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Stratifying the derived category of cochains on BG for G a compact Lie group
Journal of Pure and Applied Algebra 218 (2014) 642–650
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Stratifying the derived category of cochains on BG for G a compact Lie group David Benson a,∗ , John Greenlees b a
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK
b
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
article
info
Article history: Received 30 January 2013 Available online 23 August 2013 Communicated by C.A. Weibel MSC: Primary: 18E30; 55R35 Secondary: 20C20
abstract The main purpose of this paper is to classify the localising subcategories of the derived category D(C ∗ (BG; k)) where G is a compact Lie group and k is a field. We also prove a version of Chouinard’s theorem for D(C ∗ (BG; k)), we describe the relationship between induction and coinduction for a closed subgroup of G, and we use this to describe the relationship between Hochschild homology and cohomology of C ∗ (BG; k). © 2013 Elsevier B.V. All rights reserved.
1. Introduction If G is a finite group and k is a field then there is a close relationship between the stable module category StMod(kG), the homotopy category of complexes of injective modules K(Inj kG), and the derived category D(C ∗ (BG)) of the cochains on the classifying space C ∗ (BG) = C ∗ (BG; k). This was investigated by Dwyer, Greenlees and Iyengar [13] and by Benson and Krause [11]. If H is a subgroup of G then the restriction map StMod(kG) → StMod(kH ) corresponds under this relationship to the induction functor, namely the derived tensor product C ∗ (BH ) ⊗C ∗ (BG) − : D(C ∗ (BG)) → D(C ∗ (BH )), which happens to be isomorphic to the coinduction functor, namely the derived homs HomC ∗ (BG) (C ∗ (BH ), −) : D(C ∗ (BG)) → D(C ∗ (BH )), see Section 7 of [11]. Chouinard’s theorem [12] for StMod(kG) states that an object is zero if and only if its restriction to kE is zero for every elementary abelian p-subgroup E of G. This was extended in [11] to the categories K(Inj kG) and D(C ∗ (BG)). The translation to D(C ∗ (BG)) states that an object is zero if and only if its induction to D(C ∗ (BE )) is zero for all E. Most known proofs of Chouinard’s theorem use Serre’s theorem [22] on products of Bocksteins. If G is a compact Lie group, we no longer have an obvious analogue of the stable module category, but it still makes sense to ask whether an object in D(C ∗ (BG)) is zero if and only if its induction to each finite elementary abelian p-subgroup of G is zero. Our first theorem shows that this is indeed the case. Serre’s theorem does not seem to be the right approach in this case. Instead, in Section 3 we use methods from Quillen’s work [20,21] on the spectrum of H ∗ (BG; k) to prove the following theorem. Theorem 1.1. Let G be a compact Lie group and k a field of characteristic p. A module M is isomorphic to zero in D(C ∗ (BG)) if and only if C ∗ (BE ) ⊗C ∗ (BG) M is isomorphic to zero in D(C ∗ (BE )) for every finite elementary abelian p-subgroup E of G.
∗
Corresponding author. E-mail address: [email protected] (D. Benson).
0022-4049/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jpaa.2013.08.004
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In a series of papers by Benson, Iyengar and Krause [4–10] the machinery was set up for classifying localising subcategories of triangulated categories, and used in the cases of the closely related categories StMod(kG), K(Inj kG) and D(C ∗ (BG)) for G finite. In these cases, the classification went via reduction to elementary abelian subgroups using the appropriate version of Chouinard’s theorem. Using the version given in Theorem 1.1, we prove the following theorem in Section 4. For the definition of stratification, see Section 4.2 of [7]. Theorem 1.2. Let G be a compact Lie group. Then the category D(C ∗ (BG)) is stratified by the canonical action of H ∗ (BG). Informally, this theorem states that the theory of support gives a bijection between localising subcategories of D(C ∗ (BG)) and subsets of VG = Spec H ∗ (BG), the spectrum of homogeneous prime ideals in H ∗ (BG). We have stated Theorem 1.1 for induction. There is also a version for coinduction, which we prove at the same time. However, just as for finite groups there is a close relationship between induction and coinduction, which we investigate in Section 6. We state here a version with a mild orientation assumption. Theorem 1.3. Let H be a closed subgroup of a compact Lie group G, and let d = dim(G/H ). Suppose that the action of H on the tangent space at the identity coset eH ∈ G/H preserves orientation, or that k has characteristic two. If M is a C ∗ (BG)-module then in D(C ∗ (BH )) we have coindG,H (M ) ≃ Σ d indG,H (M ). Finally, we use the relationship between induction and coinduction to examine the relationship between Hochschild homology and cohomology. Again we state here a version with a mild orientation assumption. Theorem 1.4. Suppose that the action of G on its Lie algebra by conjugation preserves orientation, or that k has characteristic two. Then there are isomorphisms HH n (C ∗ (BG)) ∼ = HHn+dim G (C ∗ (BG)). 2. The cochains on BG and its derived category Let G be a compact Lie group and let k be a field of characteristic p. We begin by recalling the elementary properties of the derived category D(C ∗ (BG)). We use the language of S-algebras, and for definiteness we use the model of Elmendorf, Kříž, Mandell and May [14]. In this language, if X is a space then C ∗ (X ) = C ∗ (X ; k) denotes the function spectrum from X to the Eilenberg–Mac Lane spectrum of k. This has the advantage of being a commutative S-algebra, whereas the more conventional DGA of cochains on X with coefficients in k is not equivalent to a commutative DGA. We have
π−n C ∗ (X ) ∼ = H n (X ) = H n (X ; k).
(2.1)
We write D(C ∗ (X )) for the derived category of C ∗ (X ). The objects in this derived category are the C ∗ (X )-module spectra, while the arrows are obtained from the homotopy classes of module spectrum maps by inverting the quasi-isomorphisms, namely the maps which become isomorphisms after applying π∗ . One of the advantages of commutativity of C ∗ (X ) is that it makes good sense to talk of the derived tensor product A ⊗C ∗ (X ) B and derived homomorphisms HomC ∗ (X ) (A, B) of two objects A and B in D(C ∗ (X )) and to regard the result as again being an object in D(C ∗ (X )), and there is a spectral sequence ExtH ∗ (X ) (π∗ A, π∗ B) ⇒ π∗ HomC ∗ (X ) (A, B). ∗,∗
This way, D(C ∗ (X )) is a tensor triangulated category whose tensor identity is C ∗ (X ). Since D(C ∗ (X )) is generated as a triangulated category by the tensor identity C ∗ (X ), it follows that every localising subcategory of D(C ∗ (X )) is a tensor ideal. It also follows that D(C ∗ (X )) is compactly generated, and that the full subcategory of compact objects Dc (C ∗ (X )) is the thick subcategory generated by C ∗ (X ). Next we recall Quillen’s stratification theorem in the form needed for our work. See Theorem 10.2 and the discussion following Proposition 11.2 in Quillen [21]. Let VG be the spectrum of homogeneous prime ideals of H ∗ (BG) = H ∗ (BG; k). For each closed subgroup H of G there is a map BH → BG well defined up to homotopy, and hence restriction homomorphisms resG,H : C ∗ (BG) → C ∗ (BH ) and resG,H : H ∗ (BG) → H ∗ (BH ). This way we obtain a map res∗G,H : VH → VG . For each point p in VG there exists a finite elementary abelian p-subgroup E of G such that p is in the image of res∗G,E . We say that p originates in such an E if there does not exist a proper subgroup E ′ of E such that p is in the image of res∗G,E ′ . Theorem 2.2 (Quillen). For each p ∈ VG , the pairs (E , q) where p = res∗G,E (q) and such that p originates in E are all G-conjugate. This sets up a one-to-one correspondence between primes p in VG and G-conjugacy classes of such pairs (E , q).
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If ζ ∈ H n (BG) then (2.1) allows us to represent ζ by a well defined morphism in D(C ∗ (BG))
ζˆ : C ∗ (BG) → Σ n C ∗ (BG). If M is a C ∗ (BG)-module then tensoring with M gives a well defined morphism
ζˆ ⊗C ∗ (BG) IdM : M → Σ n M . This way we obtain a well defined homomorphism to the graded centre of the derived category H ∗ (BG) → Z (D(C ∗ (BG))).
