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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm125-1-6,
author = {Ibrahim Assem and Juan Carlos Bustamante and Patrick Le Meur},
title = {Special biserial algebras with no outer derivations},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {83-98},
zbl = {1258.16015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm125-1-6}
}
Ibrahim Assem; Juan Carlos Bustamante; Patrick Le Meur. Special biserial algebras with no outer derivations. Colloquium Mathematicae, Tome 122 (2011) pp. 83-98. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm125-1-6/