Grothendieck groups of polynomial and Laurent polynomial rings
In order to probe nonregularity of noetherian k-schemes X, groups like \(NK_ 0(X)=coker (K_ 0(X)\to K_ 0(X\times {\mathbb{A}}^ 1))\) and \(K_{-1}\) (a similar cokernel) have been defined. Take in particular \(k={\mathbb{C}}\) and for X a normal quasi-projective surface. It is shown that, if R is the semilocal ring of the singular locus, then there is a surjection \(K_{-1}(X)\twoheadrightarrow C1(\hat R)/C1(R)\) where \(\hat R\) is the completion of R and C1 the ideal class group. If, on the other hand, \(Y\to X\) is a resolution of singularities, the reduced exceptional divisor E of which has smooth components and normal crossings, and if, moreover, Pic(nE)\(\twoheadrightarrow Pic(E)\) is not an isomorphism for some \(n>1\), then \(NK_ 0(X)\) has infinite rank. Examples are given.
scientific article; zbMATH DE number 3995857
scientific article