On indices of local rings along local homomorphisms

Abstract

Let φ:R → Sbe a local homomorphism of Gorenstein local rings of finite flat dimension. We prove if M is a stable maximal Cohen-Macaulay R-module, then M ⊗_R Sis also a stable maximal Cohen-Macaulay S-module. By using it, we also prove that index(R) ≤ index(S).