Projective Resolutions of Cohen-Macaulay Algebras

The problem of explicitly finding a free resolution, minimal in some suitable sense, of a module over a polynomial ring is solved in principle by the algorithm of Hilbert [H]. However, this algorithm is of enormous computational difficulty. If the module happens to be finite dimensional over the ground field, and if the module structure is given by specifying the commuting linear transformations induced by the indeterminates, then a little-known result of Scheja and Storch, resumed in Sect. 1, allows one to write down an explicit free resolution of the right length without computation. The same idea can be used for many CohenMacaulay modules (Example 1.1). But although the Scheja-Storch resolution is minimal in some cases, it is not minimal in the main case of interest where the module is a factor-ring of the polynomial ring and not the ground field itself.