Cohomology of twisted tensor products

Abstract

It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extent this is still true. We give an explicit description of the Ext-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a twisted tensor product. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra.

Hochschild cohomology and homology of quantum complete intersections

“…On the other hand, in [Bergh and Oppermann 2008a] we showed that in the situation that all commutation parameters are roots of unity, the Hochschild cohomology of a quantum complete intersection is as well behaved as in the commutative case:…”

The negative side of cohomology for Calabi-Yau categories

“…We illustrate this (and Theorem 4.2) with an example, using the same ring as in the example in Section 3. [BerOp,Theorem 5.3] that the Ext-algebra of k is given by Ext * R (k, k) = k y 1 , . .…”

The stable Auslander-Reiten quiver of a quantum complete intersection

“…Suppose all the commutators q ij are roots of unity. Then by [BeO,Theorem 5.3] and [BeO,Theorem 5.5], the growth rate of the Ext-algebra of the A ac q -module k is c. Moreover, this cohomology algebra is module-finite over its own center, and the latter is a Noetherian ring. Hence, if c ≥ 3, then it follows from [BeS,Theorem 4.1] that A ac q is of wild representation type.…”