<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>F</mml:mi></mml:math>-isocrystals and homotopy types

Olsson, Martin

doi:10.1016/j.jpaa.2006.10.006

Cited by 6 publications

(15 citation statements)

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“…New examples of homotopy types which are not realizable by smooth projective varieties can be constructed that way. I should also mention [Ol1,Ol2] in which a crystalline and a p-adic analog of the constructions above have been studied. 6.…”

Section: Schematic Homotopy Typesmentioning

confidence: 99%

Higher and derived stacks: a global overview

Toën¹

2009

Proceedings of Symposia in Pure Mathematics

153

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These are expended notes of my talk at the summer institute in algebraic geometry (Seattle, July-August 2005), whose main purpose is to present a global overview on the theory of higher and derived stacks. This text is far from being exhaustive but is intended to cover a rather large part of the subject, starting from the motivations and the foundational material, passing through some examples and basic notions, and ending with some more recent developments and open questions.

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Section: Schematic Homotopy Typesmentioning

confidence: 99%

Higher and derived stacks: a global overview

Toën¹

2009

Proceedings of Symposia in Pure Mathematics

153

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“…Finally, if content in rules and processes use XML for specifications, there exists an open-source XML content translator [81] called Nikse. Fig.…”

Section: Third Refinement Levelmentioning

confidence: 99%

A reference architecture for managing dynamic inter-organizational business processes

Norta

Grefen

Narendra

2014

Data & Knowledge Engineering

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“…RAut J .X Remarks 7.37 (1) In the case when X is projective and R is a quotient of Gal $ f .X x k /, this is essentially the main formality result of [32,Section 4], which has since been extended to the general projective case in [33,Theorem 7.22], although Frobenius-equivariance is not made explicit there. The proofs also differ in that they work with minimal algebras, rather than minimal Lie algebras.…”

Section: Varieties Over Finite Fieldsmentioning

confidence: 99%

Galois actions on homotopy groups of algebraic varieties

Pridham

2011

Geom. Topol.

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We study the Galois actions on the`-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K , we show that the`-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever`is not equal to the residue characteristic p of K . For quasiprojective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When`D p , a slightly weaker result is proved by comparing the crystalline and p -adic schematic homotopy types. Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin-Mazur homotopy groups ét n .X x K /˝y Z Q`. IntroductionIn [2], Artin and Mazur introduced the étale homotopy type of an algebraic variety. This gives rise to étale homotopy groups K et n .X; x x/; these are pro-finite groups, abelian for n 2, andx/ is the usual étale fundamental group. In [49, Section 3.5.3], Toën discussed an approach for defining`-adic schematic homotopy types, giving`-adic schematic homotopy groups $ n .X; x x/; these are (pro-finite-dimensional) Q`-vector spaces when n 2. In [33], Olsson introduced a crystalline schematic homotopy type, and established a comparison theorem with the p -adic schematic homotopy type.Thus, given a variety X defined over a number field K , there are many notions of homotopy group: for each embedding K ,! C , both classical and schematic homotopy groups of the topological space X C ; the étale homotopy groups of X x K ; the`-adic schematic homotopy groups of X x K ; over localisations K p of K , the crystalline schematic homotopy groups of X K p . If X is smooth or proper and normal, then Corollary 6.7 shows that the Galois actions on the`-adic schematic homotopy groups are mixed, with Remark 6.9 indicating when the same is true for étale homotopy groups. Corollaries 6.11 and 6.16 then show how to determine`-adic schematic homotopy groups of smooth varieties over finite fields as Galois representations, by recovering them from cohomology groups of smooth Weil sheaves, thereby extending the author's paper [38] from fundamental groups to higher homotopy groups, and indeed to the whole homotopy type. Corollaries 7.4 and 7.36 give similar results for`-adic and p -adic homotopy groups of varieties over local fields.The structure of the paper is as follows.In Section 1, we recall standard definitions of pro-finite homotopy types and homotopy groups, and then establish some fundamental results. Proposition 1.29 shows how Kan's loop group can be used to construct the pro-finite completion y X of a space X , and Proposition 1.39 describes homotopy groups of y X .Section 2 reviews the pro-algebraic homotopy types of [37], with the formulation of multipointed pro-algebraic homotopy types from [34], together with some new material on hypercohomology.We adapt these results in Section 3 to define nonabelian cohomology of a variety with coefficients in a simplicial algebraic grou...

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F-isocrystals and homotopy types

Cited by 6 publications

References 29 publications

Higher and derived stacks: a global overview

Higher and derived stacks: a global overview

A reference architecture for managing dynamic inter-organizational business processes

Galois actions on homotopy groups of algebraic varieties

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