Approximations of generalized Cohen-Macaulay modules

Abstract

It is shown that any generalized Cohen-Macaulay module $M$ can be approximated by a maximal generalized Cohen-Macaulay module $X$ up to a module of finite projective dimension, and such that the local cohomology modules of $M$ and $X$ coincide for all cohomological degrees different from the dimensions of the two modules. By a theorem of Migliore there exist graded generalized Cohen-Macaulay rings which, up to a shift, have predescribed local cohomology modules. Bounds for this shift are given in terms of homological data.