Super finitely presented modules and Gorenstein projective modules

TL;DR

This paper introduces super finitely presented modules over commutative rings, explores their relation to Gorenstein projective modules, and links these concepts to $K_0$-regularity under certain conditions.

Contribution

It establishes conditions under which super finitely presented modules have finite Gorenstein projective dimension and finite projective dimension, connecting module theory to algebraic K-theory.

Findings

Finitely generated Gorenstein projective modules are super finitely presented under property (B).

Rings with property (C) are $K_0$-regular.

Super finitely presented modules have finite Gorenstein projective dimension under property (B).

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 $\cdots\rightarrow P_n\rightarrow\cdots \rightarrow P_1\rightarrow P_0\rightarrow M\rightarrow 0$ 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 $n\geq 1$.