Linckelmann, M.

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M. Linckelmann

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Induction for Interior Algebras

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Linckelmann, M. (2002). Induction for interior algebras. Quarterly Journal of Mathematics, 53(2), 195 - 200. doi: 10.1093/qjmath/53.2.195 <http://dx.doi.org/10.1093/qjmath/53.2.195>City Research OnlineOriginal citation: Linckelmann, M. (2002). Induction for interior algebras. Quarterly Journal of Mathematics, 53(2), 195 - 200. doi: 10.1093/qjmath/53.2.195 <http://dx.doi.org/10.1093/qjmath/53.2.195>Permanent City Research Online URL: http://openaccess.city.ac.uk/1908/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact  with the author(s) of this paper, please email the team at publications@city.ac.uk.INDUCTION FOR INTERIOR ALGEBRASMarkus LinckelmannJanuary 2001Abstract. The aim of this note is to give, in Section 2, a short self-contained proofof transitivity properties of non-injective induction for interior G-algebras, due to Puig[2, 3]. It turns out to be convenient to do this in a slightly more general setup, whichis described in Section 1. Section 3 contains some further formal properties of algebrainduction.1 Algebra induction - definition and examplesWe define an induction for interior algebras which generalises Puig’s concepts of in-duction, including its non-injective version, for interior G-algebras in [2], [3]. Through-out this note we fix a commutative ring R. By default, a module over an R−algebra A isa unitary left module. Any right A−module can be considered as a left module over theopposite algebra A0. Given three R−algebras A, B, C, an A−B−bimodule M and anA−C−bimodule N , we denote by HomA(M,N) the R−module of all homomorphismsfrom M to N as left A−modules; this has a canonical structure of C − B−bimodulegiven by (c.ϕ.b)(m) = ϕ(mb)c for any b ∈ B, c ∈ C and ϕ ∈ HomA(M,N). Similarly,if N ′ is a C − B−bimodule, we denote by HomB0(M,N′) the R−module consistingof all homomorphisms from M to N ′ viewed as right B−modules; again, this has acanonical structure of C −A−bimodules given by (c.ψ.a)(m) = cψ(am) for any a ∈ A,c ∈ C and ψ ∈ HomB0(M,N′).Following the terminology of [1] (which extends that of [2, 3.1]), ifA is anR−algebra,an interior A−algebra is an R−algebra B endowed with a unitary algebra homomor-phism σ : A → B. Note that in particular B becomes an A − A−bimodule throughσ.Definition 1.1 Let A, B be R−algebras, let C be an interior B−algebra, and letM be an A−B−bimodule. We setIndM (C) = EndC0(M ⊗BC) ,Typeset by AMS-TEX12 MARKUS LINCKELMANNconsidered as interior A−algebra with structural homomorphism A→ IndM (C) map-ping a ∈ A to the C0−endomorphism of M ⊗BC given by left multiplication with a onM ⊗BC.Remark 1.2 The above definition makes sense for differential graded algebras. Inorder to limit technicalities, we leave it to the reader to verify, that the content of thisnote can easily be adapted to this more general context.We briefly sketch, that this construction generalises the corresponding notions in [2,3.3] and [3, 3.3].Example 1.3 Let G be a finite group, H a subgroup of G and let C be an interiorH−algebra over R; that is, C is an R−algebra endowed with a group homomorphismH → C× (cf. [2, 3.1]). This group homomorphism extends to an algebra homo-morphism RH → C, through which C becomes an interior RH−algebra in the