INDUCTION FOR INTERIOR ALGEBRAS

Markus Linckelmann

January 2001

1 Algebra induction - definition and examples

We define an induction for interior algebras which generalises Puig’s concepts of induction, including its non-injective version, for interior G-algebras in [2], [3]. Throughout this note we fix a commutative ring R. By default, a module over an R−algebra A is a unitary left module. Any right A−module can be considered as a left module over the opposite algebra A^{0}. Given three R−algebras A, B, C, an A − B−bimodule M and an A − C−bimodule N, we denote by Hom_{A}(M, N) the R−module of all homomorphisms from M to N as left A−modules; this has a canonical structure of C − B−bimodule given by (c.φ.b)(m) = φ(mb)c for any b ∈ B, c ∈ C and φ ∈ Hom_{A}(M, N). Similarly, if N′ is a C − B−bimodule, we denote by Hom_{B^{0}}(M, N′) the R−module consisting of all homomorphisms from M to N′ viewed as right B−modules; again, this has a canonical structure of C − A−bimodules given by (c.ψ.a)(m) = cψ(am) for any a ∈ A, c ∈ C and ψ ∈ Hom_{B^{0}}(M, N′).

Following the terminology of [1] (which extends that of [2, 3.1]), if A is an R−algebra, an interior A−algebra is an R−algebra B endowed with a unitary algebra homomorphism σ : A → B. Note that in particular B becomes an A − A−bimodule through σ.

Definition 1.1 Let A, B be R−algebras, let C be an interior B−algebra, and let M be an A − B−bimodule. We set

Ind_{M}(C) = End_{C^{0}}(M ⊗_{B} C) ,

considered as interior A−algebra with structural homomorphism A → Ind_{M}(C) mapping a ∈ A to the C^{0}−endomorphism of M ⊗_{B} C given by left multiplication with a

on M ⊗_{B} C.

Remark 1.2 The above definition makes sense for differential graded algebras. In order to limit technicalities, we leave it to the reader to verify, that the content of this note 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].

Typeset by AMS-TEX

Example 1.3 Let G be a finite group, H a subgroup of G and let C be an interior H−algebra over R; that is, C is an R−algebra endowed with a group homomorphism H → C^{×} (cf. [2, 3.1]). This group homomorphism extends to an algebra homomorphism RH → C, through which C becomes an interior RH−algebra in the sense defined above. Set M = RG, considered as RG − RH−bimodule. Then

Ind_{M}(C) ≅ Ind_{H}^{G}(C) ,

where the right side is the induction defined in [2, 3.3]. To see this it suffices to apply the standard isomorphisms Ind_{M}(C) = End_{C^{0}}(RG ⊗_{RH} C) ≅ Hom_{RH^{0}}(RG, RG ⊗_{RH}

C) ≅ RG ⊗_{RH} C ⊗_{RH} Hom_{RH^{0}}(RG, RH) ≅ RG ⊗_{RH} C ⊗_{RH} RG.

Example 1.4 Let G, H be finite groups, let φ : H → G be a group homomorphism and 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 the left and with the action of RH on the right given by restriction through φ. Then

Ind_{M}(C) ≅ Ind_{φ}(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 Ind_{M}(C) =

End_{C^{0}}((RG)_{φ} ⊗_{RH} C) ≅ End_{C^{0}}(R ⊗_{RK} C) ≅ Hom_{RK^{0}}(R, R ⊗_{RK} C) ≅ (R ⊗_{RK} C)^{K}, which

coincides with the expression in [3, 3.3.1]. The general case follows from combining this with 1.3 and the transitivity theorem 2.1 below.

Example 1.5 Let A, B be R−algebras, and let M be an A − B−bimodule such that the functor M ⊗_{B} − induces an equivalence of categories Mod(B) ≅ Mod(A).

Consider B as interior B−algebra in the trivial way; that is, via the identity map on B. Then Morita’s theorem reads Ind_{M}(B) ≅ A .

2 Transitivity of algebra induction

The transitivity theorem below generalises and simplifies the corresponding statement in [3, §3]:

Theorem 2.1. Let A, B, C be R−algebras, let D be an interior C−algebra, let M be an A − B−bimodule and let N be a B − C−bimodule. There is a canonical homomorphism of interior A−algebras

Ind_{M}(Ind_{N}(D)) ⟶ Ind_{M⊗_{B}N}(D) ,

which is an isomorphism, if M is finitely generated projective as right B−module, or if N is finitely generated projective as right C−module, or if both N ⊗_{C} D and

M ⊗_{B} N ⊗_{C} D are 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−homomorphism

Φ : \{ \begin{matrix}M ⊗_{B} Hom_{C^{0}}(N, N′) & → Hom_{C^{0}}(N, M ⊗_{B} N′) \\ m ⊗ φ & ↦ (n ↦ m ⊗ φ(n)) \end{matrix}

is natural in the three variables M, N, N′. Moreover, Φ is an isomorphism, if M is finitely generated projective as right B−module, or if N is finitely generated projective as right C−module, or if both N′, M ⊗_{B} N′ are finitely generated projective as right

C−modules.

