Hochschild Cohomology and Representation-finite Algebras Ragnar ...

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Thus ρ induces an R-linear map
HH 1 (A, ρ) : HH 1 (A, A) → HH 1 (A, End B (M)) ,
which in turn allows us to define a map χ M :HH 1 (A) → Ext 1 A (M,M) through the
following commutative diagram:
HH 1 (A,ρ)
HH 1 (A)
⏐
↓
χ M
−−−−→
Ext
1
A (M,M)
↑
f ⏐⏐
B
HH 1 (A, End B (M))
g A
−−−−→ Ext
1
Bo ⊗A(M,M).
To give an explicit description of χ M , consider the commutative diagram
Der R (A, A)
⏐
∼ ⏐↓ =
ρ◦−
−−−−→
Der R (A, End B (M))
⏐
∼ ⏐↓ =
M⊗
Hom A e(Ω A ,A) −−−−−→ A−
Hom B o ⊗A(M ⊗ A Ω A ,M),
where ρ◦ denotes composition with ρ, the isomorphism on the left comes from
Lemma 3.1.(2) while that on the right comes from the diagram (∗) in the proof of
Proposition 3.2. Let δ ∈ Der R (A, A) be a derivation and let f ∈ Hom A e(Ω A ,A)be
the corresponding map. Applying M ⊗ A − to the extension ω A and pushing out
M ⊗ A ω A along the map M ⊗ A f yields a self-extension (M ⊗ A f) ∗ (M ⊗ A ω A )of
the right A-module M. Then
χ M ([δ]) = [(M ⊗ A f) ∗ (M ⊗ A ω A )]
by our earlier definition of g A and f B .
To study the map χ M further, we give an alternative and explicit description.
To this end, we define the differentiation of a morphism between projective modules
of the following form: let
φ : ⊕ n j=1 u jA →⊕ m i=1 v iA
be an A-linear map with u j ,v i some idempotents of A. Let δ ∈ Der R (A) besuch
that δ(v i Au j ) ⊆ v i Au j for all i, j. If the matrix of φ with respect to the given
decomposition is (a ij ) m×n with a ij ∈ v i Au j ,thenwecalltheA-linear map
δ(φ) :⊕ n j=1 u j A →⊕ m i=1 v i A
given by the matrix (δ(a ij )) m×n the derivative of φ along δ.
3.3. Lemma. Let M be a B-A-bimodule that is finitely presented as right A-
module. Let
ψ
P 2 −→ ⊕ n j=1 u j A −→ φ
⊕ m i=1 v i A −→ ε
M → 0
be an exact sequence of right A-modules with P 2 projective and u j ,v i idempotents
in A. Letδ ∈ Der R (A) be such that δ(v i Au j ) ⊆ v i Au j for all i, j. Then the image
of [δ] ∈ Der R (A)/Inn R (A) under χ M is the class of the self-extention of M that
corresponds to the class [εδ(φ)] in Ext 1 A (M,M).
Proof. Let f ∈ Hom A e(Ω A ,A) be such that δ(a) =f(a ⊗ 1 − 1 ⊗ a) for all a ∈ A;
see Lemma 3.1.(1). We have seen that χ M ([δ]) = [(M ⊗ A f) ∗ (M ⊗ A ω A )]. We claim
that there exist A-linear maps α, β rendering the following diagram commutative: