Finite gentle repetitions of gentle algebras and their Avella-Alaminos--Geiss invariants

Abstract

Among finite dimensional algebras over a field $K$, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras $A$, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of $A$ in terms of some values of its Avella-Alaminos--Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras $A^{(k)}$ indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos--Geiss invariants of $A$ to the dimensions of Hochschild cohomologies of $A^{(k)}$. On the way we will see that the Avella-Alaminos--Geiss invariants of $A^{(k)}$ are completely determined by those of $A$. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the latter part of this article, we deal with this problem under some assumptions.