Almost split sequences for Cohen-Macaulay-modules (Q1087590)

Let R be a commutative complete local Gorenstein ring and \(\Lambda\) an R- algebra which is a finitely generated (maximal) Cohen-Macaulay R-module. Suppose C is nonprojective indecomposable in the category CM\(\Lambda\) consisting of the finitely generated \(\Lambda\) modules which are (maximal) Cohen-Macaulay R-modules. Our main result is that \(C_ p\) is \(\Lambda_ p\)-projective for all nonmaximal prime ideals P in R if and only if there is an almost split sequence \(0\to A\to B\to C\to 0\) in CM\(\Lambda\).