$$K$$ -theory of toric varieties revisited

Abstract

After surveying higher \(K\)-theory of toric varieties, we present Totaro’s old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy \(K\)-theory and the direct sum of \(m\) copies of the \(K\)-theory of the ground ring, \(m\) being the number of maximal cones in the underlying fan. Apart from its independent interest, in retrospect, Totaro’s observations motivated some (old) and complement several other (very recent) results. We conclude with a conjecture on the nil-groups of affine monoid rings, extending the nilpotence property. The conjecture holds true for \(K_0\).