Étale 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 étale site. Tame cohomology coincides with étale 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 étale homotopy groups, which vanish for affine schemes in positive characteristic by a result of Achinger.
For every adic space $Z$ we construct a site $Z_t$, the tame site of $Z$. For a scheme $X$ over a base scheme $S$ we obtain a tame site by associating with $X/S$ an adic space $\textit{Spa}(X,S)$ and considering the tame site $\textit{Spa}(X,S)_t$. We examine the connection of the cohomology of the tame site with étale cohomology and compare its fundamental group with the conventional tame fundamental group. Finally, assuming resolution of singularities, for a regular scheme $X$ over a base scheme $S$ of characteristic $p > 0$ we prove a cohomological purity theorem for the constant sheaf $\ma…
We extend the notion of a tame covering of a pair (X,D) where X is a regular scheme and D is a normal crossing divisor (cf. SGA1), to pairs (X,Y) where X is an arbitrary scheme and Y is a closed subset in X. We show that the abelianized tame fundamental group of a regular scheme which is flat and of finite type over Spec(Z) is finite and does not depend on the choice of a particular compactification.
We compare two notions of tameness: one introduced in [CEPT96] for actions of affine group schemes and one introduced in [AOV08] for stacks. From this comparison, we deduce results on the structure of inertia groups in these tame situations.
We prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic p (as defined by Abramovich, Olsson, and Vistoli) which lift mod p^2 degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are etale locally the quotient of a smooth scheme by a finite linearly reductive group scheme.
We study the ramification theory for actions involving group schemes, focusing on the tame ramification. We consider the notion of tame quotient stack introduced in [AOV] and the one of tame action introduced in [CEPT]. We establish a local Slice theorem for unramified actions and after proving some interesting lifting properties for linearly reductive group schemes, we establish a Slice theorem for actions by commutative group schemes inducing tame quotient stacks. Roughly speaking, we show that these actions are induced from an action of an extension of the inertia group on a finitely presen…
On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups
We strengthen Gabber's $l'$-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme $X$ of finite type over a quasi-excellent threefold can be desingularized by a $\mathrm{char}(X)$-alteration, i.e. an alteration whose order is only divisible by primes non-invertible on $X$. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of $X$ can be split into a composition of a tame Galois alteration and a $\mathrm{char}(X)$-alteration. The proof of the distillation theorem is based on the …
A construction of tame sheaves and tame de Rham--Witt cohomology
Institut für Mathematik, Universität Heidelberg（海德堡大学数学研究所）；School of Mathematics and Natural Sciences, University of Wuppertal（伍珀塔尔大学数学与自然科学学院）；Graduate School of Mathematical Sciences, the University of Tokyo（东京大学数理科学研究科）
本文提出了一种构造 tame sheaves 的一般方法，应用于大 de Rham--Witt sheaves 和 reciprocity sheaves，并比较了 tame syntomic cohomology 与 tame de Rham--Witt 复合物上的 Nygaard 过滤。
We develop a theory of tame vanishing cycles for schemes over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$ in the context of étale sheaves. We show some desired properties of this formalism, among which: a compatibility with tame vanishing cycles over a (strctly) henselian trait, a compatibility with the theory of tame vanishing cycles over $\mathbb{A}^1_{S}$, a compatibility with tensor product and with duality. In the last section, we prove that monodromy-invariant vanishing cycles, introduced by the second named author, are the homotopy fixed points with respect to a canonical continuous action of…
A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an étale tame semicommutative finite group scheme $G$, we give a lower bound on the number of connected $G$-torsors of bounded height (such as discriminant).
As is well known, the Lefschetz theorems for the étale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. Let $X$ be a regular projective variety over a field $k$, and let $D\hookrightarrow X$ be a strict normal crossings divisor. Then, if $Y$ is an ample regular hyperplane intersecting $D$ transversally, the restriction functor from tame étale coverings of $X\setminus D$ to those of $Y\setminus D\cap Y$ is an equivalence if dimension $X \ge 3$, and fully faithful if dimension $X=2$. The method is dictated by Grothendieck-Murre ("Th…
The nonabelian Hodge correspondence for vector bundles over noncompact curves is adequately described by implementing a weighted filtration on the objects involved. In order to establish a full correspondence between a Dolbeault and a de Rham space for a general complex reductive group $G$, we introduce torsors given by parahoric group schemes in the sense of Bruhat--Tits. Combined with existing results on the Riemann--Hilbert correspondence for logarithmic parahoric connections, this gives a full nonabelian Hodge correspondence from Higgs bundles to fundamental group representations over a no…
We verify basic properties of the étale site of a `scheme with marking'. The coverings are étale coverings that split over all marked points. Familiar cases are the small étale site (no marking) and the Nisnevich site (all points are marked). This version 2 closes a small gap in the literature by showing that any Nisnevich covering of a quasi-compact scheme has a finite subcovering.
We generalize the classical Chevalley-Shephard-Todd theorem to the case of finite linearly reductive group schemes. As an application, we prove that every scheme X which is etale locally the quotient of a smooth scheme by a finite linearly reductive group scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli) whose stacky structure is supported on the singular locus of X.
Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f \: Y \to X$ in that Galois category satisfies that $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar…
We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. is invertible on X). Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.
We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and étale group schemes and we prove that, if $G$ is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of $G$-covers is reducible.
The aim of this work is to provide a construction of generalized local symbols on algebraic curves as morphisms of group schemes. From a closed point of a complete, irreducible and non-singular curve $C$ over a perfect field $k$ as the only data, using theta groups over Picard schemes of curves, we offer a geometric construction that allows us to define generalizations of the tame symbol and the Hilbert norm residue symbol.
Let $\calO_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with a perfect residue field. In this paper, for a semi-stable $p$-adic formal scheme $\frakX$ over $\calO_K$ with rigid generic fibre $X$ and canonical log structure $\calM_{\frakX} = \calO_{\frakX}\cap\calO_X^{\times}$, we study Hodge--Tate crystals over the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$. As an application, we give an equivalence between the category of rational Hodge--Tate crystals on the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$ and the c…
We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.
Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of $\mathcal{K}_{g,n}(\mathcal{X},d)$ using Hilbert schemes and admissible $G$-covers.
In this paper we show that it is possible to define a topology on the category of formal schemes over a ring of $p$-adic integers such that the left adjoint of the Greenberg Transform is a site cocontinuous functor when we equip the category of schemes over the residue field with the étale topology. We show furthermore that this topology allows us to give an isomorphism between the corresponding fundamental groups, and use this isomorphism to show that it is possible to geometrize the quasicharacters of a $p$-adic torus by a local system on a formal scheme over the ring of $p$-adic integers.
Let S be a Noetherian scheme and f:X -> S a proper morphism. By SGA 4 XIV, for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules R^if_*F obtained by direct image (for the etale topology) are also constructible: there is a stratification of S on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which S is generically of characteristic zero or the sheaves F are constant (with invertible torsion on S), here we study the dependency of the stratification on F. We show that a na…
A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence.
