Self-orthogonal modules over coherent rings

Abstract

Let R be a left coherent ring, S any ring and R ω S an ( R, S)-bimodule. Suppose ω S has an ultimately closed FP-injective resolution and R ω S satisfies the conditions: (1) ω S is finitely presented; (2) The natural map R→ End( ω S ) is an isomorphism; (3) Ext S i ( ω, ω)=0 for any i≥1. Then a finitely presented left R-module A satisfying Ext R i ( A, ω)=0 for any i≥1 implies that A is ω-reflexive. Let R be a left coherent ring, S a right coherent ring and R ω S a faithfully balanced self-orthogonal bimodule and n≥0. Then the FP-injective dimension of R ω S is equal to or less than n as both left R-module and right S-module if and only if every finitely presented left R-module and every finitely presented right S-module have finite generalized Gorenstein dimension at most n.