(2.3)
Lemma 2.4. In the language of Notation 4.1 of [4], the homomorphism (2.3) makes the derived category D(C ∗ (BG)) into an H ∗ (BG)-linear triangulated category. Proof. The only thing that needs to be checked is that H ∗ (BG) is Noetherian. This is a theorem of Venkov [23]. If H is a closed subgroup of G then the map BH → BG induces a map of S-algebras C ∗ (BG) → C ∗ (BH ). We therefore have a restriction map res∗G,H : D(C ∗ (BH )) → D(C ∗ (BG)). As above, an element ζ ∈ H n (BG) induces a well defined morphism ζˆ : C ∗ (BG) → Σ n C ∗ (BG) in D(C ∗ (BG)), and this in turn induces a morphism n ∗ ˆ ∗ res G,H (ζ ) = IdC ∗ (BH ) ⊗C ∗ (BG) ζ : C (BH ) → Σ C (BH )
in D(C ∗ (BH )). Restriction has a left adjoint indG,H =
C ∗ (BH ) C
∗
(BH )C ∗ (BG) ⊗C ∗ (BG) − : D(C ∗ (BG)) → D(C ∗ (BH ))
and a right adjoint coindG,H = HomC ∗ (BG) (C ∗ (BG) C ∗ (BH )C ∗ (BH ) , −) : D(C ∗ (BG)) → D(C ∗ (BH )). The following lemma summarises the standard properties of restriction, induction and coinduction. Lemma 2.6. Let M be a C ∗ (BH )-module and N , N ′ be C ∗ (BG)-modules. Then (i) res∗G,H (M ⊗C ∗ (BH ) indG,H (N )) ≃ res∗G,H (M ) ⊗C ∗ (BG) N, (ii) indG,H (N ) ⊗C ∗ (BH ) indG,H (N ′ ) ≃ indG,H (N ⊗C ∗ (BG) N ′ ), (iii) res∗G,H (HomC ∗ (BH ) (M , coindG,H (N ))) ≃ HomC ∗ (BG) (res∗G,H (M ), N ),
(iv) HomC ∗ (BH ) (indG,H (N ), coindG,H (N ′ )) ≃ coindG,H (HomC ∗ (BG) (N , N ′ )). (v) If ζ ∈ H n (BG) then n ˆ res G,H (ζ ) ⊗C ∗ (BG) IdindG,H (N ) = indG,H (ζ ⊗C ∗ (BH ) IdN ) : indG,H (N ) → Σ indG,H (N ).
Proof. Using the associativity of tensor product and the Hom–tensor adjunction, we have isomorphisms (i) of C ∗ (BG)-modules M ⊗C ∗ (BH ) (C ∗ (BH ) ⊗C ∗ (BG) N ) ≃ M ⊗C ∗ (BG) N , (ii) of C ∗ (BH )-modules
(C ∗ (BH ) ⊗C ∗ (BG) N ) ⊗C ∗ (BH ) (C ∗ (BH ) ⊗C ∗ (BG) N ′ ) ≃ C ∗ (BH ) ⊗C ∗ (BG) (N ⊗C ∗ (BG) N ′ ), (iii) of C ∗ (BG)-modules HomC ∗ (BH ) (M , HomC ∗ (BG) (C ∗ (BH ), N )) ≃ HomC ∗ (BG) (C ∗ (BH ) ⊗C ∗ (BH ) M , N ), (iv) of C ∗ (BH )-modules HomC ∗ (BH ) (C ∗ (BH ) ⊗C ∗ (BG) N , HomC ∗ (BG) (C ∗ (BH ), N ′ ))
≃ HomC ∗ (BG) (C ∗ (BH ) ⊗C ∗ (BH ) C ∗ (BH ) ⊗C ∗ (BG) N , N ′ ) ≃ HomC ∗ (BG) (C ∗ (BH ) ⊗C ∗ (BG) N , N ′ ). (v) This follows by tensoring N with (2.5).
(2.5)
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3. Chouinard’s theorem Let G be a compact Lie group and k a field of characteristic p. The first step to understanding the category of modules over C ∗ (BG) = C ∗ (BG; k) for a compact Lie group G is to show that it is controlled by elementary abelian p-subgroups. In effect we use a part of Quillen’s method for understanding the spectrum of H ∗ (BG). Choose an embedding of G into a unitary group U. We write T for the maximal torus of U, namely the subgroup of diagonal matrices. We write S for the subgroup of elements of order dividing p in T . Thus S is an elementary abelian p-subgroup of the same rank as T , and every elementary abelian p-subgroup of G is conjugate to a subgroup of S. Finally we write F = U /S, and we note that as a G-space, all its isotropy groups are elementary abelian p-groups. Theorem 3.1. Let G be a compact Lie group and k a field of characteristic p. (i) As an object in D(C ∗ (BG)), the module C ∗ (BG) is in the thick subcategory generated by the modules C ∗ (BE ) as E runs over the elementary abelian p-subgroups of G. (ii) If M is a C ∗ (BG)-module and C ∗ (BE ) ⊗C ∗ (BG) M ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ 0. (iii) If M is a C ∗ (BG)-module and HomC ∗ (BG) (C ∗ (BE ), M ) ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ 0. Proof. (i) The G-space F is built from finitely many cells G/E, and hence C ∗ (EG ×G F ) is finitely built from modules C ∗ (EG ×G (G/E )) ≃ C ∗ (BE ). For any U-spaces A and B we have a natural map C ∗ (EU ×U A) ⊗C ∗ (BU ) C ∗ (EU ×U B) → C ∗ (EU ×U (A × B)). Taking A = U ×G X and B = F = U /S, we obtain
ν : C ∗ (EG ×G X ) ⊗C ∗ (BU ) C ∗ (BS ) → C ∗ (EG ×G (X × F )) natural in X . Now observe that H ∗ (BS ) is free over H ∗ (BT ) and H ∗ (BT ) is free over H ∗ (BU ), and hence ν is an equivalence for all X (Lemma 6.5 of Quillen [20]). In particular it follows that C ∗ (EG ×G X ) is a summand of C ∗ (EG ×G (X × F )) as C ∗ (BG)-modules. Taking X = ∗ we conclude that C ∗ (BG) is a summand of C ∗ (EG ×G F ), and is hence in the thick subcategory of D(C ∗ (BG)) generated by the C ∗ (BE ). (ii) Using (i), we see that if C ∗ (BE ) ⊗C ∗ (BG) M ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ C ∗ (BG) ⊗C ∗ (BG) M ≃ 0. (iii) Similarly, if HomC ∗ (BG) (C ∗ (BE ), M ) ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ HomC ∗ (BG) (C ∗ (BG), M ) ≃ 0. Remark 3.2. In the theorem, it is clearly sufficient to use just one representative of each conjugacy class of maximal elementary abelian p-subgroups. Remark 3.3. If G is a finite p-group then D(C ∗ (BG)) is equivalent to K(Inj kG), see [11], and StMod(kG) is equivalent to the localising subcategory of K(Inj kG) consisting of acyclic objects. In this case, the above theorem gives a new proof of Chouinard’s theorem which avoids the use of Serre’s theorem on the product of Bocksteins. 4. Localising subcategories In this section we classify the localising subcategories of D(C ∗ (BG)). Most of the proof is the same as the proof of Theorem 9.7 of [8], but the role of Proposition 9.6 of that paper is played by our version of Chouinard’s theorem. Since the roles of restriction and induction are reversed, and some of the details are slightly different, we have chosen to write out the proof in full. For p ∈ VG , let Γp : D(C ∗ (BG)) → D(C ∗ (BG)) be the local cohomology functor described in [4]. Definition 4.1. The support of a C ∗ (BG)-module M is defined to be
VG (M ) = {p ∈ VG | Γp (M ) ̸≃ 0}. The following are the analogues of Lemma 9.3 and Proposition 9.4 of [8]. Lemma 4.2. Let H be a closed subgroup of G. Fix p ∈ VG and let U = (res∗G,H )−1 {p} ⊆ VH . (i) For any C ∗ (BG)-module M we have indG,H (Γp (M )) ≃
q∈U Γq (indG,H (M )). ∗ (ii) For any C ∗ (BH )-module N we have Γp (res∗G,H (N )) ≃ q∈U resG,H (Γq (N )).
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Proof. (i) This follows from Corollary 7.10 of [9], with T = D(C ∗ (BG)), U = D(C ∗ (BH )), R = H ∗ (BG), S = H ∗ (BH ) and F = indG,H with right adjoint res∗G,H and left adjoint existing by Brown representability. (ii) Using part (i) and Lemma 2.6(i), we have
Γp res∗G,H (N ) ≃ res∗G,H (N ) ⊗C ∗ (BG) Γp C ∗ (BG)
≃ res∗G,H (N ⊗C ∗ (BH ) indG,H (Γp C ∗ (BG))) ∗ ∗ Γq C (BH ) ≃ resG,H N ⊗C ∗ (BH ) q∈U
≃
resG,H Γq (N ). ∗
q∈U
Proposition 4.3. Let H be a closed subgroup of G. (i) For any C ∗ (BG)-module M we have
VG (res∗G,H indG,H (M )) ⊆ VG (M ). (ii) For any C ∗ (BH )-module N we have
VG (res∗G,H (N )) = res∗G,H VH (N ). Proof. (i) If p ∈ VG (res∗G,H indG,H (M )) then 0 ̸= Γp (C ∗ (BH ) ⊗C ∗ (BG) M ) = C ∗ (BH ) ⊗C ∗ (BG) Γp (M ) and so Γp (M ) ̸= 0. (ii) By Lemma 4.2(ii) the condition p ∈ VG (res∗G,H (N )) is equivalent to the statement that there exists q ∈ VH such that ∗ resG,H (q) = p and Γq (N ) ̸= 0. Part (ii) of the following is the elementary abelian version of the subgroup theorem for D(C ∗ (BG)), and the proof is copied from Theorem 9.5 of [8]. The full version of the subgroup theorem will be a consequence of the stratification theorem. Theorem 4.4. (i) The category D(C ∗ (BE )) is stratified by the action of H ∗ (BE ). (ii) Let E ′ ≤ E be elementary abelian p-groups. For any object X in D(C ∗ (BE )) there is an equality
VE ′ (indE ,E ′ (X )) = (res∗E ,E ′ )−1 VE (X ). Proof. (i) By Theorem 4.2 of [11] we have an equivalence K(Inj kE ) ≃ D(C ∗ (BE )).
