Cohomologie des alg\`{e}bres de Kr\"{o}necker g\'{e}n\'{e}rales

Abstract

The computation of the Hochschild cohomology $HH^*(T)=H^*(T,T)$ 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 $HH^*(T)$ splits in little pieces whenever the bimodule $M$ is decomposable. As an example, we express the Hilbert-Poincar\'{e} serie $\sum\_{i=0}^\infty dim\_K HH^i(T\_m)t^i$ of the "general" Kr\"{o}necker algebra $T\_m=\pmatrix{A&M^m\cr 0&B\cr}$ as a function of $m\geq 1$ and those of $T$ (here the ground ring $K$ is a field and $dim\_K T<+\infty$). The Lie algebra structure of $HH^1(T)$ is also considered.