Proof. The fact that Φ is a homomorphism of A − B−bimodules which is natural in M, N, N′ is a straightforward verification. In order to show the last statement, it suffices to see that Φ is an R−linear isomorphism in any of the three cases stated above. If M = B as right B−module, both sides are isomorphic to Hom_{C^{0}}(N, N′). By naturality, it follows that Φ is an isomorphism whenever M is finitely generated projective as right B−module. Similarly, Φ is an isomorphism if N = C as right C−module, and thus Φ is an isomorphism whenever N is finitely generated projective as right C−module. For the last case, if both N′ and M ⊗_{B} N′ are finitely generated

projective as right C−modules, we have canonical isomorphisms M⊗_{B}Hom_{C^{0}}(N, N′) ≅

M ⊗_{B} N′ ⊗_{C} Hom_{C^{0}}(N, C) ≅ Hom_{C^{0}}(N, M ⊗_{B} N′) by applying twice the right analogue

of 4.3. □

Proof of 2.1. Using the standard adjunction 4.2, we have a canonical R−isomorphism

Ind_{N}(D) ≅ Hom_{C^{0}}(N, N ⊗_{C} D) .

Applying this again, we get a canonical R−isomorphism

Ind_{M}(Ind_{N}(D)) ≅ Hom_{B^{0}}(M, M ⊗_{B} Hom_{C^{0}}(N, N ⊗_{C} D)) .

Applying this yet again to M ⊗_{B} N yields an R−isomorphism

Ind_{M⊗_{B}N}(D) ≅ Hom_{C^{0}}(M ⊗_{B} N, M ⊗_{B} N ⊗_{C} D) ≅ Hom_{B^{0}}(M, Hom_{C^{0}}(N, M ⊗_{B} N ⊗_{C} D)) .

Now the map Φ from 1.6 with N ⊗_{C}D instead of N′ induces the algebra homomorphism

with 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 ∈ Ind_{M}(Ind_{N}(D)) and denote by ψ the image in Ind_{M⊗_{B}N}(D). Then, for any m ∈ M, any n ∈ N and any d ∈ D we

have

ψ(m ⊗ n ⊗ d) = ∑_{i}m_{i} ⊗ φ_{i}(n ⊗ d) ,

where m_{i} ∈ M, φ_{i} ∈ Ind_{N}(D) such that φ(m ⊗ Id_{N⊗_{C}D}) = ∑_{i} m_{i} ⊗ φ_{i}, and where i

runs over a finite indexing set.

3 Functoriality of algebra induction

Proposition 3.1. Let A, B be R−algebras, let M be an A − B−bimodule, and let γ : C → C′ be a homomorphism of interior B−algebras. The functor − ⊗_{C} C′ induces

a homomorphism of interior A−algebras

Ind_{M}(γ) : Ind_{M}(C) ⟶ Ind_{M}(C′) .

Proof. Applying the functor − ⊗_{C} C′ to any C^{0}−endomorphism of M ⊗_{B} C yields a

(C′)^{0}−endomorphism of M ⊗_{B} C ⊗_{C} C′ ≅ M ⊗_{B} C′, where the left C−module structure

of C′ is given by γ. □

Let A, B be R−algebras, let M be an A − B−bimodule, and let C be an interior B−algebra. The algebra Ind_{M}(C) acts on M ⊗_{B}C by α.n = α(n) for any α ∈ Ind_{M}(C)

and any n ∈ M ⊗_{B} C. In this way, M ⊗_{B} C becomes an Ind_{M}(C) − C− bimodule.

This bimodule gives rise to induction and corestriction functors between the module categories of C and Ind_{M}(C), denoted by

Ind_{M} = (M ⊗_{B} C) ⊗_{C} − : Mod(C) ⟶ Mod(Ind_{M}(C)) ,
Res_{M} = Hom_{Ind_{M} (C)}(M ⊗_{B} C, −) : Mod(Ind_{M}(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 C be an interior B−algebra. The functor Ind_{M} is left adjoint to the functor Res_{M}.

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 be an interior B−algebra. If M ⊗_{B} C is a progenerator as right C−module, then Ind_{M}

and Res_{M} are mutually inverse equivalences between Mod(Ind_{M}(C)) and Mod(C). This case 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 as R−module, and if A is isomorphic to its R−dual A^{∗} = Hom_{R}(A, R) as A − A−bimodule.

Proposition 3.4. Let A, B be R−algebras, let M be an A − B−bimodule and let C be an interior B−algebra. If C is symmetric, B is finitely generated projective as R−module and M is finitely generated projective as right B−module, then Ind_{M}(C) is symmetric.

Proof. The hypotheses imply that M ⊗_{B} C and hence Ind_{M}(C) is finitely generated

projective as R−module. Using the isomorphism Ind_{M}(C) = End_{C^{0}}(M ⊗_{B} C) ≅

(M ⊗_{B}C)⊗_{C}(M ⊗_{B}C)^{∗} from 4.5 as well as 4.6 implies that Ind_{C}(M) is indeed symmetric.

