] and the second from [9,
Lemma 2.6].


16

BENSON, IYENGAR, KRAUSE, AND PEVTSOVA

Theorem 6.3. Let G be a finite group and k a field. Then the assignment (6.1)
induces a one to one correspondence between the tensor ideal thick subcategories of
stmod(kG) and the specialisation closed subsets of Proj H ∗ (G, k).
Proof. Let σ be the assignment (6.1). By Lemma 6.2, when C is a tensor ideal thick
subcategory of stmod(kG), the subset σ(C) of Proj H ∗ (G, k) is a union of Zarksiclosed subsets, and hence specialisation closed. Thus σ restricted to stmod G has the
desired image. Let τ be the map from specialisation closed subsets of Proj H ∗ (G, k)
to stmod G that assigns V to the subcategory with objects
{M ∈ stmod G | suppG M ⊆ V} .
This is readily seen to be a tensor ideal thick subcategory. The claim is that σ and
τ are inverses of each other.
Indeed for any V ⊆ Proj H ∗ (G, k) there is an inclusion στ (V) ⊆ V; equality holds
if V is closed, by Lemma 6.2, and hence also if V is specialisation closed.
Fix a tensor ideal thick subcategory C of stmod G. Evidently, there is an inclusion
C ⊆ τ σ(C). To prove that equality holds, it suffices to prove that if M is a finite
dimensional G-module with suppG (M ) ⊆ σ(C), then M is in C. Let C′ be the tensor
ideal localising subcategory of StMod(kG) generated by C. From the properties of
support, it is easy to verify that σ(C′ ) = σ(C), and then Theorem 6.1 implies M is
in C′ . Since M is compact when viewed as an object in StMod G, it follows by an
argument analogous to the proof of [21, Lemma 2.2] that M is in C, as desired. 
Acknowledgements. Part of this article is based on work supported by the National Science Foundation under Grant No. 0932078000, while DB, SBI, and HK
were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the 2012–2013 Special Year in Commutative Algebra. The authors
thank the Centre de Recerca Matemàtica, Barcelona, for hospitality during a visit in
April 2015 that turned out to be productive and pleasant. SBI and JP were partly
supported by NSF grants DMS-1503044 and DMS-0953011, respectively. We are
grateful to Eric Friedlander for comments on an earlier version of this paper.
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Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College,
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Srikanth B. Iyengar, Department of Mathematics, University of Utah, Salt Lake
City, UT 84112, U.S.A.
Henning Krause, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld,
Germany.
Julia Pevtsova, Department of Mathematics, University of Washington, Seattle, WA
98195, U.S.A.

