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Article Dans Une Revue Journal of Pure and Applied Algebra Année : 2016

Hochschild cohomology of relation extension algebras

Résumé

Let $B$ be the split extension of a finite dimensional algebra $C$ by a $C$-$C$-bimodule $E$. We define a morphism of associative graded algebras $\varphi^*:\HH^*(B)\rightarrow \HH^*(C)$ from the Hochschild cohomology of $B$ to that of $C$, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler. In the case of a trivial extension $B=C\ltimes E$, we give necessary and sufficient conditions for each $\varphi^n$ to be surjective. We prove the surjectivity of $\varphi^1$ for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of $\varphi^1$ for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.
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Dates et versions

hal-01179207 , version 1 (22-07-2015)
hal-01179207 , version 2 (01-02-2016)

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Citer

Ibrahim Assem, M. Andrea Gatica, Ralf Schiffler, Rachel Taillefer. Hochschild cohomology of relation extension algebras. Journal of Pure and Applied Algebra, 2016. ⟨hal-01179207v2⟩
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