Localising subcategories for cochains on the classifying space of a finite group

Abstract

The localising subcategories of the derived category of the cochains on the
classifying space of a finite group are classified. They are in one to one
correspondence with the subsets of the set of homogeneous prime ideals of the
cohomology ring $H^*(G,k)$.

Full text

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arXiv:1107.4765v2 [math.RT] 26 Aug 2011
LOCALISING SUBCATEGORIES FOR COCHAINS ON THE
CLASSIFYING SPACE OF A FINITE GROUP
DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE
Abstract. The localising subcategories of the derived category of the cochains
on the classifying space of a finite group are classified. They are in one to one
correspondence with the subsets of the set of homogeneous prime ideals of the
cohomology ring H∗(G,k).
1. Introduction
Let G be a finite group and k a field of characteristic p. Let C∗(BG;k) be
the cochains on the classifying space BG. Using the machinery of Elmendorf, Kˇ r´ ıˇ z,
Mandell and May [8], one can regard C∗(BG;k) as a strictly commutative S-algebra
over the field k. The derived category D(C∗(BG;k)) has thus a structure of a tensor
triangulated category via the left derived tensor product − ⊗L
for the tensor product is C∗(BG;k).
In this paper we apply techniques and results from [3, 4, 5, 6] to classify the
localising subcategories of D(C∗(BG;k)). More precisely, there is a notion of strat-
ification for triangulated categories via the action of a graded commutative ring
which implies that the localising subcategories are parameterised by sets of ho-
mogeneous prime ideals [4]. For D(C∗(BG;k)) we use the natural action of the
endomorphism ring of the tensor identity which is isomorphic to the cohomology
algebra H∗(G,k) of the group G.
C∗(BG;k)−. The unit
Theorem 1.1. The derived category D(C∗(BG;k)) is stratified by the ring H∗(G,k).
This yields a one to one correspondence between the localising subcategories of
D(C∗(BG;k)) and subsets of the set of homogeneous prime ideals of H∗(G,k).
It is proved in [6] that there is an equivalence of tensor triangulated categories
between D(C∗(BG;k)) and the localising subcategory of K(InjkG) generated by the
tensor identity. Here, K(InjkG) is the homotopy category of complexes of injective
(= projective) kG-modules, studied in [6, 9].
The main theorem of [5] states that K(InjkG) is stratified as a tensor triangulated
category by H∗(G,k).Theorem 1.1 is a consequence of a more general result
concerning tensor triangulated categories, which is described below.
Let (T,⊗,1) be a compactly generated tensor triangulated category, as described
in [3, §8], and R a graded commutative noetherian ring acting on T via a homomor-
phism R → End∗
T(1). In this case, for each homogeneous prime ideal p of R there
exists a local cohomology functor Γp: T → T; see [3]. The support of an object X
in T is then defined to be
suppRX = {p ∈ SpecR | ΓpX ?= 0}.
The research of the second author was undertaken during a visit to the University of Bielefeld,
supported by a research prize from the Humboldt Foundation, and by NSF grant DMS 0903493.
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2 DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE
The condition that T is stratified by the action of R means that assigning a sub-
category S of T to its support
suppRS =
?
X∈S
suppRX
yields a bijection between tensor ideal localising subcategories of T and subsets
of the homogeneous prime ideal spectrum SpecR contained in suppRT; see [4,
Theorem 4.2]. Theorem 1.1 is thus a special case of the result below that relates
tensor ideal localising subcategories of T and the localising subcategories of LocT(1),
the localising subcategory of T generated by the tensor unit. We note that LocT(1)
is a compactly generated tensor triangulated category in its own right and that R
acts on it as well.
Theorem 1.2. Suppose that the Krull dimension of R is finite. If T is stratified
by R as a tensor triangulated category, then so is LocT(1), and there is a bijection
?Tensor ideal localising
It assigns each tensor ideal localising subcategory S of T to S ∩ LocT(1).
subcategories of T
?
∼
−→
?Localising subcategories
of LocT(1)
?
.
Remark 1.3. The theorem is not true without the assumption that T is stratified
by R. For example, let T be the derived category of quasi-coherent sheaves on
the projective line P1
k. The tensor unit is O. In this example there are no proper
localising subcategories of LocT(O) since End∗
ideal localising subcategories of T.
T(O) = k, while there are many tensor
Remark 1.4. The assumption that the Krull dimension of R is finite is artificial, and
is used only to ensure that for each X ∈ T and p ∈ SpecR the object ΓpX belongs
to LocT(X). One can replace this condition by, for instance, the assumption that
T arises as the homotopy category of a Quillen model category [10, §6].
2. Localising subcategories of LocT(1)
In this section T is a triangulated category with set-indexed coproducts and the
tensor product ⊗ provides a symmetric monoidal structure with unit 1 on T, which
is exact in each variable and preserves set-indexed coproducts.
The proof of Theorem 1.2 is based on a sequence of elementary lemmas. The
first one describes the tensor ideal localising subcategory of T which is generated
by a class C of objects; we denote this by Loc⊗
T(C).
Lemma 2.1. Let C be a class of objects of T. Then
Loc⊗
T(C) = LocT({X ⊗ Y | X ∈ C,Y ∈ T}).
Proof. Set S = LocT({X ⊗ Y | X ∈ C,Y ∈ T}). It suffices to show that S is tensor
ideal. This means that FS ⊆ S for each tensor functor F = − ⊗ Y , which is an
immediate consequence of Lemma 2.2 below.
?
Lemma 2.2. Let F : U → V be an exact functor between triangulated categories
that preserves set-indexed coproducts. If C is a class of objects of U, then
F LocU(C) ⊆ LocV(FC).
Proof. The preimage F−1LocV(FC) is a localising subcategory of U containing C.
Thus it contains LocU(C), and one gets
F LocU(C) ⊆ FF−1LocV(FC) ⊆ LocV(FC).
?

