Structure Theorem of Kummer Étale K-group

Hagihara, Kei

doi:10.1023/b:kthe.0000006867.98007.5c

Cited by 14 publications

(41 citation statements)

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“…They allow to associate with

X

two sites

X_{Két}

and

X_{Kfl}

, called, respectively, the Kummer‐étale and the Kummer‐flat site; the first works well in characteristic 0 and for studying

l

‐adic cohomologies, while the second is more suited for positive characteristic. Coherent sheaves on Kato's Kummer flat site

X_{Kfl}

have been used to define the

K

‐theory of

X

in .…”

Section: Introductionmentioning

confidence: 99%

Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

Talpo

Vistoli

2018

Proc. London Math. Soc.

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We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author in Adv. Math. (231 (2012Math. (231 ( ) 1327Math. (231 ( -1363. We show in particular that the infinite root stack determines the logarithmic structure and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting. ContentsBoth authors were supported in part by the PRIN project 'Geometria delle varietà algebriche e dei loro spazi di moduli' from MIUR, and by research funds from the Scuola Normale SuperioreThe infinite root stack. In this paper we construct an algebraic version of the Kato-Nakayama space, the infinite root stack ∞ √ X of X, which is a proalgebraic stack over X. For this we build on the construction of the root stacks B √ X in [7] (denoted by X B/M X there), which in turn is based on several particular cases constructed by Olsson [29,34]. Denote by α X : M X → O X the logarithmic structure on X, a fine saturated logarithmic scheme, and set M X def = M X /O * X ; then M X is a sheaf of monoids on the smallétale site Xé t , whose geometric fibers are sharp fine saturated monoids (these are the monoids that appear in toric geometry, consisting of the integral points in a strictly convex rational polyhedral cone in some R n ). A Kummer extension B of M X is, roughly speaking, a sheaf of monoids containing M X , such that every section of B has locally a positive multiple in M X (see [7, Definition 4.2]); the typical example is the sheaf 1 d M X of fractions of sections of M X with some fixed denominator S ∞ √ X; this is a proalgebraic stack that can be described very explicitly, but is not an infinite root stack. It has some advantages over (X 0 , M X0 ); for example, the projection s 0 ×∞ √X → X 0 can be used as a substitute for (X 0 , M X0 ) in problems such as degenerations of Gromov-Witten invariants. We plan to go back to this in a later paper.This 'central fiber' of the infinite root stack, and much of the basic formalism developed in the present article, is crucial in the recent work [38] about a logarithmic version of the derived McKay correspondence, of S. Scherotzke, N. Sibilla and the first author.

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“…They allow to associate with

X

two sites

X_{Két}

and

X_{Kfl}

, called, respectively, the Kummer‐étale and the Kummer‐flat site; the first works well in characteristic 0 and for studying

l

‐adic cohomologies, while the second is more suited for positive characteristic. Coherent sheaves on Kato's Kummer flat site

X_{Kfl}

have been used to define the

K

‐theory of

X

in .…”

Section: Introductionmentioning

confidence: 99%

Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

Talpo

Vistoli

2018

Proc. London Math. Soc.

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“…In the complex analytic case, Kato and Nakayama introduced in [26] a topological space X log (where X is a log analytic space), which may be interpreted as the "underlying topological space" of X, and over which, in some cases, one can write a comparison between logarithmic de Rham cohomology and ordinary singular cohomology. In a different direction, for a log scheme X, Kato introduced two sites, the Kummer-flat site X Kf l and the Kummer-étale site X Ket , that are analogous to the small fppf and étale site of a scheme, and were used later by Hagihara and Nizio l [17,33] to study the K-theory of log schemes.…”

Section: Introductionmentioning

confidence: 99%

Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes

et al. 2017

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For a log scheme locally of finite type over C, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space [26]. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack [48]. Finally, for a log scheme not necessarily over C, another natural candidate is the profiniteétale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over C, these three notions agree. In particular, we construct a comparison map from the Kato-Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profiniteétale homotopy type of its infinite root stack.2010 Mathematics Subject Classification. MSC Primary: 14F35, 55P60; Secondary: 55U35. Key words and phrases. log scheme, Kato-Nakayama space, root stack, profinite spaces, infinity category, etale homotopy type, topological stack. 1 1.2. The comparison map and the equivalence of profinite completions. Our main result states:Theorem (see Theorem 6.4). Let X be a fine saturated log scheme locally of finite type over C. Then there is a canonical map of pro-topological stacksthat induces an equivalence upon profinite completion

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“…26 In other words,1 .A;M / D 1 A˚. A˝M gp /= where is A-linearly generated by the relation d˛.m/ D .1˝d m/ .1˝d m/ D .m/˝ .m/ for m 2 M .…”

mentioning

confidence: 99%