On the first $τ$-tilting Hochschild cohomology of an algebra
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| Format: | Preprint |
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2024
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| author | Cibils, Claude Lanzilotta, Marcelo Marcos, Eduardo N. Solotar, Andrea |
| author_facet | Cibils, Claude Lanzilotta, Marcelo Marcos, Eduardo N. Solotar, Andrea |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_06916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the first $τ$-tilting Hochschild cohomology of an algebra Cibils, Claude Lanzilotta, Marcelo Marcos, Eduardo N. Solotar, Andrea Rings and Algebras K-Theory and Homology Representation Theory 16E40 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. |
| title | On the first $τ$-tilting Hochschild cohomology of an algebra |
| topic | Rings and Algebras K-Theory and Homology Representation Theory 16E40 |
| url | https://arxiv.org/abs/2404.06916 |