K-theory of toric varieties revisited

TL;DR

This paper revisits the higher K-theory of toric varieties, presenting Totaro's unpublished results that connect homotopy K-theory with singular cohomology, and discusses implications for nil-groups of affine monoid rings.

Contribution

It provides a higher analog of the rational Chern character isomorphism for toric schemes and explores the structure of nil-groups in affine monoid rings.

Findings

Totaro's unpublished result links homotopy K-theory to singular cohomology.

Rational isomorphism between homotopy K-theory and K-theory of the ground ring for certain toric schemes.

Conjecture on nil-groups of affine monoid rings extending nilpotence property.

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After surveying higher K-theory of toric varieties, we present Totaro's old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology.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.