On relative and bi-relative algebraic 𝐾-theory of rings of finite characteristic

Geisser, Thomas; Hesselholt, Lars

doi:10.1090/s0894-0347-2010-00682-0

Cited by 10 publications

(7 citation statements)

References 24 publications

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“…0.1]. Moreover, T. Geisser and L. Hesselholt [14,15] have established a pro version of the Suslin-Wodzicki condition. In section 1 we use Geisser-Hesselholt's results to show that if I is the conductor ideal of a one-dimensional, Noetherian, reduced ring for which the normalisation map is finite, then it satisfies pro-excision, thereby resulting in long exact, Mayer-Vietoris, pro-excision sequences such as (pro-MV) above.…”

Section: Introductionmentioning

confidence: 99%

-theory of one-dimensional rings via pro-excision

Morrow¹

2013

J. Inst. Math. Jussieu

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Link to this article: http://journals.cambridge.org/abstract_S1474748013000194 How to cite this article: Matthew Morrow (2014). -theory of one-dimensional rings via pro-excision .Abstract This paper studies 'pro-excision' for the K-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller's conjecture in mixed characteristic, and we establish various finiteness results for the K-groups of singularities, covering both orders in number fields and singular curves over finite fields.

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Section: Introductionmentioning

confidence: 99%

-theory of one-dimensional rings via pro-excision

Morrow¹

2013

J. Inst. Math. Jussieu

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“…Proof. If X is affine, this follows from [29,Theorem C]. In general, we shall argue by induction on the minimal number of affine open subsets that cover X.…”

Section: 3mentioning

confidence: 99%

“…Let k be any field. In this section, we use some results of [29] to study the torsion in the bi-relative K-groups associated to certain abstract blow-ups in Sch k . The results of this section will be used in the proof of the torsion theorem for surfaces.…”

Section: Torsion In Bi-relative K-theorymentioning

confidence: 99%

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Murthy’s conjecture on 0-cycles

Krishna

2019

Invent. math.

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“…The last part of this paper (Sections [12][13][14] is devoted to the notion of cohomological descent (Definition 12.11), the proof of our Main Theorem 0.1 and its application to algebraic K-theory and topological cyclic homology.…”

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confidence: 99%

Toric varieties, monoid schemes and cdh descent

Cortiñas

Haesemeyer

Walker

et al. 2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

104

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Abstract. We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties and schemes, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop many notions for monoid schemes based on classical algebraic geometry, such as separated and proper maps and resolution of singularities.The goal of this paper is to prove Haesemeyer's Theorem [18, 3.12] for toric schemes in any characteristic. It is proven below as Corollary 14.4.Theorem 0.1. Assume k is a commutative regular noetherian ring containing an infinite field and let G be a presheaf of spectra defined on the category of schemes of finite type over k. If G satisfies the Mayer-Vietoris property for Zariski covers, finite abstract blow-up squares, and blow-ups along regularly embedded closed subschemes, then G satisfies the Mayer-Vietoris property for all abstract blow-up squares of toric k-schemes obtained from subdividing a fan.The application we have in mind is to understand the relationship between the algebraic K-theory K * (X) = π * K(X) and topological cyclic homology T C * (X) = {π * T C ν (X, p)} of a toric scheme over a regular ring of characteristic p (and in particular of toric varieties over a field of characteristic p). Thus we consider the presheaf of homotopy fibers {F ν (X)} of the map of pro-spectra from K(X) to {T C ν (X, p)}. Work of Geisser-Hesselholt [11, Thm. B], [12] shows that this homotopy fiber (regarded as a pro-presheaf of spectra) satisfies the hypotheses of Theorem 0.1 and hence a slight modification of the proof of our theorem implies that it satisfies the Mayer-Vietoris property for all abstract blow-up squares of toric schemes. We will give a rigorous proof of this in Corollary 14.8 below.One major tool in our proof will be a theorem of Bierstone-Milman [1] which says that the singularities of a toric variety (or scheme) can be resolved by a sequence of blow-ups X C → X along a center C that is a smooth, equivariant closed subscheme of X along which X is normally flat. If one only had to consider toric schemes, this would allow one to use Haesemeyer's original argument to prove Theorem 0.1,

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On relative and bi-relative algebraic 𝐾-theory of rings of finite characteristic

Cited by 10 publications

References 24 publications

-theory of one-dimensional rings via pro-excision

-theory of one-dimensional rings via pro-excision

Murthy’s conjecture on 0-cycles

Toric varieties, monoid schemes and cdh descent

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