Cohomologie des algèbres de Krönecker générales

Abstract

The computation of the Hochschild cohomology $H{H}^{\ast }(T)=H^{\ast }(T,T)$ of a triangular algebra $T=\pmatrix{A&M\cr 0&B\cr}$ was performed in {\bf[BG2]}, by the means of a certain triangular complex. We use this result here to show how $H{H}^{\ast }(T)$ splits in little pieces whenever the bimodule $M$ is decomposable. As an example, we express the Hilbert-Poincaré serie $\sum \_{i=0}^{\infty }dim\_KH{H}^i(T\_m)t^i$ of the "general" Krönecker algebra $T\_m=\pmatrix{A&M^m\cr 0&B\cr}$ as a function of $m\ge 1$ and those of $T$ (here the ground ring $K$ is a field and $dim\_KT<+\infty$). The Lie algebra structure of $H{H}^1(T)$ is also considered.