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 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 and 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 K0-regular. We then determine conditions under which our approach for dimension 2 works in arbitrary dimensions, before demonstrating that weighted projective spaces are not K1-regular, and for dimensions bigger than 2 are also not in general K0-regular.