Résumé
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 HH2(Lambda) is isomorphic to the space of morphisms Hom(kQ-kQ)(I/I-2, Lambda). This is useful to determine when HH2(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.