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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2024

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

Sur la cohomologie $\tau$-inclinée de Hochschild en degré un d'une algèbre.

Marcelo Lanzilotta
  • Fonction : Auteur
Eduardo N. Marcos
  • Fonction : Auteur
Andrea Solotar
  • Fonction : Auteur

Résumé

In this paper we introduce, according to one of the main ideas of $\tau$-tilting theory, the $\tau$-tilting 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 $\tau$-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 $\Lambda=kQ/I$, the Hochschild cohomology in degree two $HH^2(\Lambda) $ is isomorphic to the space of morphisms $\mathsf{Hom}_{kQ-kQ}(I/I^2, \Lambda).$ This may be useful to determine when $HH^2(\Lambda)=0$ 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.

Dates et versions

hal-04575478 , version 1 (14-05-2024)

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Claude Cibils, Marcelo Lanzilotta, Eduardo N. Marcos, Andrea Solotar. On the first $\tau$-tilting Hochschild cohomology of an algebra. 2024. ⟨hal-04575478⟩
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