Some homological properties of commutative semitrivial ring extensions

Abstract

LetR be a commutative ring with 1 andM anR-module. Ifφ:M⊗ _R M→R is anR-module homomorphism satisfyingφ(m⊗m′)=φ(m′⊗m) andφ(m⊗m′)m″=mφ(m′⊗m″), the additive abelian groupR⊗M becomes a commutative ring, if multiplication is defined by (r,m)(r′,m′)=(rr′+φ(m⊗m′),rm′+r′m). This ring is called the semitrivial extension ofR byM andφ and it is denoted byRα _φ M. This generalizes the notion of a trivial extension and leads to a more interesting variety of examples. The purpose of this paper is to studyRα _φ M; in particular, we are interested in some homological properties ofRα _φ M as that of being Cohen-Macaulay, Gorenstein or regular. A sample result: Let (R,m) be a local Noetherian ring,M a finitely generatedR-module and Im(φ) ⊑ m. ThenRα _φ M is Gorenstein if and only if eitherRαM is Gorenstein orR is Gorenstein,M is a maximal Cohen-Macaulay module andM≅M ^*, where the isomorphism is given by the adjoint ofφ.