sensedefined above. Set M = RG, considered as RG−RH−bimodule. ThenIndGH(C)∼= IndM (C) ,where the left side is the induction defined in [2, 3.3]. To see this it suffices to applythe standard isomorphisms IndM (C) = EndC0(RG ⊗RHC) ∼= HomRH0(RG,RG ⊗RHC) ∼=RG ⊗RHC ⊗RHHomRH0(RG,RH) ∼= RG ⊗RHC ⊗RHRG = IndGH(C). Explicitly, the mapsending x ⊗ c ⊗ y ∈ IndGH(C) to the C0-endomorphism ϕ of RG ⊗RHC defined byϕ(z ⊗ d) = x⊗ cyzd if yz ∈ H and ϕ(z ⊗ d) = 0 otherwise is an algebra isomorphism(where x, y, z ∈ G and c, d ∈ C). See also [4, 10.7, 11.2].Example 1.4 Let G, H be finite groups, let ϕ : H → G be a group homomorphismand let C be an interior H−algebra. Set M = (RG)ϕ; that is, M is the RG −RH−bimodule which is equal to RG, endowed with the regular action of RG on theleft and with the action of RH on the right given by restriction through ϕ. ThenIndM (C) ∼= Indϕ(C) ,where the right side is the non-injective induction defined in [3, 3.3]. To see this,consider first the case where ϕ is surjective. Set K = ker(ϕ). Then IndM (C) =EndC0((RG)ϕ ⊗RHC) ∼= EndC0(R ⊗RKC) ∼= HomRK0(R,R ⊗RKC) ∼= (R ⊗RKC)K , whichcoincides with the expression in [3, 3.3.1]. The general case follows from combiningthis with 1.3 and the transitivity theorem 2.1 below.Example 1.5 Let A, B be R−algebras, and letM be an A−B−bimodule such thatthe functor M ⊗B− induces an equivalence of categories Mod(B) ∼= Mod(A). ConsiderB as interior B−algebra in the trivial way; that is, via the identity map on B. ThenMorita’s theorem reads IndM (B) ∼= A .INDUCTION FOR INTERIOR ALGEBRAS 32 Transitivity of algebra inductionThe transitivity theorem below generalises and simplifies the corresponding state-ment in [3, §3]:Theorem 2.1. Let A, B, C be R−algebras, let D be an interior C−algebra, let M bean A−B−bimodule and let N be a B −C−bimodule. There is a canonical homomor-phism of interior A−algebrasIndM (IndN (D)) −→ IndM⊗BN (D) ,which is an isomorphism, if M is finitely generated projective as right B−module, or ifN is finitely generated projective as right C−module, or if both N ⊗CD and M ⊗BN ⊗CDare finitely generated projective as right C−modules.The following technical lemma is needed in the proof of 2.1.Lemma 2.2. Let A, B, C be R−algebras, let M be an A − B−bimodule and let N ,N ′ be B − C−bimodules. Then the A−B−homomorphismΦ :{M ⊗BHomC0(N,N′) → HomC0(N,M ⊗BN ′)m⊗ ϕ 7→ (n 7→ m⊗ ϕ(n))is natural in the three variables M , N , N ′. Moreover, Φ is an isomorphism, if M isfinitely generated projective as right B−module, or if N is finitely generated projectiveas right C−module, or if both N ′, M ⊗BN ′ are finitely generated projective as rightC−modules.Proof. The fact that Φ is a homomorphism of A − B−bimodules which is natural inM , N , N ′ is a straightforward verification. In order to show the last statement, itsuffices to see that Φ is an R−linear isomorphism in any of the three cases statedabove. If M = B as right B−module, both sides are isomorphic to HomC0(N,N′).By naturality, it follows that Φ is an isomorphism whenever M is finitely generatedprojective as right B−module. Similarly, Φ is an isomorphism if N = C as rightC−module, and thus Φ is an isomorphism whenever N is finitely generated projectiveas right C−module. For the last case, if both N ′ and M ⊗BN ′ are finitely generatedprojective as right C−modules, we have canonical isomorphismsM⊗BHomC0(N,N′) ∼=M ⊗BN ′⊗CHomC0(N,C) ∼= HomC0(N,M ⊗BN ′) by applying twice the right