Etale Site Research Articles

Cotangent bundle and microsupports in the mixed characteristic case

For a regular scheme and a prime number $p$, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by $p=0$, under a certain finiteness condition. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in some cases.

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.

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.

Crystalline comparison isomorphisms in p-adic Hodge theory :the absolutely unramified case

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for \'etale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-\'etale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-\'etale site. Moreover, we need to prove the Poincar\'e lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

Ramification theory and formal orbifolds in arbitrary dimension

Formal orbifolds are defined in higher dimension to study wild ramification. Their etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to approximate the etale fundamental groups of normal varieties. Etale site on formal orbifolds are also defined. This framework allows one to study wild ramification in an organized way. Brylinski–Kato filtration, Lefschetz theorem for fundamental groups and l-adic sheaves in these contexts are also studied.

The Monsky–Washnitzer and the overconvergent realizations

We construct the dagger realization functor for analytic motives over non-archimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classic \'etale site provides a new direct definition of the overconvergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from the one of Betti cohomology for smooth projective complex varieties.

Courbes nodales et ramification sauvage virtuelle

Let S be the spectrum of a discrete valuation ring of residue characteristic p>0. Let Z be a scheme separated and of finite type over S, and denote by Kc(Z,Open image in new window) (l≠p) the Grothendieck group of Open image in new window-constructible sheaves on the etale site of Z. In this article, we introduce the subgroup Open image in new window of virtual sheaves with virtual wild ramification zero, and we prove that this subgroup is preserved by the formalism of six operations “a la Grothendieck”.

Weil-�tale cohomology over finite fields

We calculate the derived functors Rγ* for the base change γ from the Weil-etale site to the etale site for a variety over a finite field. For smooth and proper varieties, we apply this to express Tate’s conjecture and Lichtenbaum’s conjecture on special values of ζ-functions in terms of Weil-etale cohomology of the motivic complex ℤ(n).

Unit L-Functions and a Conjecture of Katz

Let f: X -> Y be a separated morphism of schemes of finite type over a finite field of characteristic p, let Lambda be an artinian local Z_p-algebra with finite residue field, let m be the maximal ideal of Lambda, and let L^\bullet be a bounded constructible complex of sheaves of finite free Lambda-modules on the \'etale site of Y. We show that the ratio of L-functions L(X,L^\bullet)/L(Y,f_! L^\bullet), which is a priori an element of 1+T Lambda[[T]], in fact lies in 1+ m T Lambda [T]. This implies a conjecture of Katz predicting the location of the zeroes and poles of the L-function of a p-adic \'etale lisse sheaf on the closed unit disk in terms of \'etale cohomology with compact support.

On $p$-Adic Zeta-Functions Associated to the Positive Topology of Algebraic Number Fields

In [S] we introduced a new Grothendieck topology (called positive topology) over the rings of integers of algebraic number fields. If K is a finite extension of Q then Spec(0^) furnished with the positive topology shows a behavior very similar to the etale site over a complete curve over a finite field.

Vanishing cycles for formal schemes. II

Let k be a non-Archimedean field, and let X be a formal scheme locally finitely presented over the ring of integers k◦ (see §1). In this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre Xη of X (which is a k-analytic space) to the category of etale sheaves on the closed fibre Xs of X (which is a scheme over the residue field of k). We prove that if X is the formal completion X of a scheme X finitely presented over k◦ along the closed fibre, then the vanishing cycles sheaves of X are canonically isomorphic to those of X (as defined in [SGA7], Exp. XIII). In particular, the vanishing cycles sheaves of X depend only on X , and any morphism φ : Ŷ → X induces a homomorphism from the pullback of the vanishing cycles sheaves of X under φs : Ys → Xs to those of Y. Furthermore, we prove that, for each X , one can find a nontrivial ideal of k◦ such that if two morphisms φ, ψ : Ŷ → X coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by φ and ψ coincide. These facts were conjectured by P. Deligne. In §1 we associate with a formal scheme X locally finitely presented over k◦ a k-analytic space Xη (in the sense of [Ber1] and [Ber2]). In §2 we find that the morphism φη : Yη → Xη, which is induced by an etale morphism of formal schemes φ : Y → X, possesses a certain property. Morphisms of k-analytic spaces with this property are called quasi-etale, and they give rise to a quasi-etale site Xqet of a k-analytic space X. There is a canonical morphism of sites Xqet → Xet, where Xet is the etale site introduced in [Ber2]. We show that the inverse image functor identifies the category of etale sheaves X et with a full subcategory of

ÉTALE TOPOLOGIES OF SCHEMES OVER FIELDS OF FINITE TYPE OVERQ

The author proves a conjecture of Grothendieck concerning the possibility of recovering a normal scheme over a field of finite type over Q from its etale site.