The nilpotence conjecture in K-theory of toric varieties

Joseph Gubeladze

备注：: final version, accepted for publication in Inventiones mathematicae

英文摘要：

It is shown that all nontrivial elements in higher $K$-groups 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The coefficient field in the nilpotence conjecture for toric varieties

The main result of the work ``The nilpotence conjecture in K-theory of toric varieties'' is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties.

2002-11-11math.AGmath.KT

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.

2007-02-05math.ACmath.KT

Support varieties and the Hochschild cohomology ring modulo nilpotence

This is a survey paper based on my talks at the 41st Symposium on Ring Theory and Representation Theory, held in Shizuoka University, Japan in September 2008, and will appear in the conference proceedings. The paper begins with a brief introduction to the use of Hochschild cohomology in developing the theory of support varieties for a module over an artin algebra, by Snashall and Solberg (Proc. London Math. Soc.(3) 88 (2004), 705-732). The paper then describes the current status of research concerning the structure of the Hochschild cohomology ring modulo nilpotence.

2008-12-05math.KTmath.RT

Some remarks concerning Voevodsky's nilpotence conjecture

In this note we extend Voevodsky's nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky's original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds.

2014-11-04math.AGmath.ATmath.KT

Oriented Borel-Moore homologies of toric varieties

We generalize the Künneth formula for Chow groups to an arbitrary OBM-homology theory satisfying descent (e.g. algebraic cobordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a description for the homology groups of smooth toric varieties, that allows the calculation of algebraic cobordism groups of singular toric varieties. Some computations are carried out.

2020-11-28math.AGmath.KT

The $K$-theory of toric varieties

Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data.

2007-09-13math.AGmath.KT

更多相关论文收起更多论文

Toric varieties with huge Grothendieck group

In each dimension n>2 there are many projective simplicial toric varieties whose Grothendieck groups of vector bundles are at least as big as the ground field. In particular, the conjecture that the Grothendieck groups of locally trivial sheaves and coherent sheaves on such varieties are rationally isomorphic fails badly.

2003-07-04math.AGmath.KT

The equivariant K-theory of toric varieties

This paper contains two results concerning the equivariant K-theory of toric varieties. The first is a formula for the equivariant K-groups of an arbitrary affine toric variety, generalizing the known formula for smooth ones. In fact, this result is established in a more general context, involving the K-theory of graded projective modules. The second result is a new proof of a theorem due to Vezzosi and Vistoli concerning the equivariant K-theory of smooth (not necessarily affine) toric varieties.

2008-09-19math.AGmath.KT

K-theory of toric varieties revisited

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…

2012-12-13math.AGmath.ATmath.KT