On the first $\tau$-Hochschild cohomology of an algebra

Abstract

In this paper, we introduce, according to one of the main ideas of $\tau$-tilting theory, the $\tau$-Hochschild cohomology in degree one of a finite-dimensional $k$-algebra $\Lambda$, where $k$ is a field. We define the excess of $\Lambda$ as the difference between the dimensions of the $\tau$-Hochschild cohomology in degree one and the dimension of the usual Hochschild cohomology in degree one. One of the main results is that for a zero excess bound quiver algebra $\Lambda =kQ/I$, the Hochschild cohomology in degree 2 ${\mathsf{HH}}^2(\Lambda )$ is isomorphic to the space of morphisms ${\mathsf{Hom}}_{kQ-kQ}(I/I^2,\Lambda )$. This is useful to determine when ${\mathsf{HH}}^2(\Lambda )=0$ for these algebras. We compute the excess for hereditary, radical square zero and monomial triangular algebras. For a bound quiver algebra $\Lambda$, a formula for the excess of $\Lambda$ is obtained. We also give a criterion for $\Lambda$ to be $\tau$-rigid.