Generalized tilting modules and Frobenius extensions

Abstract

Let $ A/S $ be a ring extension with $ S $ commutative. We prove that $ \omega{\otimes}_SA_A $ is a generalized tilting module if $ \omega_S $ is a generalized tilting module. In this case, we obtain that $ ^\bot \omega $-resol.dim$ _S(M) $ and $ ^\bot (\omega\otimes_SA) $-resol.dim$ _A(M) $ are identical for any $ A $-module $ M $. As an application, we show that $ S $ satisfies gorenstein symmetric Conjecture if and only if so does $ A $. Furthermore, we introduce the concept of $ ^\bot\omega $-Gorenstein projective modules, and we obtain that the relative Gorenstein projectivity is invariant under Frobenius extensions.

Keywords

• generalized tilting modules
• frobenius extensions
• separable extensions
• ^⊥ω-dimensions
• ^⊥ω-gorenstein projective modules