Higher K-Theory of Toric Varieties

Gubeladze, Joseph

doi:10.1023/a:1026237927244

Cited by 7 publications

(22 citation statements)

References 27 publications

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“…On the other hand there is a big difference between the simplicial and general cases. A further support for the nilpotence conjecture, obtained in [Gu3], is the implication indicated by the north-west arrow in the diagram above (see also [Gu6,Proposition 3.5(c)]). This is a higher analog of Vorst's result [V2] on Serre's problem for discrete Hodge algebras.…”

Section: Introductionsupporting

confidence: 63%

“…Actually, the south-west arrow in our diagram says that for p = 0 the nilpotence conjecture is equivalent to the formally stronger equalities K 0 (R) = K 0 (R [M]) where M runs through all normal monoids 1 . This is proved in [Gu6,Proposition 3.5(a,b)].…”

Section: Introductionmentioning

confidence: 80%

“…That for a restricted class of nonsimplicial toric varieties 2 over number fields 3 the aforementioned 'almost proof' can be converted into an actual proof was also shown in [Gu6], based on the Bloch-Stienstra-Weibel action of the big Witt vectors on the nil-K-theory ( [Bl][St] [W]) -an idea suggested by Burt Totaro, and Goodwillie's rational isomorphism between relative K-groups and relative cyclic homology groups for a nilpotent ideal [Go] -a technique of crucial importance also for this paper. The mentioned action of the big Witt vectors led us in [Gu6] to a question on certain filtration on certain relative K-groups, which, if answered in the positive, completes the proof of Conjecture 1.1 when the coefficient ring is a characteristic 0 field [Gu6,Question 10.2].…”

Section: Introductionmentioning

confidence: 88%

“…The decisive step towards Theorem 1.2 has been our previous work [Gu6] which, using Thomason's Mayer-Vietoris sequence for singular varieties [TT] and virtually all of our previous results 'almost proves' the nilpotence conjecture when the coefficient ring is a field. This 'almost proves' is made precise in Section 4 below.…”

Section: Introductionmentioning

confidence: 99%

“…This 'almost proves' is made precise in Section 4 below. One of the main features of [Gu6] is the passage to auxiliary non-affine toric varieties (with two affine toric charts) and the endomorphism rings of certain rank 2 vector bundles on them. This explains the heavy use of 2 × 2 matrix rings in this paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The nilpotence conjecture in K-theory of toric varieties

Gubeladze

2005

Invent. math.

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Section: Introductionsupporting

confidence: 63%

Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The nilpotence conjecture in K-theory of toric varieties

Gubeladze

2005

Invent. math.

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The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic

Cortiñas

Haesemeyer

Walker

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

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We show that if X is a toric scheme over a regular commutative ring k then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when k is replaced by an appropriate K-regular, not necessarily commutative k-algebra.

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$$K$$ -theory of toric varieties revisited

Gubeladze

2013

J. Homotopy Relat. Struct.

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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 .1. K-theory of toric varieties: survey 1.1. Conventions. All our rings and monoids are commutative. Unless specified otherwise, the monoid operation is written additively.Our monoid and convex geometry terminology follows [3]. In particular, an affine monoid is a finitely generated submonoid of a free abelian group. For an affine monoid M its largest subgroup will be denoted by U(M). An affine monoid M is called (i) positive if U(M) = 0, (ii) normal if M is isomorphic to a monoid of the form C ∩ Z d for a finite rational cone C ⊂ R d , and (iii) seminormal if, for every element x ∈ gp(M), the inclusions 2x, 3x ∈ M imply x ∈ M, where gp(M) is the group of differences of M, i.e., gp(M) is the universal group to which M maps. We say that M is simplicial if the coneis such. For a functor, defined on rings, a natural number c, and a ring R we denote by c * the endomorphism, induced by the monoid ring endomorphismc , where the monoid operation is written multiplicatively. For generalities on toric varieties the reader is referred to [3, Chap. 10] and [7]. All fans considered below are assumed to be finite and rational. Let F be a fan. One calls F simplicial if the cones in F are such. The set of maximal cones in F is denoted by max(F ). The toric scheme over a ring R, associated with F , will be denoted by V R (F ).For a fan F in R d , the scheme V R (F ) is (i) complete if and only if F is complete, i.e., max(F ) σ = R d , (ii) projective if and only if F is projective, i.e., there is a full dimensional polytope P in the dual space (R d ) op such that the set of duals of the corner cones of P is exactly max(F ), and (iii) quasi-projective if and only if

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Higher K-Theory of Toric Varieties

Cited by 7 publications

References 27 publications

The nilpotence conjecture in K-theory of toric varieties

The nilpotence conjecture in K-theory of toric varieties

The K-Theory of Toric Schemes Over Regular Rings of Mixed Characteristic

$$K$$ -theory of toric varieties revisited

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