Hochschild Cohomology and Representation-finite Algebras Ragnar ...

15
Hom B (M, N)
⏐
∼ ⏐↓ =
ad
−−−−→
Der R (A, Hom B (M,N))
⏐
∼ ⏐↓ =
Hom A e(A ⊗ A, Hom B (M,N))
⏐
∼ ⏐↓ =
(j A, (M,N))
−−−−−−−−→ Hom A e(Ω A , Hom B (M,N))
∼ =
⏐ ⏐↓
(∗)
Hom B o ⊗A(M ⊗ A, N)
(M⊗j A,N)
−−−−−−−→ Hom B o ⊗A(M ⊗ A Ω A , N).
Since ω A splits as an extension of left A-modules, the sequence
M⊗ Aj A
M ⊗ A ω A : 0 −−−−→ M ⊗ A Ω A −−−−−→ M ⊗ A
M⊗ Aµ A
−−−−−→ M −−−−→ 0
is an extension of B-A-bimodules that splits as an extension of left B-modules. For
δ ∈ Der R (A, Hom B (M,N)), let ˜δ ∈ Hom Bo ⊗A(M ⊗ A Ω A ,N) be the corresponding
map under the isomorphisms given in the above diagram (∗). Pushing out M ⊗ A ω A
along the map ˜δ, we get an exact commutative diagram:
M⊗ Aj A
M ⊗ A ω A : 0 −−−−→ M ⊗ A Ω A −−−−−→ M ⊗ A
⏐
⏐
˜δ ↓
↓
M⊗ Aµ A
−−−−−→ M −−−−→ 0
∥
˜δ ∗ (M ⊗ A ω A ): 0 −−−−→ N −−−−→ E −−−−→ M −−−−→ 0,
where the rows are extensions of B-A-bimodules that split as extensions of left B-
modules. Now the lower row splits as an extension of B-A-bimodules if and only if
˜δ factors through M ⊗ A j A if and only if δ ∈ Inn R (A, Hom B (M,N)). This shows
that associating to the class [δ] ∈ HH 1 (A, Hom B (M,N)) of a derivation δ the class
g A ([δ]) = [˜δ ∗ (M ⊗ A ω A )] ∈ Ext 1 B o ⊗A (M,N)
yields a well-defined injective map g A with image contained in the kernel of f A .
It remains to show that the kernel of f A is contained in the image of g A . For
this purpose, let η : 0 → N → E −→ p
M → 0 be an extension of B-A-bimodules
that splits as an extension of left B-modules. Let q : M → E be a B-linear map
such that pq =1I M ,andletσ : M ⊗ A → E be the B-A-bilinear map such that
σ(m ⊗ a) =q(m)a. These data define an exact commutative diagram of B-Abimodules
as follows:
M⊗ Aj A
M⊗ Aµ A
M ⊗ A ω A : 0 −−−−→ M ⊗ A Ω A −−−−−→ M ⊗ A −−−−−→ M −−−−→ 0
⏐
⏐
ζ↓
σ↓
∥
η : 0 −−−−→ N −−−−→ E
p
−−−−→ M −−−−→ 0,
where ζ is induced by σ. In view of the isomorphisms given in (∗), there exists a
derivation δ ∈ Der R (A, Hom B (M,N)) whose image is ζ. Hence g A ([δ]) = [η]. This
completes the proof of the proposition.
Now we specialize the preceding proposition to the case M = N. The canonical
algebra anti-homomorphism ρ : A → End B (M) is a homomorphism of A-bimodules.