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 \(HH^*(T)=H^*(T,T)\).


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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