Approximations of generalized Cohen-Macaulay modules (Q1419651)

The first main result of this paper shows that for any generalized Cohen-Macaulay module \(M\) over a local Gorenstein ring \((R,m)\) with \(d = \dim M < n = \dim R\), there exists an exact sequence \(0 \to Y \to X \to M \to 0\) , where \(X\) is a maximal generalized Cohen-Macaulay module with \(H_m^d(X) = 0\), \(Y\) is of projective dimension \(n-d-1\), and the epimorphism \(X \to M\) induces isomorphisms \(H_m^i(X) \cong H_m^i(M)\) for \(i < n\), \(i \neq d\). Given a maximal Cohen-Macaulay module \(X\), this technique yields codimension 2 ideals \(I\) for which \(H_m^i(X) \cong H_m^i(R/I)\) for \(i < n-1\) so that one can construct generalized Cohen-Macaulay local rings with prescribed local cohomology. The second main result of this paper deals with the question whether there exists a graded ideal \(I\) in a polynomial ring \(S\) in \(n\) variables over a field such that \(H_m^i(S/I) \cong M_i\) for \(i = 0,\dots,d-1\), where \(M_0,\dots,M_d\) are given graded \(S\)-modules of finite length, \(d < n\). Due to a result of \textit{J. C. Migliore} [in ``Introduction to liaison theory and deficiency modules'', Prog. Math. 165 (1998; Zbl 0921.14033)], this question has only a positive answer if we shift the degree of the modules \(M_i\) up to a certain number. The authors give a clever estimate for this number in terms of numerical data of the graded resolutions of the modules \(M_i\).