Weight decompositions on étale fundamental groups

Pridham, J. P.

doi:10.1353/ajm.0.0055

Cited by 8 publications

(15 citation statements)

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“…In [38], W $ 1 .X; x x/ was defined to be the universal object classifying continuous W .X k ; x/-equivariant homomorphisms 1 .X; x x/ ! G.Q`/ to algebraic groups.…”

Section: Definition 522mentioning

confidence: 99%

“…G.Q`/ to algebraic groups. In the terminology of [38,Definition 1.3], W $ 1 .X; x x/ is the maximal quotient of $ 1 .X; x x/ on which Frobenius acts algebraically.…”

Section: Definition 522mentioning

confidence: 99%

“…Since m is of strictly negative weights, we may adapt [38,Corollary 1.21] by observing that O.R Ë exp.m//=O.R/ is of strictly positive weights, and that the weight 0 part…”

Section: Varieties Over Finite Fieldsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…We also explore what it means for a pro-discrete group to act algebraically on a homotopy type. In Section 5, we investigate properties of homotopy types endowed with algebraic Galois actions.In Section 6, the techniques of [38] for studying Galois actions on algebraic groups then extend the finite characteristic results of the author in [35] to nonnilpotent and higher pro-Q`-algebraic homotopy groups. The results are similar to [34], substituting Frobenius actions for Hodge structures.…”

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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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“…In [38], W $ 1 .X; x x/ was defined to be the universal object classifying continuous W .X k ; x/-equivariant homomorphisms 1 .X; x x/ ! G.Q`/ to algebraic groups.…”

Section: Definition 522mentioning

confidence: 99%

“…G.Q`/ to algebraic groups. In the terminology of [38,Definition 1.3], W $ 1 .X; x x/ is the maximal quotient of $ 1 .X; x x/ on which Frobenius acts algebraically.…”

Section: Definition 522mentioning

confidence: 99%

“…Since m is of strictly negative weights, we may adapt [38,Corollary 1.21] by observing that O.R Ë exp.m//=O.R/ is of strictly positive weights, and that the weight 0 part…”

Section: Varieties Over Finite Fieldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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