Hochschild Cohomology and Representation-finite Algebras Ragnar ...

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Proof. Let δ ∈ Der R (A) be a derivation and f ∈ Hom A e(Ω A ,A) the corresponding
map; see Lemma 3.1.(1). For each i, themapf defines an exact commutative
diagram:
M i⊗ Aj A
M i ⊗ A ω A : 0 −−−−→ M i ⊗ A Ω A −−−−−−→ Mi ⊗ A −−−−−−→ Mi⊗AµA
M i −−−−→ 0
⏐ ∥
⏐
M i⊗ Af↓ σ i
⏐↓ ∥∥
η i : 0 −−−−→ M i −−−−→ E i −−−−→ M i −−−−→ 0.
Summing up yields the following exact commutative diagram:
M⊗ Aj A
M⊗ Aµ A
M ⊗ A ω A : 0 −−−−→ M ⊗ A Ω A −−−−−→ M ⊗ A −−−−−→ M −−−−→ 0
⏐ ∥
⏐
M⊗ Af↓ ⊕ ⏐↓ ∥∥
iσ i
⊕ i η i : 0 −−−−→ M −−−−→ ⊕ i E i −−−−→ M −−−−→ 0.
Therefore, χ M ([δ]) = [⊕ i ζ i ] ∈ ⊕ n
i=1 Ext1 A (M i,M i ). This completes the proof of the
lemma.
In our main application, we shall consider the case where M is a right A-module
and B =End A (M), whence M is endowed with the canonical B-A-bimodule structure.
Moreover, there exists a morphism of B-A-bimodules ρ : A → End B (M),
where the map ρ(a) fora ∈ A is given by ρ(a)(x) =xa, for all x ∈ M. Recall that
M is a faithfully balanced B-A-bimodule if ρ is an isomorphism.
3.5. Theorem. Let A be an algebra over a commutative ring R. Let M be
arightA-module and set B =End A (M). Assume that M is a faithfully balanced
B-A-bimodule with Ext 1 B (M,M) =0. Then there exists an exact sequence
0 −→ HH 1 (B) −→ HH 1 (A) −→ χM
Ext 1 A (M,M)
of R-modules. Moreover, if M = ⊕ n i=1 M i is a decomposition of right A-modules,
then the image of χ M lies in the diagonal part ⊕ n i=1 Ext1 A(M i ,M i ).
Proof. By hypothesis, HH 1 (A, ρ) :HH 1 (A) → HH 1 (A, End B (M)) is an isomorphism.
Furthermore, the map g A in Proposition 3.2.(1) is an isomorphism since
Ext 1 B (M,M) =0. Thusγ = g A ◦ HH 1 (A, ρ) is an isomorphism. Set c = γ −1 ◦ g B .
It follows from the definition of χ M that the diagram
0 −−−−→ HH 1 c
(B) −−−−→ HH 1 χ M
(A) −−−−→ Ext
1
A (M, M)
⏐
∥
γ↓ ∼ =
∥
0 −−−−→ HH 1 (B)
g B
−−−−→ Ext
1
B o ⊗A (M,M)
f B
−−−−→ Ext
1
A (M, M)
is commutative. By Proposition 3.2.(2), the lower row is exact, and hence the
upper row is an exact sequence as desired. The last part of the theorem follows
from Lemma 3.4. This completes the proof.
For a right A-module, denote by add(M) the full subcategory of the category of
right A-modules generated by the direct summands of direct sums of finitely many
copies of M. WecallM an A-generator if A lies in add(M). We shall now study the