K-theory of blow-ups and vector bundles on the cone over a surface
The \(K\)-theory of varieties with isolated singularities is studied in this paper. Let \(K^ Y_ 0 (X)\) be the zero-th homotopy group of the \(K\)-theory spectrum of Waldhausen's category of perfect complexes on \(X\) which are acyclic on \(Y\) where \(Y\) is a finite subset of the variety \(X\) containing all singularities. This group is computed and shown to contain \(SK_ 1(U)\) as a direct summand for \(U = \text{Spec} ({\mathcal O}_{X,Y}) - Y\). More explicit computations are given in the special case where \(X\) is the cone over a nonsingular projective surface of degree \(d \geq 2\) in \(\mathbb{P}^{d + 1}\) not contained in a hyperplane and \(Y\) is the cone point. In particular the vanishing of the relative Chow groups \(\text{CH}^ i (X,Y)\) is proven for \(i \geq 2\). Finally special cases of a conjecture of Srinivas are verified that for a projective normal nonsingular surface \(T\) over \(\mathbb{C}\) with affine cone \(X\) and cone point \(Y\) the vanishing of \(H^ 2 (T, {\mathcal O}_ T(1))\) is equivalent to the vanishing of \(\text{CH}^ 3 (X,Y)\).
homotopy group of the \(K\)-theory spectrum
scientific article; zbMATH DE number 488351