Global Coefficient Ring in the Nilpotence Conjecture

In this note we show that the nilpotence conjecture for toric varieties is true over any regular coefficient ring containing Q. In [G] we showed that for any additive submonoid M of a rational vector space with the trivial group of units and a field k with chark = 0 the multiplicative monoid N acts nilpotently on the quotient Ki(k[M ])/Ki(k) of the ith K-groups, i ≥ 0. In other words, for any sequence of natural numbers c1, c2, . . . ≥ 2 and any element x ∈ Ki(k[M ]) we have (c1 · · · cj)∗(x) ∈ Ki(k) for all j ≫ 0 (potentially depending in x). Here c∗ refers to the group endomorphism of Ki(k[M ]) induced by the monoid endomorphism M → M , m 7→ m, writing the monoid operation multiplicatively. The motivation of this result is that it includes the known results on (stable) triviality of vector bundles on affine toric varieties and higher K-homotopy invariance of affine spaces. Here we show how the mentioned nilpotence extends to all regular coefficient rings containing Q, thus providing the last missing argument in the long project spread over many papers. See the introduction of [G] for more details. Using Bloch-Stienstra’s actions of the big Witt vectors on the NKi-groups [St] (that has already played a crucial role in [G], but in a different context), Lindel’s technique of étale neighborhoods [L], van der Kallen’s étale localization [K], and Popescu’s desingularization [Sw], we show Theorem 1. Let M be an additive submonoid of a Q-vector space. Then for any regular ring R with Q ⊂ R the multiplicative monoid N acts nilpotently on Ki(R[M ])/Ki(R), i ≥ 0. Conventions. All our monoids and rings are assumed to be commutative. X is a variable. The monoid operation is written mutliplicatively, denoting by e the neutral element. Z+ is the additive monoid of nonnegative integers. For a sequence of natural numbers c = c1, c2, . . . ≥ 2 and an additive submonoid N of a rational space V we put