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 , where 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 codimensionr 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 Φ1. If r=2 and if R is a normal domain of dimension two, then we can show that Φ2 is a surjective map if and only if R is a unique factorization domain. Several explicit computations for hypersurface rings are also given.