The maps from H ∗ (BE ) to the endomorphisms of the tensor identities of these two categories agree, so by Lemma 3.10 of [8], D(C ∗ (BE )) is stratified by the action of H ∗ (BE ) and the equivalence preserves support. (ii) Fix a prime q in VE ′ and set p = res∗E ,E ′ (q). Proposition 4.3(ii) shows that
VE (res∗E ,E ′ Γq C ∗ (BE ′ )) = {p} = VE (Γp C ∗ (BE )). By (i) we therefore have Loc(res∗E ,E ′ Γq C ∗ (BE ′ )) = Loc(Γp C ∗ (BE )).
This implies that if M is a C ∗ (BE )-module then res∗E ,E ′ Γq C ∗ (BE ′ ) ⊗C ∗ (BE ) M ̸= 0 if and only if Γp C ∗ (BE ) ⊗C ∗ (BE ) M ̸= 0. Thus
Γq indE ,E ′ (M ) ̸= 0 ⇐⇒ Γq C ∗ (BE ′ ) ⊗C ∗ (BE ′ ) indE ,E ′ (M ) ̸= 0
⇐⇒ res∗E ,E ′ (Γq C ∗ (BE ′ )) ⊗C ∗ (BE ) M ̸= 0 ⇐⇒ Γp C ∗ (BE ) ⊗C ∗ (BE ) M ̸= 0 ⇐⇒ Γp M ̸= 0. The second implication follows from Lemma 2.6(i) and the fact that res∗E ,E ′ is faithful. The following is the main theorem of this section. Theorem 4.5. Let G be a compact Lie group. The triangulated category D(C ∗ (BG)) is stratified by the canonical action of the cohomology algebra H ∗ (BG). In particular, there is a bijection between the localising subcategories of D(C ∗ (BG)) and the subsets of VG . The localising subcategory corresponding to V is the full subcategory of C ∗ (BG)-modules M such that VG (M ) ⊆ V .
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Proof. For the statement in the first paragraph of the theorem, we need to prove that for p ∈ VG , the subcategory Γp D(C ∗ (BG)) is minimal among localising subcategories of D(C ∗ (BG)). Let M be a non-zero module in Γp D(C ∗ (BG)). Then by Theorem 3.1(ii), there exists an elementary abelian p-subgroup E0 ≤ G such that indG,E0 (M ) ̸≃ 0. Choose a prime q0 ∈ VE0 (indG,E0 (M )). By Proposition 4.3(i) we have res∗G,E0 (q0 ) ∈ VG (res∗G,E0 indG,E0 (M )) ⊆ VG (M ) = {p} and so res∗G,E0 (q0 ) = p. It follows that for some pair (E , q) corresponding to p as in Theorem 2.2, we have E0 ≥ E and q0 = res∗E0 ,E (q). It then follows from Theorem 4.4(ii) that q ∈ VE (indG,E (M )). By Theorem 2.2 all pairs (E , q) where p originates in E are conjugate. So if we choose one, then each non-zero M ∈ Γp D(C ∗ (BG)) has Γq (indG,E (M )) ̸= 0. Now let N be another non-zero object in Γp D(C ∗ (BG)). Then Hom∗C ∗ (BG) (C ∗ (BE ) ⊗C ∗ (BG) M , N ) ≃ Hom∗C ∗ (BE ) (indG,E (M ), indG,E (N )).
(4.6)
By Lemma 4.2(i), Γq indG,E (M ) and Γq indG,E (N ) are non-zero direct summands of indG,E (M ) and indG,E (N ) respectively. Now it follows from Theorem 4.4(i) that Γq D(C ∗ (BE )) is a minimal localising subcategory of D(C ∗ (BE )). So by Lemma 4.1 of [7] Hom∗C ∗ (BE ) (Γq indG,E (M ), Γq indG,E (N )) ̸= 0. Thus by (4.6) we have Hom∗C ∗ (BG) (C ∗ (BE ) ⊗C ∗ (BG) M , N ) ̸= 0. Now D(C ∗ (BG)) is generated by the tensor identity C ∗ (BG), so C ∗ (BE ) ⊗C ∗ (BG) M is in the localising subcategory generated by M. Hence we have HomC ∗ (BG) (M , N ) ̸= 0. Again using Lemma 4.1 of [7], we conclude that Γp D(C ∗ (BG)) is a minimal localising subcategory of D(C ∗ (BG)). The statement in the second paragraph of the theorem follows from the first paragraph together with Theorem 4.2 of [7]. Remark 4.7. The case of the above theorem where G is a finite group is the main theorem of [5]. The methods are somewhat different. Question 4.8. For what spaces X is D(C ∗ (X )) stratified by H ∗ (X )? Our main theorem shows that this is the case for X = BG, G a compact Lie group. 5. Consequences of the stratification We begin with the tensor product theorem. Theorem 5.1. Let M and N be C ∗ (BG)-modules. Then
VG (M ⊗C ∗ (BG) N ) = VG (M ) ∩ VG (N ). Proof. This follows from Theorem 4.5 together with Theorem 7.3 of [7]. Next, we have the subgroup theorem. Theorem 5.2. Let H be a closed subgroup of G and let M be a C ∗ (BG)-module. Then
VH (indG,H (M )) = (res∗G,H )−1 VG (M ). Proof. The proof of this is the same as the proof of Theorem 4.4(ii) except that one uses Theorem 4.5 instead of Theorem 4.4(i). The following is the analogue in this situation of the main theorem of Benson, Carlson and Rickard [2]. Theorem 5.3. There is a bijection between the thick subcategories of the compact objects Dc (C ∗ (BG)) and the specialisation closed subsets of VG . The thick subcategory corresponding to V ⊆ VG is the full subcategory of C ∗ (BG)-modules M in Dc (C ∗ (BG)) such that VG (M ) ⊆ V . Proof. This follows from Theorem 4.5 together with Theorem 6.1 of [7]. A localising subcategory C of a triangulated category T is said to be strictly localising if the inclusion has a right adjoint. This is equivalent to the existence of a localisation functor L : T → T such that an object X of T is in C if and only if LX = 0. Lemma 5.4. Every localising subcategory of D(C ∗ (BG)) is strictly localising. Proof. The proof is the same as the proof of Lemma 11.11 of [8].
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A strictly localising subcategory is smashing if the localisation functor L preserves coproducts. For a tensor triangulated category generated by the tensor unit 1, this is equivalent to the statement that the natural transformation L1 ⊗ X → LX is an isomorphism for all X in T . The following is the telescope conjecture for D(C ∗ (BG)). Theorem 5.5. Let C be a localising subcategory of D(C ∗ (BG)). Then the following are equivalent. (1) The localising subcategory C is smashing. (2) The localising subcategory C is generated by compact objects. (3) The support of C is specialisation closed. Proof. This is proved in the same way as Theorem 11.12 of [8]. 6. Induction and coinduction We have proved the main theorem without mentioning that there is a close relationship between induction and coinduction for compact Lie groups, just as there is for finite groups, but with an extra twist. Namely, there is an invertible C ∗ (BH )-module, tensoring with which takes one to the other. Theorem 6.1. Let G be a compact Lie subgroup and let H be a closed subgroup. If M is a C ∗ (BG)-module then coindG,H (M ) ∼ = coindG,H (C ∗ (BG)) ⊗C ∗ (BH ) indG,H (M ). Proof. Since G/H is a finite CW complex, C ∗ (BH ) is small as a C ∗ (BG)-module. So the evaluation map HomC ∗ (BG) (C ∗ (BH ), C ∗ (BG)) ⊗C ∗ (BG) M → HomC ∗ (BG) (C ∗ (BH ), M ) is an equivalence. The invertibility of the module coindG,H (C ∗ (BG)) depends on the Wirthmüller isomorphism, as we now describe. We begin with some generalities on equivariant stable homotopy theory. We work in the G-equivariant stable homotopy category GS . This is a compactly generated tensor triangulated category whose tensor product is X ∧ Y . This has an internal function spectrum which we denote Hom, characterised by the adjunction HomGS (X , Hom(Y , Z )) ∼ = HomGS (X ∧ Y , Z ). Furthermore, we have equivalences Hom(X , Hom(Y , Z )) ≃ Hom(X ∧ Y , Z ).
