Necessary conditions for the depth formula over Cohen–Macaulay local rings

Abstract

Let R be a Cohen–Macaulay local ring and let M and N be finitely generated R-modules. In this paper we investigate some of the necessary conditions for the depth formula depth(M)+depth(N)=depth(R)+depth(M⊗RN) to hold. We show that, under certain conditions, M and N satisfy the depth formula if and only if ToriR(M,N)=0 for all i≥1. We also examine the relationship between the depth of M⊗RN and the vanishing of Ext modules with various applications. One of them extends partially a result by Auslander on even dimensional regular local rings to complete intersections. In another application, we show that there is no nontrivial semidualizing module over Veronese subrings of the formal power series ring k[[X1,…,Xn]] over a field k.