The KH-Theory of Complete Simplicial Toric Varieties and the Algebraic K-Theory of Weighted Projective Spaces

We show that, for a complete simplicial toric variety $X$, we can determine its homotopy \KH-theory entirely in terms of the torus pieces of open sets forming an open cover of $X$. We then construct conditions under which, given two complete simplicial toric varieties, the two spectra \KH(X) \otimes \Q and \KH(Y) \otimes \Q are weakly equivalent. We apply this result to determine the rational \KH-theory of weighted projective spaces. We next examine \K-regularity for complete toric surfaces; in particular, we show that complete toric surfaces are \K_{0}-regular. We then determine conditions under which our approach for dimension 2 works in arbitrary dimensions, before demonstrating that weighted projective spaces are not \K_{1}-regular, and for dimensions bigger than 2 are also not in general \K_{0}-regular.Comment: 14 pages. Updated version, strengthening the proofs to hold true over any regular ring. To appear in the Journal of Pure and Applied Algebr

The obstruction to excision in K-theory and in cyclic homology

Let $f:A \to B$ be a ring homomorphism of not necessarily unital rings and $I\triangleleft A$ an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups $K_*(A:I) \to K_*(B:f(I))$ to be an isomorphism; it is measured by the birelative groups $K_*(A,B:I)$. We show that these are rationally isomorphic to the corresponding birelative groups for cyclic homology up to a dimension shift. In the particular case when A and B are \Q-algebras we obtain an integral isomorphism.Comment: Final version to appear in Inventione