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

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 two $\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.