## Cohomology of the triangular algebra and applications. (Cohomologie de l’algèbre triangulaire et applications.)(English)Zbl 1032.16007

Let $$K$$ be a commutative ring, $$A$$ and $$B$$ be two associative and unitary $$K$$-algebras and $$M$$ be an $$(A\otimes B^o)$$-module. We suppose $$A$$, $$B$$ and $$M$$ $$K$$-projective. The triangular algebra associated to this triple $$A$$, $$B$$ and $$M$$ is the $$K$$-module $$T=\left(\begin{smallmatrix} A&M\\ 0&B\end{smallmatrix}\right)$$ with multiplication law given by: $\left(\begin{smallmatrix} a&m\\ 0&b\end{smallmatrix}\right)\left(\begin{smallmatrix} a'&m'\\ 0&b'\end{smallmatrix}\right)=\left(\begin{smallmatrix} aa'&am'+mb'\\ 0&bb'\end{smallmatrix}\right).$ Let $$H^*(T,\Lambda)$$ be the Hochschild cohomology of $$T$$ with coefficients in a $$T$$-bimodule $$\Lambda$$. The authors recover some classical exact sequences when they describe $$H^*(T,\Lambda)$$ as the cohomology of a cone complex of a simple to construct morphism. In particular, when $$M$$ is $$A$$-projective, these exact sequences give better insights for $$HH^*(T)=H^*(T,T)$$.

### MSC:

 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16S50 Endomorphism rings; matrix rings 16E05 Syzygies, resolutions, complexes in associative algebras
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