There is a fixed point functor GS → S sending X to X G . We write HomG (X , Y ) for the function spectrum of G-maps Hom(X , Y )G . There is also a quotient functor which is a homotopy functor on free spectra. This gives the homotopy quotient functor GS → S sending X to XhG = EG+ ∧G X . If H is a closed subgroup of G then there is a forgetful functor from GS to H S . It has a left adjoint sending X to G+ ∧H X and a right adjoint sending X to HomH (G+ , X ). Theorem 6.2 (Wirthmüller Isomorphism). Let H be a closed subgroup of a compact Lie group G, and let L be the tangent representation of H at the identity coset eH ∈ G/H. If X is an object in H S , then HomH (G+ , X ) is equivalent to G+ ∧H (Σ −L X ) as objects in GS . Proof. See Theorem 1.1 of May [19], or Chapter II of Lewis, May and Steinberger [15]. Substituting k for X , we obtain the following. Corollary 6.3. The function spectrum Hom(G/H+ , k) is equivalent to G+ ∧H Σ −L k.
Definition 6.4. We define the Borel G-spectrum b by b = Hom(EG+ , k). This is a commutative k-algebra object in GS . Lemma 6.5. For a based G-space X we have Hom(X , b)G ≃ C˜ ∗ (EG+ ∧G X ).
Thus b is the representing spectrum for Borel cohomology. Proof. From the definition of b we have Hom(X , b) = Hom(X , Hom(EG+ , k))
≃ Hom(EG+ ∧ X , k). Taking G-fixed points gives the result. It is worth noting a special case. Corollary 6.6. Let H be a closed subgroup of G. Then we have an equivalence Hom(G/H+ , b)G ≃ C ∗ (BH ).
D. Benson, J. Greenlees / Journal of Pure and Applied Algebra 218 (2014) 642–650
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Remark 6.7. This is an equivalence of C ∗ (BG)-modules. In particular, taking H = G we find C ∗ (BG) ≃ bG . Theorem 6.8. Let H be a closed subgroup of a compact Lie group G. Then there is an equivalence of C ∗ (BH )-modules C ∗ (BH −L ) ≃ HomC ∗ (BG) (C ∗ (BH ), C ∗ (BG)). Proof. We have Homb (Hom(G/H+ , b), b) ≃ Homb (G/H+ ∧ S −L ∧ b, b)
≃ Hom(G/H+ ∧ S , b) −L
(by the Wirthmüller isomorphism) (by extension of scalars).
Taking fixed points we find Homb (Hom(G/H+ , b), b)G ≃ C ∗ (BH −L ).
Next, passage to fixed points gives a map Homb (Hom(G/H+ , b), b)G → HombG (Hom(G/H+ , b)G , bG ) = HomC ∗ (BG) (C ∗ (BH ), C ∗ (BG))
which we claim is an equivalence. Indeed, we have a natural transformation Homb (M , b)G → HombG (M G , bG ).
It is an equivalence for M = b and hence for any small b-module, including the module M = Hom(G/H+ , b).
Corollary 6.9. coindG,H (C (BG)) ∼ = C ∗ (BH −L ) is an invertible C ∗ (BH )-module. ∗
Combining this with Theorem 6.1 we obtain the following. Theorem 6.10. coindG,H (M ) ≃ C ∗ (BH −L ) ⊗C ∗ (BH ) indG,H (M ). Remark 6.11. If the tangent representation L of H at the identity coset eH ∈ G/H is orientable then the Thom isomorphism gives C ∗ (BH −L ) ≃ C ∗ (Σ −d BH ) ≃ Σ d C ∗ (BH ) where d is the codimension of H in G. In this case the relationship between induction and coinduction is simply a degree shift of d. Even if it is not orientable then there is a subgroup of index two in H for which it is orientable. In this case there is a sign representation ε of H such that C ∗ (BH −L ) ≃ Σ d C ∗ (BH , ε). 7. Hochschild homology and cohomology In this section we use the relationship between induction and coinduction established in the last section in order to relate Hochschild homology and cohomology. We consider G × G acting on G by left and right multiplication:
(g1 , g2 )(g ) = g1 gg2−1 . This may be identified with the coset space (G × G)/∆(G) where ∆(G) is the diagonal copy of G in G × G:
∆(G) = {(g , g ) | g ∈ G} ≤ G × G. This is a closed subgroup, and the action of G on the tangent space at e∆(G) ∈ (G × G)/∆(G) may be identified with the conjugation action of G on its Lie algebra. Definition 7.1. The Hochschild homology HH∗ (C ∗ (BG)) is defined to be the homology of C ∗ (BG) ⊗C ∗ (B(G×G)) C ∗ (BG). The Hochschild cohomology HH ∗ (C ∗ (BG)) is defined to be the cohomology of HomC ∗ (B(G×G)) (C ∗ (BG), C ∗ (BG)). The relevant observation here is that this is an instance of induction and coinduction: indG×G,∆(G) (C ∗ (BG)) = C ∗ (BG) ⊗C ∗ (B(G×G)) C ∗ (BG) coindG×G,∆(G) (C ∗ (BG)) = HomC ∗ (B(G×G)) (C ∗ (BG), C ∗ (BG)). Therefore, applying Theorem 6.10 and Remark 6.11, we have the following theorem. Theorem 7.2. Suppose that the action of G on its Lie algebra by conjugation preserves orientation (this is true if G is finite, and also if G is connected) or that k has characteristic two. Then there are natural isomorphisms HH n (C ∗ (BG)) ∼ = HHn+dim G (C ∗ (BG)). If the action does not preserve orientation then HH n (C ∗ (BG)) ∼ = HHn+dim G (C ∗ (BG, ε)) where ε is the sign representation corresponding to the subgroup of G of index two that preserves orientation.
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Now Eilenberg–Moore gives an equivalence ∧ C ∗ (ΛBGp ) ≃ C ∗ (BG) ⊗C ∗ (B(G×G)) C ∗ (BG) ∧
where ΛBGp is the free loop space on the p-completion of BG. Therefore we have ∧ HH∗ (C ∗ (BG)) ∼ = H −∗ (ΛBGp ).
The fibration ∧
∧
∧
Ω BGp → ΛBGp → BGp
gives rise to a spectral sequence ∧
∧
H ∗ (BG; H ∗ (Ω BGp )) ⇒ H ∗ (ΛBGp ). ∧
It follows that if we can compute H ∗ (Ω BGp ), which is the subject of Levi [16–18], Benson [1], Benson, Greenlees and Shamir [3], then we are in a position to begin the computation of HH∗ (C ∗ (BG)) and therefore also of HH ∗ (C ∗ (BG)). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
∧
D.J. Benson, An algebraic model for chains on Ω (BGp ), Trans. Amer. Math. Soc. 361 (2009) 2225–2242. D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59–80. D.J. Benson, J.P.C. Greenlees, S. Shamir, Complete intersections and mod p cochains, Algebr. Geom. Topol. 13 (1) (2013) 61–114. D.J. Benson, S.B. Iyengar, H. Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008) 575–621. D.J. Benson, S.B. Iyengar, H. Krause, Localizing subcategories for cochains on the classifying space of a finite group, C. R. Acad. Sci. Paris, Série I 349 (2011) 953–956. D.J. Benson, S.B. Iyengar, H. Krause, Module categories for finite group algebras, in: A. Skowroński, K. Yamagata (Eds.), Representations of algebras and related topics, EMS Ser. Congr. Rep., 2011, pp. 55–83. D.J. Benson, S.B. Iyengar, H. Krause, Stratifying triangulated categories, J. Topol. 4 (2011) 641–666. D.J. Benson, S.B. Iyengar, H. Krause, Stratifying modular representations of finite groups, Ann. of Math. 174 (2011) 1643–1684. D.J. Benson, S.B. Iyengar, H. Krause, Colocalising subcategories and cosupport, J. Reine Angew. Math. 673 (2012) 161–207. D.J. Benson, S.B. Iyengar, H. Krause, Representations of finite groups: local cohomology and support, in: Oberwolfach Reports, Birkhäuser Verlag, Basel, 2012. D.J. Benson, H. Krause, Complexes of injective kG-modules, Algebra Number Theory 2 (2008) 1–30. L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976) 278–302. W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357–402. A.D. Elmendorf, I. Kříž, M.A. Mandell, J.P. May, Rings, modules and algebras in stable homotopy theory, in: Surveys and Monographs, vol. 47, American Math. Society, 1996. L.G. Lewis Jr., J.P. May, M. Steinberger, Equivariant stable homotopy theory, in: Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin/New York, 1986. R. Levi, On finite groups and homotopy theory, in: Mem. AMS, vol. 567, American Math. Society, 1995. R. Levi, A counter-example to a conjecture of Cohen, in: Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guíxols, 1994), Birkhäuser Verlag, Basel, 1996, pp. 261–269. ∧ R. Levi, On homological rate of growth and the homotopy type of Ω BGp , Math. Z. 226 (1997) 429–444. J.P. May, The Wirthmüller isomorphism revisited, Theory Appl. Categ. 11 (2003) 132–142. D.G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971) 549–572. D.G. Quillen, The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971) 573–602. J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965) 413–420. B.B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSSR 127 (1959) 943–944.
