Grothendieck groups, convex cones and maximal Cohen–Macaulay points.

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