Alg\`ebre homologique des N-complexes et homologie de Hochschild aux racines de l'unit\'e

We set up a homological algebra for N-complexes, which are graded modules together with a degree -1 endomorphism d satisfying d^N=0. We define Tor- and Ext-groups for N-complexes and we compute them in terms of their classical counterparts (N=2). As an application, we get an alternative definition of the Hochschild homology of an associative algebra out of an N-complex whose differential is based on a primitive N-th root of unity.