Galois actions on homotopy groups of algebraic varieties

Abstract

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...

Nonabelian reciprocity laws and higher Brauer–Manin obstructions

“…We now need to consider fibre sequences, because G F,Σ does not explicitly act on our model forXé t . If the G F,Σ -representation H * (X, V ) is an extension of T -representations for all R-representations V , then [Pri6,Theorem 3.32] combines with [Fri1] to give a fibre sequencē…”

“…F,Σ → 1 of pro-algebraic homotopy groups; in particular we will have an exact sequence of completed fundamental groups whenever ̟ 2 (BG F,Σ ) T,Mal = 0, i.e. if G F,Σ is 2-good relative to T in the sense of [Pri6,Definition 3.35] and [Pri7,§1.2.3].…”

“…The filtration [U ] n corresponds to the good truncation filtration on RΓ(X, O(R)), but there are variants for other filtrations, replacing H 1+• (X, O(R)) with the E 1 page of the associated spectral sequence. For the case of the weight filtration on a quasiprojective variety, with representations tamely ramified around the divisor, see [Pri6,Corollary 6.16].…”

“…Then the Zariski closure R of the image of πé t 1 (X,x) is just SL 2 × {1}, and the quotient T := S/R is the Zariski closure of the representation G F,Σ → G m (Q ℓ ) × m GL(H 1 (Γ, V m )). The conditions of [Pri6,Theorem 3.32] are then satisfied by construction.…”

“…Applying relative pro-Σ completion over G F,Σ levelwise (cf. [Pri6,§1]) to the pro-simplicial group G(Xé t ,x) of Example 2.20 gives a pro-(finite simplicial group)Ĝ(Xé t ,x) as in the proof of Proposition 2.2; up to homotopy, this is independent of the choices made, by [Pri6,Proposition 1.32]. When Σ is the set of all primes (corresponding to O F,Σ = F ), note thatĜ(Xé t ,x) is just the profinite completion of G(Xé t ,x).…”