Grothendieck groups, convex cones and maximal Cohen–Macaulay points
R Takahashi�- Mathematische Zeitschrift, 2021 - Springer
Mathematische Zeitschrift, 2021•Springer
Let R be a commutative noetherian ring. Let H (R) 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 R-vector space H (R) _ R= H (R) ⊗ _ ZRH (R) R= H (R)⊗ ZR has finite
dimension. Let C (R) C (R)(resp. C _r (R) C r (R)) be the convex cone in H (R) _ RH (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�…
group of finitely generated R-modules by the subgroup generated by pseudo-zero modules.
Suppose that the R R-vector space H (R) _ R= H (R) ⊗ _ ZRH (R) R= H (R)⊗ ZR has finite
dimension. Let C (R) C (R)(resp. C _r (R) C r (R)) be the convex cone in H (R) _ RH (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�…
Abstract
Let R be a commutative noetherian ring. Let be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the -vector space has finite dimension. Let (resp. ) be the convex cone in 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 . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in in terms of topological properties of . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group .
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