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LOCALISING SUBCATEGORIES FOR COCHAINS ON BG
3
Lemma 2.3. Let Γ : T → T be a colocalization functor that preserves set-indexed
coproducts. Then for any X ∈ T and Y ∈ LocT(1), there is a natural isomorphism
ΓX ⊗ Y
∼
− → Γ(X ⊗ Y ).
Remark 2.4. There is an analogous result for a localization functor L: T → T that
preserves set-indexed coproducts: For any X ∈ T and Y ∈ LocT(1), there is a
natural isomorphism L(X ⊗ Y ) − → LX ⊗ Y .
∼
Proof. A colocalisation functor Γ comes with a natural morphism ΓX → X. Ten-
soring this with an object Y ∈ LocT(1) gives a morphism ΓX ⊗ Y → X ⊗ Y that
factors through the natural morphism Γ(X ⊗ Y ) → X ⊗ Y . Here, one uses that
ΓX ⊗Y belongs to ΓT, since the objects Y′∈ T with ΓX ⊗Y′∈ ΓT form a local-
ising subcategory containing 1. The induced morphism φY: ΓX ⊗Y → Γ(X ⊗Y )
is an isomorphism. To see this, observe that the objects Y′∈ T such that φY′ is
an isomorphism form a localising subcategory containing 1.
?
Proposition 2.5. Suppose that the unit 1 is compact in T and let Γ : T → LocT(1)
denote the right adjoint of the inclusion LocT(1) → T. If S is a localising subcate-
gory of LocT(1), then
Loc⊗
T(S) ∩ LocT(1) = Γ(Loc⊗
T(S)) = S.
Proof. We verify each of the following inclusions
S ⊆ Loc⊗
T(S) ∩ LocT(1) ⊆ Γ(Loc⊗
T(S)) ⊆ S.
The first one is clear. Composing the functor Γ with the inclusion LocT(1) → T
yields a colocalisation functor that preserves set-indexed coproducts, since 1 is
compact. For an object X in Loc⊗
second inclusion. Applying Lemma 2.3 together with the description of Loc⊗
from Lemma 2.1 yields the third inclusion.
T(S)∩LocT(1), we have ΓX∼= X. This gives the
T(S)
?
Corollary 2.6. Suppose that the unit 1 is a compact object in T. Assigning each
localising subcategory S of LocT(1) to Loc⊗
T(S) gives a bijection
?Localising subcategories
of LocT(1)
?
∼
−→
?Tensor ideal localising subcategories of
T generated by objects from LocT(1)
?
.
Proof. The inverse map sends U ⊆ T to U ∩ LocT(1).
?
We are now ready to prove Theorem 1.2. Note that in this T is a compactly gen-
erated tensor triangulated category, which entails a host of additional requirements;
see [3, §8] for a list.
Proof of Theorem 1.2. It follows from Proposition 2.5 that the assignment
S ?−→ Loc⊗
T(S)
is an injective map from the localising subcategories of LocT(1) to the tensor ideal
localising subcategories of T. In general, it is not bijective, as the example of
Remark 1.3 shows. However, since T is stratified by R as a tensor triangulated
category, it follows from [4, §7] that each tensor ideal localising subcategory is
generated by a set of objects of the form Γp1. Since R has finite Krull dimension,
[4, Theorem 3.4] yields that Γp1 is in LocT(1). Therefore, given a tensor ideal
localising subcategory U of T, the localising subcategory
U′= LocT({Γp1 | p ∈ SuppRU}) ⊆ LocT(1)

