Gorenstein Orders Associated with Modules

Abstract

Let R be a commutative Gorenstein ring. We call a Noether R-algebra Λ a Gorenstein R-order provided Λ has Gorenstein dimension zero as an R-module and Λ ≅ Hom R (Λ, R) as Λ-bimodules. Let Λ be a Gorenstein R-order. We provide several ways to extend Λ to a Gorenstein R-order Δ containing an idempotent e such that Λ ≅ eΔe. In particular, for a finitely generated right Λ-module M having Gorenstein dimension zero as an R-module, setting Δ = EndΛ(M ⊕ Λ), we show that EndΛ(Ω n M ⊕ Λ) is derived equivalent to Δ for all n ∈ ℤ, and hence if Δ is a Gorenstein R-order then so is every EndΛ(Ω n M ⊕ Λ), and that Δ is a Gorenstein R-order if M 𝔭 is projective as a right Λ𝔭-module for every prime ideal 𝔭 of R with ht 𝔭 ≤ 1 and , 1 ≤ i ≤ ht 𝔭 −2, for every prime ideal 𝔭 of R with ht 𝔭 ≥3.