analogue of4.3. A straightforward verification shows that the composition of the two isomorphismscoincides with Φ. 4 MARKUS LINCKELMANNProof of 2.1. Using the standard adjunction 4.2, we have a canonical R−isomorphismIndN (D) ∼= HomC0(N,N ⊗CD) .Applying this again, we get a canonical R−isomorphismIndM (IndN (D)) ∼= HomB0(M,M ⊗BHomC0(N,N ⊗CD)) .Applying this yet again to M ⊗BN yields an R−isomorphismIndM⊗BN (D) ∼= HomC0(M ⊗BN,M ⊗BN ⊗CD) ∼= HomB0(M,HomC0(N,M ⊗BN ⊗CD)) .Now the map Φ from 1.6 with N⊗CD instead of N ′ induces the algebra homomorphismwith the required properties. Remark 2.3 The algebra homomorphism in 2.1 can explicitly be described as follows(using the explicit formulae from §4). Let ϕ ∈ IndM (IndN (D)) and denote by ψ theimage in IndM⊗BN (D). Then, for any m ∈M , any n ∈ N and any d ∈ D we haveψ(m⊗ n⊗ d) =∑imi ⊗ ϕi(n⊗ d) ,where mi ∈ M , ϕi ∈ IndN (D) such that ϕ(m ⊗ IdN⊗CD) =∑imi ⊗ ϕi, and where iruns over a finite indexing set.3 Functoriality of algebra inductionProposition 3.1. Let A, B be R−algebras, let M be an A − B−bimodule, and letγ : C → C′ be a homomorphism of interior B−algebras. The functor − ⊗CC′ inducesa homomorphism of interior A−algebrasIndM (γ) : IndM (C) −→ IndM (C′) .Proof. Applying the functor − ⊗CC′ to any C0−endomorphism of M ⊗BC yields a(C′)0−endomorphism of M ⊗BC ⊗CC′ ∼= M ⊗BC′, where the left C−module structureof C′ is given by γ. Let A, B be R−algebras, let M be an A − B−bimodule, and let C be an interiorB−algebra. The algebra IndM (C) acts on M ⊗BC by α.n = α(n) for any α ∈ IndM (C)INDUCTION FOR INTERIOR ALGEBRAS 5and any n ∈ M ⊗BC. In this way, M ⊗BC becomes an IndM (C) − C− bimodule.This bimodule gives rise to induction and corestriction functors between the modulecategories of C and IndM (C), denoted byIndM = (M ⊗BC)⊗C− : Mod(C) −→ Mod(IndM (C)) ,ResM = HomIndM (C)(M ⊗BC,−) : Mod(IndM (C)) −→Mod(C) .The following is just a particular case of the standard adjunction 4.2:Proposition 3.2. Let A, B be R−algebras, let M be an A − B−bimodule and let Cbe an interior B−algebra. The functor IndM is left adjoint to the functor ResM .The following converse of 1.5 is again a particular case of Morita’s theorem:Proposition 3.3. Let A, B be R−algebras, let M be an A−B−bimodule and let C bean interior B−algebra. If M ⊗BC is a progenerator as right C−module, then IndM andResM are mutually inverse equivalences between Mod(IndM (C)) and Mod(C). Thiscase occurs in particular, if M is a progenerator as right B−module.An R−algebra A is called it symmetric, if A is finitely generated projective asR−module, and ifA is isomorphic to itsR−dual A∗ = HomR(A,R) as A−A−bimodule.Proposition 3.4. Let A, B be R−algebras, let M be an A − B−bimodule and letC be an interior B−algebra. If C is symmetric, B is finitely generated projective asR−module and M is finitely generated projective as right B−module, then IndM (C) issymmetric.Proof. The hypotheses imply that M ⊗BC and hence IndM (C) is finitely generatedprojective as R−module. Using the isomorphism IndM (C) = EndC0(M ⊗BC) ∼= (M ⊗BC)⊗C(M ⊗BC)∗ from 4.5 as well as 4.6 implies that IndC(M) is indeed symmetric. 4 AppendixWe collect in this section some of the (well-known) standard isomorphisms betweenmodules and homomorphism spaces as we need them in this note. Let A, B, C beR−algebras, let M be an A − B−bimodule and let N be an A − C−bimodule. Re-call from §1 that the space of homomorphisms HomA(M,N) from M to N as leftA−modules becomes a B − C−bimodule via (b.