On the first $τ$-tilting Hochschild cohomology of an algebra

Abstract

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