Grothendieck groups, convex cones and maximal Cohen-Macaulay points
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) Z R has finite dimension. Let C(A) (resp. Cr(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 Cr(A) in terms of topological properties of Cr(A). Finally, we consider maximal Cohen-Macaulay modules of rank one as elements of the divisor class group Cl(A).