ϕ.c)(m) = ϕ(mb)c; similarly, thespace of homomorphisms HomB0(M,N′) of homomorphisms from M to N ′ as rightB−modules becomes a C − A−bimodule via (c.ψ.a)(m) = cψ(am). There is an iso-morphism of bifunctors6 MARKUS LINCKELMANN4.1.HomA(M ⊗B−,−) ∼= HomB(−,HomA(M,−))on Mod(B) ×Mod(A) given, for any A−module U and any B−module V , by themutually inverse isomorphisms4.2. HomA(M ⊗BV, U) ∼= HomB(V,HomA(M,U))ϕ 7→ (v 7→ (m 7→ ϕ(m⊗ v)))(m⊗ v 7→ ψ(v)(m)) ← ψ.This isomorphism of bifuctors has an obvious analogue for right modules, namelythe isomorphism HomB0(− ⊗AM,−) ∼= HomA0(−,HomB0(M,−)). There is a homo-morphism of B − C−bimodules4.3. {HomA(M,A)⊗AN → HomA(M,N)ϕ⊗ n 7→ (m 7→ ϕ(m)n)which is functorial in M and N . This is an isomorphism if one of M , N is finitelygenerated projective as A−module. Again, there is an obvious right analogue N ′ ⊗BHomB0(M,B) → HomB0(M,N′), which is an isomorphism if one of M , N ′ is finitelygenerated projective as right B−modules.If A is symmetric and s ∈ A∗ = HomR(A,R) is the image of 1A under some A −A−bimodule isomorphism A ∼= A∗, there is an isomorphism of B − A−bimodules4.4. {HomA(M,A) ∼= M∗ϕ 7→ s ◦ ϕwhich is functorial in M . Similarly, if B is symmetric, there is an isomorphism ofB − A−bimodules HomB0(M,B) ∼= M∗ which is functorial in M . In other words, ifA and B are symmetric, the three duality functors HomA(−, A), HomB0(−, B) andHomR(−, R) are naturally isomorphic through isomorphisms depending on the choicesof symmetrising forms on A and B.Combining 4.3 and 4.4, if A is symmetric and if one of M , N is finitely generatedprojective as left A−module, there is a natural isomorphism of B − C−bimodules4.5.HomA(M,N) ∼=M∗ ⊗AN .Similarly, if B is symmetric and one of M , N ′ is finitely generated projective asright B−modules, we have a natural isomorphism HomB0(M,N′) ∼= N ′ ⊗BM∗ as C −A−bimodules. Finally, if A is symmetric and if M is finitely generated projective asA−module, there is a natural isomorphism of C −B−bimodulesINDUCTION FOR INTERIOR ALGEBRAS 74.6.(M∗ ⊗AN)∗ ∼= N∗ ⊗AM .Sketch of proof. The proof of 4.2 is a straightforward verification. The given map in4.3 is easily seen to be functorial, and thus, in order to show that it is an isomor-phism if one of M , N is finitely generated projective as left A−module, it sufficesto verify this for M = A or N = A, which is trivial. For the proof of 4.4 we usethat A ∼= A∗ = HomR(A,R), together with the adjunction 4.2, from which we getHomA(M,A) ∼= HomA(M,HomR(A,R)) ∼= HomA(A ⊗AM,R) ∼= M∗. Statement 4.5follows from combining 4.3 and 4.4; finally, statement 4.6 is obtained from 4.2 and 4.5by (M∗ ⊗AN)∗ = HomR(M∗ ⊗AN,R) ∼= HomA(N,HomR(M∗, R)) ∼= HomA(N,M) ∼=N∗ ⊗AM . References1. B. Ku¨lshammer, T. Okuyama, A. Watanabe, A lifting theorem with applications to blocks andsource algebras, J. Algebra 232 (2000), 299–309.2. L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265–292.3. L. Puig, On the local structure of Morita and Rickard equivalences between Brauer blocks, Progressin Mathematics 178, Boston (Mass.), Basel: Birkha¨user, 1999.4. B. Stalder, Une extension de l’alge`bre du groupe en the´orie des repre´sentations modulaires, The`se,Universite´ Lausanne (2000).Markus LinckelmannCNRS, Universite´ Paris 7UFR Mathe´matiques2, place Jussieu75251 Paris Cedex 05FRANCE

Induction for interior algebras

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