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Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa
Stratifying the derived category of cochains on BG for G a compact Lie group David Benson a,∗ , John Greenlees b a
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, UK
b
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
article
info
Article history: Received 30 January 2013 Available online 23 August 2013 Communicated by C.A. Weibel MSC: Primary: 18E30; 55R35 Secondary: 20C20
abstract The main purpose of this paper is to classify the localising subcategories of the derived category D(C ∗ (BG; k)) where G is a compact Lie group and k is a field. We also prove a version of Chouinard’s theorem for D(C ∗ (BG; k)), we describe the relationship between induction and coinduction for a closed subgroup of G, and we use this to describe the relationship between Hochschild homology and cohomology of C ∗ (BG; k). © 2013 Elsevier B.V. All rights reserved.
1. Introduction If G is a finite group and k is a field then there is a close relationship between the stable module category StMod(kG), the homotopy category of complexes of injective modules K(Inj kG), and the derived category D(C ∗ (BG)) of the cochains on the classifying space C ∗ (BG) = C ∗ (BG; k). This was investigated by Dwyer, Greenlees and Iyengar [13] and by Benson and Krause [11]. If H is a subgroup of G then the restriction map StMod(kG) → StMod(kH ) corresponds under this relationship to the induction functor, namely the derived tensor product C ∗ (BH ) ⊗C ∗ (BG) − : D(C ∗ (BG)) → D(C ∗ (BH )), which happens to be isomorphic to the coinduction functor, namely the derived homs HomC ∗ (BG) (C ∗ (BH ), −) : D(C ∗ (BG)) → D(C ∗ (BH )), see Section 7 of [11]. Chouinard’s theorem [12] for StMod(kG) states that an object is zero if and only if its restriction to kE is zero for every elementary abelian p-subgroup E of G. This was extended in [11] to the categories K(Inj kG) and D(C ∗ (BG)). The translation to D(C ∗ (BG)) states that an object is zero if and only if its induction to D(C ∗ (BE )) is zero for all E. Most known proofs of Chouinard’s theorem use Serre’s theorem [22] on products of Bocksteins. If G is a compact Lie group, we no longer have an obvious analogue of the stable module category, but it still makes sense to ask whether an object in D(C ∗ (BG)) is zero if and only if its induction to each finite elementary abelian p-subgroup of G is zero. Our first theorem shows that this is indeed the case. Serre’s theorem does not seem to be the right approach in this case. Instead, in Section 3 we use methods from Quillen’s work [20,21] on the spectrum of H ∗ (BG; k) to prove the following theorem. Theorem 1.1. Let G be a compact Lie group and k a field of characteristic p. A module M is isomorphic to zero in D(C ∗ (BG)) if and only if C ∗ (BE ) ⊗C ∗ (BG) M is isomorphic to zero in D(C ∗ (BE )) for every finite elementary abelian p-subgroup E of G.
∗
Corresponding author. E-mail address: [email protected] (D. Benson).
0022-4049/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jpaa.2013.08.004
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In a series of papers by Benson, Iyengar and Krause [4–10] the machinery was set up for classifying localising subcategories of triangulated categories, and used in the cases of the closely related categories StMod(kG), K(Inj kG) and D(C ∗ (BG)) for G finite. In these cases, the classification went via reduction to elementary abelian subgroups using the appropriate version of Chouinard’s theorem. Using the version given in Theorem 1.1, we prove the following theorem in Section 4. For the definition of stratification, see Section 4.2 of [7]. Theorem 1.2. Let G be a compact Lie group. Then the category D(C ∗ (BG)) is stratified by the canonical action of H ∗ (BG). Informally, this theorem states that the theory of support gives a bijection between localising subcategories of D(C ∗ (BG)) and subsets of VG = Spec H ∗ (BG), the spectrum of homogeneous prime ideals in H ∗ (BG). We have stated Theorem 1.1 for induction. There is also a version for coinduction, which we prove at the same time. However, just as for finite groups there is a close relationship between induction and coinduction, which we investigate in Section 6. We state here a version with a mild orientation assumption. Theorem 1.3. Let H be a closed subgroup of a compact Lie group G, and let d = dim(G/H ). Suppose that the action of H on the tangent space at the identity coset eH ∈ G/H preserves orientation, or that k has characteristic two. If M is a C ∗ (BG)-module then in D(C ∗ (BH )) we have coindG,H (M ) ≃ Σ d indG,H (M ). Finally, we use the relationship between induction and coinduction to examine the relationship between Hochschild homology and cohomology. Again we state here a version with a mild orientation assumption. Theorem 1.4. Suppose that the action of G on its Lie algebra by conjugation preserves orientation, or that k has characteristic two. Then there are isomorphisms HH n (C ∗ (BG)) ∼ = HHn+dim G (C ∗ (BG)). 2. The cochains on BG and its derived category Let G be a compact Lie group and let k be a field of characteristic p. We begin by recalling the elementary properties of the derived category D(C ∗ (BG)). We use the language of S-algebras, and for definiteness we use the model of Elmendorf, Kříž, Mandell and May [14]. In this language, if X is a space then C ∗ (X ) = C ∗ (X ; k) denotes the function spectrum from X to the Eilenberg–Mac Lane spectrum of k. This has the advantage of being a commutative S-algebra, whereas the more conventional DGA of cochains on X with coefficients in k is not equivalent to a commutative DGA. We have
π−n C ∗ (X ) ∼ = H n (X ) = H n (X ; k).
(2.1)
We write D(C ∗ (X )) for the derived category of C ∗ (X ). The objects in this derived category are the C ∗ (X )-module spectra, while the arrows are obtained from the homotopy classes of module spectrum maps by inverting the quasi-isomorphisms, namely the maps which become isomorphisms after applying π∗ . One of the advantages of commutativity of C ∗ (X ) is that it makes good sense to talk of the derived tensor product A ⊗C ∗ (X ) B and derived homomorphisms HomC ∗ (X ) (A, B) of two objects A and B in D(C ∗ (X )) and to regard the result as again being an object in D(C ∗ (X )), and there is a spectral sequence ExtH ∗ (X ) (π∗ A, π∗ B) ⇒ π∗ HomC ∗ (X ) (A, B). ∗,∗
This way, D(C ∗ (X )) is a tensor triangulated category whose tensor identity is C ∗ (X ). Since D(C ∗ (X )) is generated as a triangulated category by the tensor identity C ∗ (X ), it follows that every localising subcategory of D(C ∗ (X )) is a tensor ideal. It also follows that D(C ∗ (X )) is compactly generated, and that the full subcategory of compact objects Dc (C ∗ (X )) is the thick subcategory generated by C ∗ (X ). Next we recall Quillen’s stratification theorem in the form needed for our work. See Theorem 10.2 and the discussion following Proposition 11.2 in Quillen [21]. Let VG be the spectrum of homogeneous prime ideals of H ∗ (BG) = H ∗ (BG; k). For each closed subgroup H of G there is a map BH → BG well defined up to homotopy, and hence restriction homomorphisms resG,H : C ∗ (BG) → C ∗ (BH ) and resG,H : H ∗ (BG) → H ∗ (BH ). This way we obtain a map res∗G,H : VH → VG . For each point p in VG there exists a finite elementary abelian p-subgroup E of G such that p is in the image of res∗G,E . We say that p originates in such an E if there does not exist a proper subgroup E ′ of E such that p is in the image of res∗G,E ′ . Theorem 2.2 (Quillen). For each p ∈ VG , the pairs (E , q) where p = res∗G,E (q) and such that p originates in E are all G-conjugate. This sets up a one-to-one correspondence between primes p in VG and G-conjugacy classes of such pairs (E , q).
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If ζ ∈ H n (BG) then (2.1) allows us to represent ζ by a well defined morphism in D(C ∗ (BG))
ζˆ : C ∗ (BG) → Σ n C ∗ (BG). If M is a C ∗ (BG)-module then tensoring with M gives a well defined morphism
ζˆ ⊗C ∗ (BG) IdM : M → Σ n M . This way we obtain a well defined homomorphism to the graded centre of the derived category H ∗ (BG) → Z (D(C ∗ (BG))).
