Etale Homotopy Research Articles

English

A new approach to etale homotopy theory is presented which applies to a much broader class of objects than previously existing approaches, namely it applies not only to all schemes (without any local Noetherian hypothesis), but also to arbitrary higher stacks on the etale site of such schemes, and in particular to all algebraic stacks. This approach also produces a more refined invariant, namely a pro-object in the infinity category of spaces, rather than in the homotopy category. We prove a profinite comparison theorem at this level of generality, which states that if $\mathcal{X}$ is an arbitrary higher stack on the etale site of affine schemes of finite type over $\mathbb{C},$ then the etale homotopy type of $\mathcal{X}$ agrees with the homotopy type of the underlying stack $\mathcal{X}_{top}$ on the topological site, after profinite completion. In particular, if $\mathcal{X}$ is an Artin stack locally of finite type over $\mathbb{C}$, our definition of the etale homotopy type of $\mathcal{X}$ agrees up to profinite completion with the homotopy type of the underlying topological stack $\mathcal{X}_{top}$ of $\mathcal{X}$ in the sense of Noohi. In order to prove our comparison theorem, we provide a modern reformulation of the theory of local systems and their cohomology using the language of $\infty$-categories which we believe to be of independent interest.

Betti realization of varieties defined by formal Laurent series

We give two constructions of functorial topological realizations for schemes of finite type over the field $\mathbb{C}(\!(t)\!)$ of formal Laurent series with complex coefficients, with values in the homotopy category of spaces over the circle. The problem of constructing such a realization was stated by D. Treumann, motivated by certain questions in mirror symmetry. The first construction uses spreading out and the usual Betti realization over $\mathbb{C}$. The second uses generalized semistable models and log Betti realization defined by Kato and Nakayama, and applies to smooth rigid analytic spaces as well. We provide comparison theorems between the two constructions and relate them to the \'etale homotopy type and de Rham cohomology. As an illustration of the second construction, we treat two examples, the Tate curve and the non-archimedean Hopf surface.

The tame site of a scheme

\'Etale cohomology with non-invertible coefficients has some unpleasant properties, e.g., it is not A^1-homotopy invariant and for constructible coefficients the expected finiteness properties do not hold. In this paper we introduce the `tame site' which is slightly coarser than the \'etale site. Tame cohomology coincides with \'etale cohomology for invertible coefficients but is better behaved in the general case. The fundamental group of the tame site is the (curve-)tame fundamental group of Wiesend and Kerz/Schmidt. The higher tame homotopy groups hopefully have a better behaviour than the higher \'etale homotopy groups, which vanish for affine schemes in positive characteristic by a result of Achinger.

Nonabelian reciprocity laws and higher Brauer–Manin obstructions

We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular curve, these take values in Galois cohomology of modular forms, and give obstructions to an adelic elliptic curve with global Tate module underlying a global elliptic curve.

Aspherical neighborhoods on arithmetic surfaces: the local case

On arithmetic surfaces over local rings of integers we examine whether any geometric point has a basis of \'etale neighborhoods whose $\mathfrak{c}$-completed \'etale homotopy types are of type $K(\pi,1)$ for a given full class~$\mathfrak{c}$ of finite groups.

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

For a log scheme locally of finite type over $\mathbb{C}$, a natural candidate for its profinite homotopy type is the profinite completion of its Kato-Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over $\mathbb{C}$, another natural candidate is the profinite \'etale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over $\mathbb{C}$, 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 profinite \'etale homotopy type of its infinite root stack.

Étale homotopy types of moduli stacks of polarised abelian schemes

We determine the Artin–Mazur etale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the etale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an etale homotopy point of view.

Anabelian geometry with étale homotopy types

Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field k which is finitely generated over Q has a fundamental system of (affine) anabelian Zariski-neighbourhoods. This was predicted by Grothendieck in his letter to Faltings.

