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

Abstract

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.