(2.3)
Lemma 2.4. In the language of Notation 4.1 of [4], the homomorphism (2.3) makes the derived category D(C ∗ (BG)) into an H ∗ (BG)-linear triangulated category. Proof. The only thing that needs to be checked is that H ∗ (BG) is Noetherian. This is a theorem of Venkov [23]. If H is a closed subgroup of G then the map BH → BG induces a map of S-algebras C ∗ (BG) → C ∗ (BH ). We therefore have a restriction map res∗G,H : D(C ∗ (BH )) → D(C ∗ (BG)). As above, an element ζ ∈ H n (BG) induces a well defined morphism ζˆ : C ∗ (BG) → Σ n C ∗ (BG) in D(C ∗ (BG)), and this in turn induces a morphism n ∗ ˆ ∗ res G,H (ζ ) = IdC ∗ (BH ) ⊗C ∗ (BG) ζ : C (BH ) → Σ C (BH )
in D(C ∗ (BH )). Restriction has a left adjoint indG,H =
C ∗ (BH ) C
∗
(BH )C ∗ (BG) ⊗C ∗ (BG) − : D(C ∗ (BG)) → D(C ∗ (BH ))
and a right adjoint coindG,H = HomC ∗ (BG) (C ∗ (BG) C ∗ (BH )C ∗ (BH ) , −) : D(C ∗ (BG)) → D(C ∗ (BH )). The following lemma summarises the standard properties of restriction, induction and coinduction. Lemma 2.6. Let M be a C ∗ (BH )-module and N , N ′ be C ∗ (BG)-modules. Then (i) res∗G,H (M ⊗C ∗ (BH ) indG,H (N )) ≃ res∗G,H (M ) ⊗C ∗ (BG) N, (ii) indG,H (N ) ⊗C ∗ (BH ) indG,H (N ′ ) ≃ indG,H (N ⊗C ∗ (BG) N ′ ), (iii) res∗G,H (HomC ∗ (BH ) (M , coindG,H (N ))) ≃ HomC ∗ (BG) (res∗G,H (M ), N ),
(iv) HomC ∗ (BH ) (indG,H (N ), coindG,H (N ′ )) ≃ coindG,H (HomC ∗ (BG) (N , N ′ )). (v) If ζ ∈ H n (BG) then n ˆ res G,H (ζ ) ⊗C ∗ (BG) IdindG,H (N ) = indG,H (ζ ⊗C ∗ (BH ) IdN ) : indG,H (N ) → Σ indG,H (N ).
Proof. Using the associativity of tensor product and the Hom–tensor adjunction, we have isomorphisms (i) of C ∗ (BG)-modules M ⊗C ∗ (BH ) (C ∗ (BH ) ⊗C ∗ (BG) N ) ≃ M ⊗C ∗ (BG) N , (ii) of C ∗ (BH )-modules
(C ∗ (BH ) ⊗C ∗ (BG) N ) ⊗C ∗ (BH ) (C ∗ (BH ) ⊗C ∗ (BG) N ′ ) ≃ C ∗ (BH ) ⊗C ∗ (BG) (N ⊗C ∗ (BG) N ′ ), (iii) of C ∗ (BG)-modules HomC ∗ (BH ) (M , HomC ∗ (BG) (C ∗ (BH ), N )) ≃ HomC ∗ (BG) (C ∗ (BH ) ⊗C ∗ (BH ) M , N ), (iv) of C ∗ (BH )-modules HomC ∗ (BH ) (C ∗ (BH ) ⊗C ∗ (BG) N , HomC ∗ (BG) (C ∗ (BH ), N ′ ))
≃ HomC ∗ (BG) (C ∗ (BH ) ⊗C ∗ (BH ) C ∗ (BH ) ⊗C ∗ (BG) N , N ′ ) ≃ HomC ∗ (BG) (C ∗ (BH ) ⊗C ∗ (BG) N , N ′ ). (v) This follows by tensoring N with (2.5).
(2.5)
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3. Chouinard’s theorem Let G be a compact Lie group and k a field of characteristic p. The first step to understanding the category of modules over C ∗ (BG) = C ∗ (BG; k) for a compact Lie group G is to show that it is controlled by elementary abelian p-subgroups. In effect we use a part of Quillen’s method for understanding the spectrum of H ∗ (BG). Choose an embedding of G into a unitary group U. We write T for the maximal torus of U, namely the subgroup of diagonal matrices. We write S for the subgroup of elements of order dividing p in T . Thus S is an elementary abelian p-subgroup of the same rank as T , and every elementary abelian p-subgroup of G is conjugate to a subgroup of S. Finally we write F = U /S, and we note that as a G-space, all its isotropy groups are elementary abelian p-groups. Theorem 3.1. Let G be a compact Lie group and k a field of characteristic p. (i) As an object in D(C ∗ (BG)), the module C ∗ (BG) is in the thick subcategory generated by the modules C ∗ (BE ) as E runs over the elementary abelian p-subgroups of G. (ii) If M is a C ∗ (BG)-module and C ∗ (BE ) ⊗C ∗ (BG) M ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ 0. (iii) If M is a C ∗ (BG)-module and HomC ∗ (BG) (C ∗ (BE ), M ) ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ 0. Proof. (i) The G-space F is built from finitely many cells G/E, and hence C ∗ (EG ×G F ) is finitely built from modules C ∗ (EG ×G (G/E )) ≃ C ∗ (BE ). For any U-spaces A and B we have a natural map C ∗ (EU ×U A) ⊗C ∗ (BU ) C ∗ (EU ×U B) → C ∗ (EU ×U (A × B)). Taking A = U ×G X and B = F = U /S, we obtain
ν : C ∗ (EG ×G X ) ⊗C ∗ (BU ) C ∗ (BS ) → C ∗ (EG ×G (X × F )) natural in X . Now observe that H ∗ (BS ) is free over H ∗ (BT ) and H ∗ (BT ) is free over H ∗ (BU ), and hence ν is an equivalence for all X (Lemma 6.5 of Quillen [20]). In particular it follows that C ∗ (EG ×G X ) is a summand of C ∗ (EG ×G (X × F )) as C ∗ (BG)-modules. Taking X = ∗ we conclude that C ∗ (BG) is a summand of C ∗ (EG ×G F ), and is hence in the thick subcategory of D(C ∗ (BG)) generated by the C ∗ (BE ). (ii) Using (i), we see that if C ∗ (BE ) ⊗C ∗ (BG) M ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ C ∗ (BG) ⊗C ∗ (BG) M ≃ 0. (iii) Similarly, if HomC ∗ (BG) (C ∗ (BE ), M ) ≃ 0 for every elementary abelian p-subgroup E of G then M ≃ HomC ∗ (BG) (C ∗ (BG), M ) ≃ 0. Remark 3.2. In the theorem, it is clearly sufficient to use just one representative of each conjugacy class of maximal elementary abelian p-subgroups. Remark 3.3. If G is a finite p-group then D(C ∗ (BG)) is equivalent to K(Inj kG), see [11], and StMod(kG) is equivalent to the localising subcategory of K(Inj kG) consisting of acyclic objects. In this case, the above theorem gives a new proof of Chouinard’s theorem which avoids the use of Serre’s theorem on the product of Bocksteins. 4. Localising subcategories In this section we classify the localising subcategories of D(C ∗ (BG)). Most of the proof is the same as the proof of Theorem 9.7 of [8], but the role of Proposition 9.6 of that paper is played by our version of Chouinard’s theorem. Since the roles of restriction and induction are reversed, and some of the details are slightly different, we have chosen to write out the proof in full. For p ∈ VG , let Γp : D(C ∗ (BG)) → D(C ∗ (BG)) be the local cohomology functor described in [4]. Definition 4.1. The support of a C ∗ (BG)-module M is defined to be
VG (M ) = {p ∈ VG | Γp (M ) ̸≃ 0}. The following are the analogues of Lemma 9.3 and Proposition 9.4 of [8]. Lemma 4.2. Let H be a closed subgroup of G. Fix p ∈ VG and let U = (res∗G,H )−1 {p} ⊆ VH . (i) For any C ∗ (BG)-module M we have indG,H (Γp (M )) ≃
q∈U Γq (indG,H (M )). ∗ (ii) For any C ∗ (BH )-module N we have Γp (res∗G,H (N )) ≃ q∈U resG,H (Γq (N )).
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Proof. (i) This follows from Corollary 7.10 of [9], with T = D(C ∗ (BG)), U = D(C ∗ (BH )), R = H ∗ (BG), S = H ∗ (BH ) and F = indG,H with right adjoint res∗G,H and left adjoint existing by Brown representability. (ii) Using part (i) and Lemma 2.6(i), we have
Γp res∗G,H (N ) ≃ res∗G,H (N ) ⊗C ∗ (BG) Γp C ∗ (BG)
≃ res∗G,H (N ⊗C ∗ (BH ) indG,H (Γp C ∗ (BG))) ∗ ∗ Γq C (BH ) ≃ resG,H N ⊗C ∗ (BH ) q∈U
≃
resG,H Γq (N ). ∗
q∈U
Proposition 4.3. Let H be a closed subgroup of G. (i) For any C ∗ (BG)-module M we have
VG (res∗G,H indG,H (M )) ⊆ VG (M ). (ii) For any C ∗ (BH )-module N we have
VG (res∗G,H (N )) = res∗G,H VH (N ). Proof. (i) If p ∈ VG (res∗G,H indG,H (M )) then 0 ̸= Γp (C ∗ (BH ) ⊗C ∗ (BG) M ) = C ∗ (BH ) ⊗C ∗ (BG) Γp (M ) and so Γp (M ) ̸= 0. (ii) By Lemma 4.2(ii) the condition p ∈ VG (res∗G,H (N )) is equivalent to the statement that there exists q ∈ VH such that ∗ resG,H (q) = p and Γq (N ) ̸= 0. Part (ii) of the following is the elementary abelian version of the subgroup theorem for D(C ∗ (BG)), and the proof is copied from Theorem 9.5 of [8]. The full version of the subgroup theorem will be a consequence of the stratification theorem. Theorem 4.4. (i) The category D(C ∗ (BE )) is stratified by the action of H ∗ (BE ). (ii) Let E ′ ≤ E be elementary abelian p-groups. For any object X in D(C ∗ (BE )) there is an equality
VE ′ (indE ,E ′ (X )) = (res∗E ,E ′ )−1 VE (X ). Proof. (i) By Theorem 4.2 of [11] we have an equivalence K(Inj kE ) ≃ D(C ∗ (BE )).
