Hochschild Cohomology and Representation-finite Algebras Ragnar ...

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The bar resolution of A over R is given by the complex:
···→A ⊗(i+2)
b
−→
i
A ⊗(i+1) →···→A ⊗3 b
−→
1
A
⊗2
−→ µA
A → 0 ,
where µ A is the multiplication map on A, andb i is the map given by
b i (a 0 ⊗ a 1 ⊗···⊗a i+1 )= ∑ i
j=0 (−1)j a 0 ⊗···⊗a j a j+1 ⊗···⊗a i+1 .
Each term in the bar resolution of A over R is naturally an A-bimodule and the
differentials respect that structure, whence the bar resolution can be viewed as a
complex of right A e -modules. Let B denote the truncated bar resolution of A over
R. The i-th cohomology group of the complex Hom A e(B, X) is called the i-th
Hochschild cohomology group of A over R with coefficients in X and is denoted by
HH i R (A, X). In case X = A, wewriteHHi R (A) =HHi R (A, A). If R is understood,
we shall simply write HH i (A, X) forHH i R (A, X) andHHi (A) forHH i R(A).
The following semicontinuity result on Hochschild cohomology is a straightforward
application of Theorem 1.2.
1.3. Proposition. Let R be a noetherian commutative ring. Let A be an R-
algebra and X an A-bimodule, and assume that both A and X are finitely generated
projective as R-modules. For each i ≥ 0, the dimension function
p ↦→ dim k(p) HH i k(p) (k(p) ⊗ R A, k(p) ⊗ R X)
is then upper semicontinuous on Spec(R) and takes on only finitely many values.
Proof. As A is finitely generated projective over R, soisA ⊗i for each i ≥ 0.
There is thus an isomorphism of functors on ModR as follows:
−⊗X ⊗ Hom R (A ⊗i ,R) ∼ = Hom A e(A ⊗(i+2) , −⊗X) ,
where the right A e -module structure on −⊗X is inherited from the one on X.
Consequently, for each R-module M, thecomplexHom A e(B, M⊗X) is isomorphic
toacomplexP· of the following form:
0−→M ⊗ X ⊗ Hom R (R, R)−→···−→M ⊗ X ⊗ Hom R (A ⊗i ,R)−→··· ,
and so HH i R (A, M ⊗ X) ∼ = H i (P·). As X is assumed to be finitely generated
projective over R, the same holds for X ⊗ Hom R (A ⊗i , R), whence the functor
HH i R(A, −⊗X) satisfies the assumptions of (1.2) for each i ≥ 0.
To conclude, let S be a commutative R-algebra and consider the S-algebra A S =
S⊗A. WithB the truncated bar resolution of A over R, the truncated bar resolution
of A S over S is isomorphic to B S = S ⊗ B. Moreover,X S = S ⊗ X is naturally an
A S -bimodule. Now adjunction gives rise to the following isomorphism of complexes:
Hom (AS) e(B S,X S ) ∼ = Hom A e(B,X S ) ,
whence HH i S(A S ,X S ) ∼ = HH i R(A, X S )foreachi ≥ 0. Applying this isomorphism
to S = k(p) forp ∈ Spec(R) completes the proof.
Now we turn our attention to the first Hochschild cohomology group. Note that
HH 1 R(A, X) ∼ = Ext 1 Ae(A, X) for any R-algebra A and any A-bimodule X. Asusual,
it is useful to interpret this group through (classes of) outer derivations. To this