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12
The nilpotence conjecture in Ktheory of toric varieties
, 2002
"... It is shown that all nontrivial elements in higher Kgroups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. ..."
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It is shown that all nontrivial elements in higher Kgroups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties.
A 1Homotopy of Chevalley Groups
, 2008
"... In this paper, we describe the sheaves of A1homotopy groups of a simplyconnected Chevalley group G. The A1homotopy group sheaves can be identified with the sheafification of the unstable KaroubiVillamayor Kgroups. ..."
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In this paper, we describe the sheaves of A1homotopy groups of a simplyconnected Chevalley group G. The A1homotopy group sheaves can be identified with the sheafification of the unstable KaroubiVillamayor Kgroups.
Improved stability for SK1 and WMSd of a nonsingular affine algebra. Ktheory
 Asterisque
, 1992
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Global coefficient ring in the nilpotence conjecture
 Proc. AMS, available at arXiv:math/0701430v3
, 2007
"... Abstract. 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 mono ..."
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Abstract. 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 Kgroups, 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 ↦ → 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 Khomotopy 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 BlochStienstra’s actions of the big Witt vectors on the NKigroups [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 Qvector 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 N c = lim
Injective Stability for K1 of Classical Modules
, 909
"... Abstract: In [13], the second author and W. van der Kallen showed that the injective stabilization bound for K1 of general linear group is d+1 over a regular affine algebra over a perfect C1field, where d is the krull dimension of the base ring and it is finite and at least 2. In this article we pr ..."
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Abstract: In [13], the second author and W. van der Kallen showed that the injective stabilization bound for K1 of general linear group is d+1 over a regular affine algebra over a perfect C1field, where d is the krull dimension of the base ring and it is finite and at least 2. In this article we prove that the injective stabilization bound for K1 of the symplectic group is d + 1 over a geometrically regular ring containing a field, where d is the stable dimension of the base ring and it is finite and at least 2. Then using the LocalGlobal Principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d + 1 for K1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above. 1
On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme
, 2011
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Stability result for projective modules over blowup rings
, 2004
"... Let R be a normal affine domain of dimension n ≥ 3 over an algebraically closed field k. Suppose char k = 0 or char k = p ≥ n. Let g,f1,...,fr be a Rregular sequence and A = R[f1/g,...,fr/g]. Let P be a stably free Amodule of rank n − 1. Then, Murthy proved that there exists a projective Rmodule ..."
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Let R be a normal affine domain of dimension n ≥ 3 over an algebraically closed field k. Suppose char k = 0 or char k = p ≥ n. Let g,f1,...,fr be a Rregular sequence and A = R[f1/g,...,fr/g]. Let P be a stably free Amodule of rank n − 1. Then, Murthy proved that there exists a projective Rmodule Q such that Q ⊗ RA = P and ∧ n−1 Q = R