The maps from H ∗ (BE ) to the endomorphisms of the tensor identities of these two categories agree, so by Lemma 3.10 of [8], D(C ∗ (BE )) is stratified by the action of H ∗ (BE ) and the equivalence preserves support. (ii) Fix a prime q in VE ′ and set p = res∗E ,E ′ (q). Proposition 4.3(ii) shows that
VE (res∗E ,E ′ Γq C ∗ (BE ′ )) = {p} = VE (Γp C ∗ (BE )). By (i) we therefore have Loc(res∗E ,E ′ Γq C ∗ (BE ′ )) = Loc(Γp C ∗ (BE )).
This implies that if M is a C ∗ (BE )-module then res∗E ,E ′ Γq C ∗ (BE ′ ) ⊗C ∗ (BE ) M ̸= 0 if and only if Γp C ∗ (BE ) ⊗C ∗ (BE ) M ̸= 0. Thus
Γq indE ,E ′ (M ) ̸= 0 ⇐⇒ Γq C ∗ (BE ′ ) ⊗C ∗ (BE ′ ) indE ,E ′ (M ) ̸= 0
⇐⇒ res∗E ,E ′ (Γq C ∗ (BE ′ )) ⊗C ∗ (BE ) M ̸= 0 ⇐⇒ Γp C ∗ (BE ) ⊗C ∗ (BE ) M ̸= 0 ⇐⇒ Γp M ̸= 0. The second implication follows from Lemma 2.6(i) and the fact that res∗E ,E ′ is faithful. The following is the main theorem of this section. Theorem 4.5. Let G be a compact Lie group. The triangulated category D(C ∗ (BG)) is stratified by the canonical action of the cohomology algebra H ∗ (BG). In particular, there is a bijection between the localising subcategories of D(C ∗ (BG)) and the subsets of VG . The localising subcategory corresponding to V is the full subcategory of C ∗ (BG)-modules M such that VG (M ) ⊆ V .
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Proof. For the statement in the first paragraph of the theorem, we need to prove that for p ∈ VG , the subcategory Γp D(C ∗ (BG)) is minimal among localising subcategories of D(C ∗ (BG)). Let M be a non-zero module in Γp D(C ∗ (BG)). Then by Theorem 3.1(ii), there exists an elementary abelian p-subgroup E0 ≤ G such that indG,E0 (M ) ̸≃ 0. Choose a prime q0 ∈ VE0 (indG,E0 (M )). By Proposition 4.3(i) we have res∗G,E0 (q0 ) ∈ VG (res∗G,E0 indG,E0 (M )) ⊆ VG (M ) = {p} and so res∗G,E0 (q0 ) = p. It follows that for some pair (E , q) corresponding to p as in Theorem 2.2, we have E0 ≥ E and q0 = res∗E0 ,E (q). It then follows from Theorem 4.4(ii) that q ∈ VE (indG,E (M )). By Theorem 2.2 all pairs (E , q) where p originates in E are conjugate. So if we choose one, then each non-zero M ∈ Γp D(C ∗ (BG)) has Γq (indG,E (M )) ̸= 0. Now let N be another non-zero object in Γp D(C ∗ (BG)). Then Hom∗C ∗ (BG) (C ∗ (BE ) ⊗C ∗ (BG) M , N ) ≃ Hom∗C ∗ (BE ) (indG,E (M ), indG,E (N )).
(4.6)
By Lemma 4.2(i), Γq indG,E (M ) and Γq indG,E (N ) are non-zero direct summands of indG,E (M ) and indG,E (N ) respectively. Now it follows from Theorem 4.4(i) that Γq D(C ∗ (BE )) is a minimal localising subcategory of D(C ∗ (BE )). So by Lemma 4.1 of [7] Hom∗C ∗ (BE ) (Γq indG,E (M ), Γq indG,E (N )) ̸= 0. Thus by (4.6) we have Hom∗C ∗ (BG) (C ∗ (BE ) ⊗C ∗ (BG) M , N ) ̸= 0. Now D(C ∗ (BG)) is generated by the tensor identity C ∗ (BG), so C ∗ (BE ) ⊗C ∗ (BG) M is in the localising subcategory generated by M. Hence we have HomC ∗ (BG) (M , N ) ̸= 0. Again using Lemma 4.1 of [7], we conclude that Γp D(C ∗ (BG)) is a minimal localising subcategory of D(C ∗ (BG)). The statement in the second paragraph of the theorem follows from the first paragraph together with Theorem 4.2 of [7]. Remark 4.7. The case of the above theorem where G is a finite group is the main theorem of [5]. The methods are somewhat different. Question 4.8. For what spaces X is D(C ∗ (X )) stratified by H ∗ (X )? Our main theorem shows that this is the case for X = BG, G a compact Lie group. 5. Consequences of the stratification We begin with the tensor product theorem. Theorem 5.1. Let M and N be C ∗ (BG)-modules. Then
VG (M ⊗C ∗ (BG) N ) = VG (M ) ∩ VG (N ). Proof. This follows from Theorem 4.5 together with Theorem 7.3 of [7]. Next, we have the subgroup theorem. Theorem 5.2. Let H be a closed subgroup of G and let M be a C ∗ (BG)-module. Then
VH (indG,H (M )) = (res∗G,H )−1 VG (M ). Proof. The proof of this is the same as the proof of Theorem 4.4(ii) except that one uses Theorem 4.5 instead of Theorem 4.4(i). The following is the analogue in this situation of the main theorem of Benson, Carlson and Rickard [2]. Theorem 5.3. There is a bijection between the thick subcategories of the compact objects Dc (C ∗ (BG)) and the specialisation closed subsets of VG . The thick subcategory corresponding to V ⊆ VG is the full subcategory of C ∗ (BG)-modules M in Dc (C ∗ (BG)) such that VG (M ) ⊆ V . Proof. This follows from Theorem 4.5 together with Theorem 6.1 of [7]. A localising subcategory C of a triangulated category T is said to be strictly localising if the inclusion has a right adjoint. This is equivalent to the existence of a localisation functor L : T → T such that an object X of T is in C if and only if LX = 0. Lemma 5.4. Every localising subcategory of D(C ∗ (BG)) is strictly localising. Proof. The proof is the same as the proof of Lemma 11.11 of [8].
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A strictly localising subcategory is smashing if the localisation functor L preserves coproducts. For a tensor triangulated category generated by the tensor unit 1, this is equivalent to the statement that the natural transformation L1 ⊗ X → LX is an isomorphism for all X in T . The following is the telescope conjecture for D(C ∗ (BG)). Theorem 5.5. Let C be a localising subcategory of D(C ∗ (BG)). Then the following are equivalent. (1) The localising subcategory C is smashing. (2) The localising subcategory C is generated by compact objects. (3) The support of C is specialisation closed. Proof. This is proved in the same way as Theorem 11.12 of [8]. 6. Induction and coinduction We have proved the main theorem without mentioning that there is a close relationship between induction and coinduction for compact Lie groups, just as there is for finite groups, but with an extra twist. Namely, there is an invertible C ∗ (BH )-module, tensoring with which takes one to the other. Theorem 6.1. Let G be a compact Lie subgroup and let H be a closed subgroup. If M is a C ∗ (BG)-module then coindG,H (M ) ∼ = coindG,H (C ∗ (BG)) ⊗C ∗ (BH ) indG,H (M ). Proof. Since G/H is a finite CW complex, C ∗ (BH ) is small as a C ∗ (BG)-module. So the evaluation map HomC ∗ (BG) (C ∗ (BH ), C ∗ (BG)) ⊗C ∗ (BG) M → HomC ∗ (BG) (C ∗ (BH ), M ) is an equivalence. The invertibility of the module coindG,H (C ∗ (BG)) depends on the Wirthmüller isomorphism, as we now describe. We begin with some generalities on equivariant stable homotopy theory. We work in the G-equivariant stable homotopy category GS . This is a compactly generated tensor triangulated category whose tensor product is X ∧ Y . This has an internal function spectrum which we denote Hom, characterised by the adjunction HomGS (X , Hom(Y , Z )) ∼ = HomGS (X ∧ Y , Z ). Furthermore, we have equivalences Hom(X , Hom(Y , Z )) ≃ Hom(X ∧ Y , Z ).
