Modules of finite length and Chow groups of surfaces with rational double points
type: Item
instance of: scholarly article
title: Modules of finite length and Chow groups of surfaces with rational double points
zbMATH Open document ID: 0603.14008
published in: Illinois Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: For a local ring R, let \({\mathcal C}_ R\) be the category of R-modules of finite length and finite projective dimension, and denote by \(K_ 0({\mathcal C}_ R)\) its Grothendieck group. The main result of this paper is that if Spec(R) is a rational double point of a surface over an algebraically closed field of characteristic zero, then \(K_ 0({\mathcal C}_ R)={\mathbb{Z}}\). The second result is that if \(f: X\to Y\) is a resolution of singularities of a normal quasiprojective surface X with only quotient singularities, then the induced map \(f^*: K_ 0(X)\to K_ 0(Y)\) of Grothendieck groups (of vector bundles) induces an isomorphism \(F_ 0K_ 0(X)\to F_ 0K_ 0(Y)\) of Chow groups, where \(F_ 0K_ 0(X)\subset K_ 0(X)\) is the subgroup generated by the classes of smooth points. The author makes some remarks about the situation in characteristic \(p,\) and computes \(K_ 0({\mathcal C}_ R)\) up to p-torsion for a particular class of examples.
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 13E15
Mathematics Subject Classification ID: 13D15
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 3973028
zbMATH Keywords: modules of finite length
zbMATH Keywords: Grothendieck group
zbMATH Keywords: rational double point of a surface
zbMATH Keywords: resolution of singularities
zbMATH Keywords: Chow groups
author: V. Srinivas
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3973028