Super finitely presented modules and Gorenstein projective modules

Abstract

Let $R$ be a commutative ring. An $R$-module $M$ is said to be super finitely presented if there is an exact sequence of $R$-modules $\cdots \to P_n\to \cdots \to P_1\to P_0\to M\to 0$ where each $P_i$ is finitely generated projective. In this paper it is shown that if $R$ has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if $R$ has the property (C) that every super finitely presented module has finite projective dimension, then $R$ is $K_0$-regular, i.e., $K_0(R[x_1,\cdots ,x_n])\cong K_0(R)$ for all $n\ge 1$.

15 pages

References in corpus (2)

• Global Gorenstein dimensions of polynomial rings and of direct products of rings
• Global Gorenstein dimensions of polynomial rings and of direct product of rings