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4 DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE
satisfies Loc⊗
we have shown that each localising subcategory of LocT(1) is generated by objects
of the form Γp1, so LocT(1) is stratified by the action of R; see [4, Theorem 4.2].
T(U′) = U. This proves the surjectivity of the assignment. Moreover,
?
3. The cohomological nucleus
Let (T,⊗,1) be a compactly generated tensor triangulated category and let
R be a graded commutative noetherian ring acting on T via a homomorphism
R → End∗
T(1). Suppose in addition that R has finite Krull dimension.
We define the cohomological nucleus of T as the set of homogeneous prime ideals
p of R such that there exists an object X ∈ T satisfying Hom∗
ΓpX ?= 0. This definition is motivated by work of Benson, Carlson, and Robinson
in the context of modular group representations [2].
For p in SpecR consider the tensor ideal localising subcategory
ΓpT = {Y ∈ T | Y∼= ΓpX for some X ∈ T}.
Note that an object X ∈ T belongs to ΓpT if and only if Hom∗
p-torsion for every compact C ∈ T, by [3, Corollary 4.10]. The result below gives a
local description of the cohomological nucleus.
T(1,X) = 0 and
T(C,X) is p-local and
Proposition 3.1. Let p be a homogeneous prime ideal of R. The following condi-
tions are equivalent:
(1) Every object X in T with Hom∗
T(1,X) = 0 satisfies ΓpX = 0.
(2) One has LocT(Γp1) = ΓpT.
(3) Every localising subcategory of ΓpT is a tensor ideal of T.
Proof. The Krull dimension of R is finite, so ΓpX is in LocT(X) for each X in T,
by [4, Theorem 3.4]. This fact is used without further comment.
(1) ⇒ (2): Set S = LocT(Γp1). Note that S ⊆ ΓpT; we claim that equality holds.
Indeed, S ⊆ LocT(1) and also Loc⊗
T(S) = ΓpT, since Γp= Γp1 ⊗ −. Thus, for any
X in ΓpT from Proposition 2.5 one gets an exact triangle ΓX → X → X′→ with
ΓX ∈ S and Hom∗
(2) ⇒ (3): Let S be a localising subcategory of ΓpT. Using (2) and the fact that
ΓpT is a tensor ideal of T, one has Loc⊗
T(S) ⊆ LocT(1). Then it follows, again from
Proposition 2.5, that S is a tensor ideal of T.
(3) ⇒ (1): Assume Hom∗
Condition (3) implies that LocT(Γp1) = ΓpT. Thus ΓpX belongs to LocT(Γp1) and
therefore also to LocT(1). So one obtains Hom∗
ΓpX = 0.
T(1,X) = 0. Then (1) implies X′= 0 and hence X ∈ S.
T(1,X) = 0; then Hom∗
T(1,ΓpX) = 0, as 1 is compact.
T(ΓpX,ΓpX) = 0, which implies
?
Consider as an example for T the stable module category StModkG of a finite
group G with the canonical action of R = H∗(G,k). We refer to [1, 2] for the
discussion of two variations of the nucleus, namely the group theoretic and the
representation theoretic nucleus. There it is shown that LocT(1) = T if and only
if the centraliser of every element of order p in G is p-nilpotent and every block is
either principal or semisimple, where p denotes the characteristic of the field k.
It is convenient to define for any class C of objects of T
C⊥= {Y ∈ T | Hom∗
T(X,Y ) = 0 for all X ∈ C},
⊥C = {X ∈ T | Hom∗
T(X,Y ) = 0 for all Y ∈ C}.
Now let S = LocT(1). The representation theoretic nucleus is by definition
?
X∈S⊥∩Tc
suppRX.

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LOCALISING SUBCATEGORIES FOR COCHAINS ON BG
5
Clearly, this is contained in the cohomological nucleus. It is a remarkable fact that
the representation theoretic nucleus is non-empty if S⊥?= 0; this is proved in [1, 2].
Moreover, Question 13 of [7] asks whether S =⊥(S⊥∩ Tc). Note that S =⊥(S⊥)
follows from general principles.
Acknowledgments. It is a pleasure to thank Greg Stevenson for helpful comments
on this work.
References
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Math. 1 (1994/95), 196–205, electronic.
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Algebra 131 (1990), no. 1, 40–73.
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categories, Ann. Scient.´Ec. Norm. Sup. (4) 41 (2008), 575–621.
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groups, Annals of Math 175 (2012), to appear.
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(Bielefeld, 1998), 285–296, Trends Math, Birkh¨ auser, Basel, 2000.
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1128–1162.
[10] G. Stevenson, Support theory via actions of tensor triangulated categories, arXiv:1105.4692v1.
Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College,
Aberdeen AB24 3UE, Scotland U.K.
Srikanth B. Iyengar, Department of Mathematics, University of Nebraska, Lincoln,
NE 68588, U.S.A.
Henning Krause, Fakult¨ at f¨ ur Mathematik, Universit¨ at Bielefeld, 33501 Bielefeld,
Germany.