Hochschild Cohomology and Representation-finite Algebras Ragnar ...

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behaviour of Hochschild cohomology under (r-)pseudo-tilting, a notion we define
as follows.
3.6. Definition. Let A be an algebra over a commutative ring R and let
r ≥ 0 be an integer. A right A-module M is called r-pseudo-tilting if it satisfies the
following conditions :
(1) Ext i A (M,M) =0for1≤ i ≤ r;
(2) there exists an exact sequence
0 → A → M 0 → M 1 →···→M r → 0
of right A-modules, where M 0 ,...,M r ∈ add(M).
Moreover, the module M is called pseudo-tilting if Ext i A (M,M) = 0 for all i ≥ 1
and (2) holds for some r ≥ 0; in other words, M is r-pseudo-tilting for all sufficiently
large r.
Note that a 0-pseudo-tilting A-module is nothing but an A-generator. Moreover,
a tilting or co-tilting module of projective or injective dimension at most r,
respectively, in the sense of [23] is an r-pseudo-tilting module, and consequently, a
pseudo-tilting module. We recall now the following result from [23, (1.4)].
3.7. Lemma (Miyashita). Let A be an R-algebra and r ≥ 0 an integer. Let
M be an r-pseudo-tilting right A-module and set B =End A (M). Then M is a
faithfully balanced B-A-bimodule and Ext i A (M,M) =0for all i ≥ 1.
As an immediately consequence of Lemma 3.7 and Theorem 3.5, we get the
following result.
3.8. Proposition. Let A be an algebra over a commutative ring R. Let M be
an r-pseudo-tilting right A-module and set B =End A (M).
(1) There exists an exact sequence of R-modules
0 −→ HH 1 (B) −→ HH 1 (A) −→ χM
Ext 1 A(M,M) .
(2) If r ≥ 1, thenHH 1 (A) ∼ = HH 1 (B).
So far we have not put any restriction on the R-algebra structure. As usual in
Hochschild theory, much sharper results can be obtained if the algebras involved
areprojectiveasR-modules, for example, when R is a field. First we recall the
following result from [13, p.346].
3.9. Proposition (Cartan-Eilenberg). Let A, B be algebras over a commutative
ring R, andletM,N be B-A-bimodules. If A and B are projective as
R-modules, then there exist spectral sequences
and
HH i (A, Ext j B
(M,N)) =⇒ Exti+j
B o ⊗A (M,N)
HH i (B,Ext j A
(M,N)) =⇒ Exti+j
B o ⊗A (M,N) .