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 char k = 0 the multiplicative monoid N acts nilpotently on the quotient K i (k[M])/K i (k) of the ith K-groups, i ≥ 0. In other words, for any sequence of natural numbers c 1 , c 2 , . . . ≥ 2 and any element x ∈ K i (k[M]) we have (c 1 · · · c j ) * (x) ∈ K i (k) for all j ≫ 0 (potentially depending in x). Here c * refers to the group endomorphism of K i (k[M]) induced by the monoid endomorphism M → M, m → m c , 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 NK i -groups [St] (that has already played a crucial role in [G], but in a different context), Lindel's technique ofétale neighborhoods [L], van der Kallen'sétale localization [K], and Popescu's desingularization [Sw], we show Theorem 1. Let M be an additive submonoid of a Q-vector space with trivial group of units. Then for any regular ring R with Q ⊂ R the multiplicative monoid N acts 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 = c 1 , c 2 , . . . ≥ 2 and an additive submonoid N of a rational space V we put Lemma 2. Let M be a finitely generated submonoid of a rational vector space with the trivial group of units. Then M embeds into a free commutative monoid Z r + .
For the stronger version of Lemma 2 with r = dim Q (Q ⊗ M) see, for instance, [BG,Proposition 2.15(e)]. (In [BG] the monoids as in Lemma 2 are called the affine positive monoids.) Lemma 3. Let F be a functor from rings to abelian groups, H be a monoid with the trivial group of units, and Then we have the implication The special case of Lemma 3 when H = Z + is known as the Swan-Weibel homotopy trick and the proof of the general cases makes no real difference, see [G,Proposition 8.2].
Lemma 4. Theorem 1 is true for any coefficient ring of the form In the special case when k is a number field Lemma 4 is proved in Step 2 in [G, §8], but word-by-word the same argument goes through for a field k provided the nilpotence conjecture is true for the monoid rings with coefficients in k.
Notice. The reason we state the result in [G] only for number fields is that the preceding result in [G] is the validity of the nilpotence conjecture for such coefficient fields. Actually, the proof of Theorem 1 is anétale version of the idea of interpreting the globalization problem for coefficient rings in terms of the K-homotopy invariance, used for Zariski topology in [G, §8].
Finally, in order to explain one formula we now summarize very briefly the Bloch-Stienstra action of the ring of big Witt vectors W(Λ) on For the details the reader is referred to [St].
The additive group of W(R) can be thought of as the multiplicative group of formal power series 1+XΛ [[X]]. It has the decreasing filtration by the ideals I p (R) = (1+X p+1 Λ[[X]]), p = 1, 2, . . . , and every element α(X) ∈ W(Λ) admits a convergent series expansion in the corresponding additive topology α(X) = Π N (1 − λ m X m ), λ m ∈ Λ. To define a continuous W(Λ)-module structure on NK i (Λ) it is enough to define the appropriate action of the Witt vectors of type 1 − λX m , satisfying the condition that every element of NK i (Λ) is annihilated by some ideal I p (W(Λ)). Finally, such an action of 1 − λX m on NK i (Λ) is provided by the composite map in the upper row of the following commutative diagram with exact vertical columns: (1) m * corresponds to scalar extension through the Λ-algebra endomorphism Λ[X] → Λ[X], X → X m , (2) m * corresponds to scalar restriction through the same endomorphism Λ[X] → Λ[X], (3) λ * corresponds to scalar extension through the Λ-algebra endomorphism Λ[X] → Λ[X], X → λX. (4) m · − is multiplication by m. A straightforward check of the commutativity of the appropriate diagrams, based on the description above, shows that for a ring homomorphism f : Λ 1 → Λ 2 we have where the same f * is used for the both induced homomorphisms Proof of Theorem 1. Since K-groups commute with filtered colimits there is no loss of generality in assuming that M is finitely generated. Then by Lemma 2 R[M] admits a Z + -grading In particular, by the Quillen local-global patching for higher K-groups [V], we can without loss of generality assume that R is local.
Notice. Actually, the local-global patching proved in [V] is for the special case of polynomial extensions. However, the more general version for graded rings is a straightforward consequence via the Swan-Weibel homotopy trick, discussed above.
By Popescu's desingularization [Sw] and the same filtered colimit argument we can further assume that R is a regular localization of an affine k-algebra for a field k with char k = 0. In this situation Lindel has shown [L,Proposition 2 Notice. Lindel's result is valid in arbitrary characteristic under the conditions that the residue field of R is a simple separable extension of k, which is automatic in our situation because char k = 0.
Using again that K-groups commute with filtered colimits, the validity of Theorem 1 for R is easily seen to be equivalent to the equality for every sequence of natural numbers c = c 1 , c 2 , . . . ≥ 2.
Next we show that (3) follows from the condition In fact, by the filtered colimit argument we have On the other hand, by Lemma 2 the ring R[M c ] has a Z c + -grading:

So by Lemma 3 we have (3).
To complete the proof it is enough to show (4) assuming (2)