Étale homotopy equivalence of rational points on algebraic varieties

It is possible to talk about the \'etale homotopy equivalence of rational points on algebraic varieties by using a relative version of the \'etale homotopy type. We show that over $p$-adic fields rational points are homotopy equivalent in this sense if and only if they are \'etale-Brauer equivalent. We also show that over the real field rational points on projective varieties are \'etale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Ch\^atelet surfaces.

Étale contractible varieties in positive characteristic

Unlike in characteristic 0, there are no non-trivial smooth varieties over an algebraically closed field k of characteristic p>0 that are contractible in the sense of etale homotopy theory.

Étale homotopy types and bisimplicial hypercovers

Suppose $(C,x)$ is a pointed locally connected small Grothendieck site, and let $(X,z)$ denote any connected locally fibrant simplicial sheaf $X$ equipped with a geometric point $z$. Following Artin-Mazur, an etale homotopy type of $X$ may then be defined via the geometrically pointed hypercovers of $X$ to yield a pro-object of the homotopy category, but this is not the only possible definition. In Etale homotopy of simplicial schemes, Friedlander defined another etale homotopy type of a simplicial scheme $X$ by taking diagonals of geometrically pointed bisimplicial hypercovers. In this paper, these two types are shown to be pro-isomorphic by means of a direct comparison of the associated cocycle categories. Friedlander's construction of etale homotopy types as actual pro-simplicial sets relies on a rigidity property of the etale topology that may not always be available for arbitrary sites; the cocycle methods employed here do not have this limitation. By consequence, the associated homotopy types constructed from hypercovers and bisimplicial hypercovers are shown to be pro-isomorphic on any locally connected small Grothendieck site, and the comparison at the level of cocycles shows, in particular, that both abelian and non-abelian sheaf cohomology may be computed via bisimplicial hypercovers on arbitrary small Grothendieck sites.

On the homotopy type of the Deligne–Mumford compactification

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and the etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.

Étale Homotopy Types of Moduli Stacks of Algebraic Curves with Symmetries

Using the machinery of etale homotopy theory a' la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\Q}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite group G of automorphisms, which comes with a fixed embedding in the mapping class group, such that in the associated complex analytic situation the action of G is precisely the differentiable action induced by this specified embedding of G in the mapping class group.

Calculating limits and colimits in pro-categories

We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are essential for a theorem about \'etale homotopy types. Also, we show that cofiltered limits in pro-categories commute with finite colimits.

Becker-Gottlieb Transfer in Etale Homotopy

Introduction. In this paper we establish a version of the BeckerGottlieb transfer in the context of etale homotopy. The transfer map we obtain generalises the Becker-Gottlieb transfer to smooth quasi-projective varieties over any algebraically closed field of arbitrary characteristic. We next recall the Becker-Gottlieb transfer in topology for fiber bundles with compact fiber. (See [B-G-3] p. 107.) Let

Spanier-Whitehead duality in étale homotopy

We construct a ( mod - l ) (\bmod {\text {-}}l) Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a ( mod - l ) (\bmod {\text {-}}l) Spanier-Whitehead dual.

The Etale Homotopy Type of Varieties over R

The Etale Homotopy Type of Varieties over R

Etale homotopy theory of algebraic groups

Etale homotopy theory of algebraic groups

The etale homotopy theory of a geometric fibration

A geometric fibration, f:X→Y, is a smooth map of schemes which locally on Y admits a smooth, relative compactification. The etale homotopy type of the geometric fibre when completed away from the residue characteristics of Y,\((X_y )_{\hat et} \), is shown to be weakly homotopy equivalent to the completion of the Hurewicz fibre of the etale homotopy type of f,F(fetr)^. This implies a homotopy sequence for f. A key topological fact is verified in the appendix: for any pointed Hurewicz fibre triple F→E→B, the action of πl(F) on the homotopy type of various covering spaces of F extends to an action of πl(E).

Fibrations in etale homotopy theory

Fibrations in etale homotopy theory