Suite exacte de Mayer-Vietoris d'une extension triangulaire

Abstract

In this article, we complete the results published in [2] using new proofs: we obtain a more general long exact sequence for the extension groups $\operatorname {Ext}_T^*$ of the triangular algebra $T=\left[\begin{smallmatrix} A & M \\ 0 & B\end{smallmatrix}\right]$. One may find such a long exact sequence in the works of Palmér and Roos [22], [23], but only for very special instances of $T$-modules. This extra effort allows us to show that the well-known Hochschild cohomological exact sequence of Happel ([16] and, more generally,[8] and [20]) is just a particular instance of that of Palmér-Roos, even if it seems we really need an assumption on $M$ (the same trick also appears for the homological version using the ${\operatorname {Tor}}^T_*$ functor). With this method, we obtain a sensible improvement for the isomorphism of [3], which computes the Hochschild cohomology of triangular algebras in several cases. At the end of the article, we relate the Palmér-Roos exact sequence with the classical (i.e. topological) Mayer-Vietoris exact sequence.