Modules of finite length and Chow groups of surfaces with rational double points
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.
scientific article; zbMATH DE number 3973028