  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).
