Grothendieck groups, convex cones and maximal Cohen-Macaulay points

TL;DR

This paper investigates the structure of the Grothendieck group and convex cones associated with maximal Cohen-Macaulay modules over a commutative noetherian ring, providing conditions for finiteness and exploring their geometric properties.

Contribution

It introduces new criteria linking the finiteness of maximal Cohen-Macaulay points to topological and geometric properties of convex cones in the Grothendieck group.

Findings

Characterization of finiteness conditions for Cohen-Macaulay modules

Analysis of the convex cone structure and its boundaries

Connections between Cohen-Macaulay modules and divisor class groups

Abstract

Let A be a commutative noetherian ring. Let H(A) be the quotient of the Grothendieck group of finitely generated A-modules by the subgroup generated by pseudo-zero modules. Suppose that the real vector space H(A)_R = H(A) \otimes_Z R has finite dimension. Let C(A) (resp. C_r(A)) be the convex cone in H(A)_R spanned by maximal Cohen-Macaulay A-modules (resp. maximal Cohen-Macaulay A-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of C(A). We provide various equivalent conditions for A to have only finitely many rank r maximal Cohen-Macaulay points in C_r(A) in terms of topological properties of C_r(A). Finally, we consider maximal Cohen-Macaulay modules of rank one as elements of the divisor class group Cl(A).