Let R be a commutative noetherian ring. Let \(\mathsf {H}(R)\) be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the \(\mathbb {R}\)-vector space \(\mathsf {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb {R}\) has finite dimension. Let \(\mathsf {C}(R)\) (resp. \(\mathsf {C}_r(R)\)) be the convex cone in \(\mathsf {H}(R)_\mathbb {R}\) 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 \(\mathsf {C}(R)\). We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in \(\mathsf {C}_r(R)\) in terms of topological properties of \(\mathsf {C}_r(R)\). Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group \({\text {Cl}}(R)\).

令R为可交换的无醚环。令\（\ mathsf {H}（R）\）为由伪零模块生成的子组的有限生成R模块的Grothendieck组的商。假设\（\ mathbb {R} \）-向量空间\（\ mathsf {H}（R）_ \ mathbb {R} = \ mathsf {H}（R）\ otimes _ \ mathbb {Z} \ mathbb {R} \）具有有限的尺寸。令\（\ mathsf {C}（R）\）（分别为\（\ mathsf {C} _r（R）\））为\（\ mathsf {H}（R）_ \ mathbb {R } \）由最大Cohen–Macaulay R-模块（分别为最大Cohen–Macaulay R-秩为r的模块）跨越）。我们探索\（\ mathsf {C}（R）\）的内部，闭合和边界以及凸多面体次锥。我们提供各种等价条件- [R仅具有有限多个秩ř最大科恩-麦考个\（\ mathsf {C} _r（R）\）中的拓扑性质方面\（\ mathsf {C} _r（R） \）。最后，我们将排名第一的最大Cohen–Macaulay模块视为除数类组\（{\ text {Cl}}（R）\）的元素。