We deene two (n + 1) graded Lie brackets on spaces of multilinear mappings. The rst one is able to recognize n-graded as-sociative algebras and their modules and gives immediately the correct diierential for Hochschild cohomology. The second one recognizes n-graded Lie algebra structures and their modules and gives rise to the notion of Chevalley cohomology.

We construct a minimal projective bimodule resolution for every finite dimensional quantum complete intersection of codimension two. Then we use this resolution to compute both the Hochschild cohomology and homology for such an algebra. In particular, we show that the cohomology vanishes in high degrees, while the homology is always nonzero.

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection.

We give decompositions of the Hochschild and cyclic homology of a com-mutative algebra in characteristic p into p?1 parts, and show these decom-positions are compatible with the shuue product structures. We also give a counterexample to a conjecture attributed to Barr, which asserts that a modiied version of Harrison cohomology coincides with Andr e/Quillen cohomology.