Linkage of Cohen-Macaulay modules over a gorenstein ring

Let R be a Gorenstein complete local ring. We say that finitely generated modules M and N are linked if Hom_R/λR(M,R/λR) ≅ Ω¹_R/λR(N), where λ is a regular sequence contained in both of the annihilators of M and N. We shall show that the Cohen-Macaulay approximation functor gives rise to a map Φr from the set of even linkage classes of Cohen-Macaulay modules of codimension r to the set of isomorphism classes of maximal Cohen-Macaulay modules. When r - 1, we give a condition for two modules to have the same image under the map Φr. If r - 2 and if R is a normal domain of dimension two, then we can show that Φr is a surjective map if and only if R is a unique factorization domain. Several explicit computations for hypersurface rings are also given.