□

4 Appendix

We collect in this section some of the (well-known) standard isomorphisms between modules and homomorphism spaces as we need them in this note. Let A, B, C be R−algebras, let M be an A − B−bimodule and let N be an A − C−bimodule. Recall from §1 that the space of homomorphisms Hom_{A}(M, N) from M to N as left A−modules becomes a B − C−bimodule via (b.φ.c)(m) = φ(mb)c; similarly, the space of homomorphisms Hom_{B^{0}}(M, N′) of homomorphisms from M to N′ as right B−modules becomes a C − A−bimodule via (c.ψ.a)(m) = cψ(am). There is an isomorphism of bifunctors

4.1.

Hom_{A}(M ⊗_{B} −, −) ≅ Hom_{B}(−, Hom_{A}(M, −))

on Mod(B) × Mod(A) given, for any A−module U and any B−module V, by the mutually inverse isomorphisms

4.2.

\{ \begin{matrix}Hom_{A}(M ⊗_{B} V, U) & ≅ & Hom_{B}(V, Hom_{A}(M, U)) \\ φ & ↦ & (v ↦ (m ↦ φ(m ⊗ v))) \\ (m ⊗ v ↦ ψ(v)(m)) & ← & ψ \end{matrix} .

This isomorphism of bifuctors has an obvious analogue for right modules, namely the isomorphism Hom_{B^{0}}(− ⊗_{A} M, −) ≅ Hom_{A^{0}}(−, Hom_{B^{0}}(M, −)). There is a homo-

morphism of B − C−bimodules

4.3.

\{ \begin{matrix}Hom_{A}(M, A) ⊗_{A} N & → & Hom_{A}(M, N) \\ φ ⊗ n & ↦ & (m ↦ φ(m)n) \end{matrix}

which is functorial in M and N. This is an isomorphism if one of M, N is finitely generated projective as A−module. Again, there is an obvious right analogue N′ ⊗_{B}

Hom_{B^{0}}(M, B) → Hom_{B^{0}}(M, N′), which is an isomorphism if one of M, N′ is finitely generated projective as right B−modules.

If A is symmetric and s ∈ A^{∗} = Hom_{R}(A, R) is the image of 1_{A} under some A − A−bimodule isomorphism A ≅ A^{∗}, there is an isomorphism of B − A−bimodules

4.4.

\{ \begin{matrix}Hom_{A}(M, A) & ≅ & M^{∗} \\ φ & ↦ & s ∘ φ \end{matrix}

which is functorial in M. Similarly, if B is symmetric, there is an isomorphism of B − A−bimodules Hom_{B^{0}}(M, B) ≅ M^{∗} which is functorial in M. In other words, if A and B are symmetric, the three duality functors Hom_{A}(−, A), Hom_{B^{0}}(−, B) and Hom_{R}(−, R) are naturally isomorphic through isomorphisms depending on the choices of symmetrising forms on A and B.

Combining 4.3 and 4.4, if A is symmetric and if one of M, N is finitely generated projective as left A−module, there is a natural isomorphism of B − C−bimodules 4.5.

Hom_{A}(M, N) ≅ M^{∗} ⊗_{A} N .

Similarly, if B is symmetric and one of M, N′ is finitely generated projective as right B−modules, we have a natural isomorphism Hom_{B^{0}}(M, N′) ≅ N′ ⊗_{B} M^{∗} as

C − A−bimodules. Finally, if A is symmetric and if M is finitely generated projective as A−module, there is a natural isomorphism of C − B−bimodules

4.6.

(M^{∗} ⊗_{A} N)^{∗} ≅ N^{∗} ⊗_{A} M .

Sketch of proof. The proof of 4.2 is a straightforward verification. The given map in 4.3 is easily seen to be functorial, and thus, in order to show that it is an isomorphism if one of M, N is finitely generated projective as left A−module, it suffices to verify this for M = A or N = A, which is trivial. For the proof of 4.4 we use that A ≅ A^{∗} = Hom_{R}(A, R), together with the adjunction 4.2, from which we get Hom_{A}(M, A) ≅ Hom_{A}(M, Hom_{R}(A, R)) ≅ Hom_{A}(A ⊗_{A} M, R) ≅ M^{∗}. Statement 4.5 follows from

combining 4.3 and 4.4; finally, statement 4.6 is obtained from 4.2 and 4.5 by (M^{∗} ⊗_{A}

N)^{∗} = Hom_{R}(M^{∗} ⊗_{A} N, R) ≅ Hom_{A}(N, Hom_{R}(M^{∗}, R)) ≅ Hom_{A}(N, M) ≅ N^{∗} ⊗_{A} M.

□

References

1. B. Külshammer, T. Okuyama, A. Watanabe, A lifting theorem with applications to blocks and source algebras, preprint (1999).

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, Progress in Mathematics 178, Boston (Mass.), Basel: Birkhäuser, 1999.

Markus Linckelmann CNRS, Université Paris 7 UFR Mathématiques 2, place Jussieu 75251 Paris Cedex 05

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