There is a fixed point functor GS → S sending X to X G . We write HomG (X , Y ) for the function spectrum of G-maps Hom(X , Y )G . There is also a quotient functor which is a homotopy functor on free spectra. This gives the homotopy quotient functor GS → S sending X to XhG = EG+ ∧G X . If H is a closed subgroup of G then there is a forgetful functor from GS to H S . It has a left adjoint sending X to G+ ∧H X and a right adjoint sending X to HomH (G+ , X ). Theorem 6.2 (Wirthmüller Isomorphism). Let H be a closed subgroup of a compact Lie group G, and let L be the tangent representation of H at the identity coset eH ∈ G/H. If X is an object in H S , then HomH (G+ , X ) is equivalent to G+ ∧H (Σ −L X ) as objects in GS . Proof. See Theorem 1.1 of May [19], or Chapter II of Lewis, May and Steinberger [15]. Substituting k for X , we obtain the following. Corollary 6.3. The function spectrum Hom(G/H+ , k) is equivalent to G+ ∧H Σ −L k.
Definition 6.4. We define the Borel G-spectrum b by b = Hom(EG+ , k). This is a commutative k-algebra object in GS . Lemma 6.5. For a based G-space X we have Hom(X , b)G ≃ C˜ ∗ (EG+ ∧G X ).
Thus b is the representing spectrum for Borel cohomology. Proof. From the definition of b we have Hom(X , b) = Hom(X , Hom(EG+ , k))
≃ Hom(EG+ ∧ X , k). Taking G-fixed points gives the result. It is worth noting a special case. Corollary 6.6. Let H be a closed subgroup of G. Then we have an equivalence Hom(G/H+ , b)G ≃ C ∗ (BH ).
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Remark 6.7. This is an equivalence of C ∗ (BG)-modules. In particular, taking H = G we find C ∗ (BG) ≃ bG . Theorem 6.8. Let H be a closed subgroup of a compact Lie group G. Then there is an equivalence of C ∗ (BH )-modules C ∗ (BH −L ) ≃ HomC ∗ (BG) (C ∗ (BH ), C ∗ (BG)). Proof. We have Homb (Hom(G/H+ , b), b) ≃ Homb (G/H+ ∧ S −L ∧ b, b)
≃ Hom(G/H+ ∧ S , b) −L
(by the Wirthmüller isomorphism) (by extension of scalars).
Taking fixed points we find Homb (Hom(G/H+ , b), b)G ≃ C ∗ (BH −L ).
Next, passage to fixed points gives a map Homb (Hom(G/H+ , b), b)G → HombG (Hom(G/H+ , b)G , bG ) = HomC ∗ (BG) (C ∗ (BH ), C ∗ (BG))
which we claim is an equivalence. Indeed, we have a natural transformation Homb (M , b)G → HombG (M G , bG ).
It is an equivalence for M = b and hence for any small b-module, including the module M = Hom(G/H+ , b).
Corollary 6.9. coindG,H (C (BG)) ∼ = C ∗ (BH −L ) is an invertible C ∗ (BH )-module. ∗
Combining this with Theorem 6.1 we obtain the following. Theorem 6.10. coindG,H (M ) ≃ C ∗ (BH −L ) ⊗C ∗ (BH ) indG,H (M ). Remark 6.11. If the tangent representation L of H at the identity coset eH ∈ G/H is orientable then the Thom isomorphism gives C ∗ (BH −L ) ≃ C ∗ (Σ −d BH ) ≃ Σ d C ∗ (BH ) where d is the codimension of H in G. In this case the relationship between induction and coinduction is simply a degree shift of d. Even if it is not orientable then there is a subgroup of index two in H for which it is orientable. In this case there is a sign representation ε of H such that C ∗ (BH −L ) ≃ Σ d C ∗ (BH , ε). 7. Hochschild homology and cohomology In this section we use the relationship between induction and coinduction established in the last section in order to relate Hochschild homology and cohomology. We consider G × G acting on G by left and right multiplication:
(g1 , g2 )(g ) = g1 gg2−1 . This may be identified with the coset space (G × G)/∆(G) where ∆(G) is the diagonal copy of G in G × G:
∆(G) = {(g , g ) | g ∈ G} ≤ G × G. This is a closed subgroup, and the action of G on the tangent space at e∆(G) ∈ (G × G)/∆(G) may be identified with the conjugation action of G on its Lie algebra. Definition 7.1. The Hochschild homology HH∗ (C ∗ (BG)) is defined to be the homology of C ∗ (BG) ⊗C ∗ (B(G×G)) C ∗ (BG). The Hochschild cohomology HH ∗ (C ∗ (BG)) is defined to be the cohomology of HomC ∗ (B(G×G)) (C ∗ (BG), C ∗ (BG)). The relevant observation here is that this is an instance of induction and coinduction: indG×G,∆(G) (C ∗ (BG)) = C ∗ (BG) ⊗C ∗ (B(G×G)) C ∗ (BG) coindG×G,∆(G) (C ∗ (BG)) = HomC ∗ (B(G×G)) (C ∗ (BG), C ∗ (BG)). Therefore, applying Theorem 6.10 and Remark 6.11, we have the following theorem. Theorem 7.2. Suppose that the action of G on its Lie algebra by conjugation preserves orientation (this is true if G is finite, and also if G is connected) or that k has characteristic two. Then there are natural isomorphisms HH n (C ∗ (BG)) ∼ = HHn+dim G (C ∗ (BG)). If the action does not preserve orientation then HH n (C ∗ (BG)) ∼ = HHn+dim G (C ∗ (BG, ε)) where ε is the sign representation corresponding to the subgroup of G of index two that preserves orientation.
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Now Eilenberg–Moore gives an equivalence ∧ C ∗ (ΛBGp ) ≃ C ∗ (BG) ⊗C ∗ (B(G×G)) C ∗ (BG) ∧
where ΛBGp is the free loop space on the p-completion of BG. Therefore we have ∧ HH∗ (C ∗ (BG)) ∼ = H −∗ (ΛBGp ).
The fibration ∧
∧
∧
Ω BGp → ΛBGp → BGp
gives rise to a spectral sequence ∧
∧
H ∗ (BG; H ∗ (Ω BGp )) ⇒ H ∗ (ΛBGp ). ∧
It follows that if we can compute H ∗ (Ω BGp ), which is the subject of Levi [16–18], Benson [1], Benson, Greenlees and Shamir [3], then we are in a position to begin the computation of HH∗ (C ∗ (BG)) and therefore also of HH ∗ (C ∗ (BG)). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
∧
D.J. Benson, An algebraic model for chains on Ω (BGp ), Trans. Amer. Math. Soc. 361 (2009) 2225–2242. D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59–80. D.J. Benson, J.P.C. Greenlees, S. Shamir, Complete intersections and mod p cochains, Algebr. Geom. Topol. 13 (1) (2013) 61–114. D.J. Benson, S.B. Iyengar, H. Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008) 575–621. D.J. Benson, S.B. Iyengar, H. Krause, Localizing subcategories for cochains on the classifying space of a finite group, C. R. Acad. Sci. Paris, Série I 349 (2011) 953–956. D.J. Benson, S.B. Iyengar, H. Krause, Module categories for finite group algebras, in: A. Skowroński, K. Yamagata (Eds.), Representations of algebras and related topics, EMS Ser. Congr. Rep., 2011, pp. 55–83. D.J. Benson, S.B. Iyengar, H. Krause, Stratifying triangulated categories, J. Topol. 4 (2011) 641–666. D.J. Benson, S.B. Iyengar, H. Krause, Stratifying modular representations of finite groups, Ann. of Math. 174 (2011) 1643–1684. D.J. Benson, S.B. Iyengar, H. Krause, Colocalising subcategories and cosupport, J. Reine Angew. Math. 673 (2012) 161–207. D.J. Benson, S.B. Iyengar, H. Krause, Representations of finite groups: local cohomology and support, in: Oberwolfach Reports, Birkhäuser Verlag, Basel, 2012. D.J. Benson, H. Krause, Complexes of injective kG-modules, Algebra Number Theory 2 (2008) 1–30. L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976) 278–302. W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357–402. A.D. Elmendorf, I. Kříž, M.A. Mandell, J.P. May, Rings, modules and algebras in stable homotopy theory, in: Surveys and Monographs, vol. 47, American Math. Society, 1996. L.G. Lewis Jr., J.P. May, M. Steinberger, Equivariant stable homotopy theory, in: Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin/New York, 1986. R. Levi, On finite groups and homotopy theory, in: Mem. AMS, vol. 567, American Math. Society, 1995. R. Levi, A counter-example to a conjecture of Cohen, in: Algebraic Topology: New Trends in Localization and Periodicity (Sant Feliu de Guíxols, 1994), Birkhäuser Verlag, Basel, 1996, pp. 261–269. ∧ R. Levi, On homological rate of growth and the homotopy type of Ω BGp , Math. Z. 226 (1997) 429–444. J.P. May, The Wirthmüller isomorphism revisited, Theory Appl. Categ. 11 (2003) 132–142. D.G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971) 549–572. D.G. Quillen, The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971) 573–602. J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965) 413–420. B.B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk. SSSR 127 (1959) 943–944.