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Pré-Publication, Document De Travail Année : 2024

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

Résumé

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

hal-04555253 , version 1 (22-04-2024)

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Claude Cibils, Marcelo Lanzilotta, Eduardo Marcos, Andrea Solotar. On the first $τ$-tilting Hochschild cohomology of an algebra. 2024. ⟨hal-04555253⟩
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