Tensor products of perfect modules and maximal surjective Buchsbaum modules (Q1380041)

Let \(A\) be a Cohen-Macaulay (shortly CM) local ring and \(M\) a finite \(A\)-module with finite \(\text{pd}_AM\). Then \(M\) is perfect iff \(M\otimes_AN\) is a CM \(A\)-module with \(\dim M\otimes_AN=\dim M\) for some (resp. any) maximal CM \(A\)-module \(N\). This result extends a Kawasaki result [cf. \textit{T. Kawasaki}, Math. Z. 218, No. 2, 191-205 (1995; Zbl 0814.13017)] obtained for the case when \(N\) is a canonical module of \(A\). Moreover if \(M\) is perfect and \(N\) is a maximal surjective Buchsbaum \(A\)-module then \(M\otimes_AN\) is Buchsbaum, \(\dim M\otimes_AN= \dim M\) and \(\text{depth }M\otimes_AN= \max\{\text{depth }N- \text{pd}_AM,0\}\).