Special biserial algebras with no outer derivations

TL;DR

This paper characterizes special biserial algebras with no outer derivations by linking their Hochschild cohomology, representation finiteness, and topological properties of their quivers, showing that certain cohomology groups vanish under these conditions.

Contribution

It establishes a precise criterion connecting the vanishing of Hochschild cohomology to the algebra's representation type and quiver topology for special biserial algebras.

Findings

First Hochschild cohomology vanishes iff the algebra is representation finite and simply connected.

Higher Hochschild cohomology groups also vanish under these conditions.

The Euler characteristic of the quiver equals the number of specific indecomposable modules.

Abstract

Let $A$ be a special biserial algebra over an algebraically closed field. We show that the first Hohchshild cohomology group of $A$ with coefficients in the bimodule $A$ vanishes if and only if $A$ is representation finite and simply connected (in the sense of Bongartz and Gabriel), if and only if the Euler characteristic of $Q$ equals the number of indecomposable non uniserial projective injective $A$-modules (up to isomorphism). Moreover, if this is the case, then all the higher Hochschild cohomology groups of $A$ vanish.