Frobenius split subvarieties pull back in almost all characteristics

Réalisation de Betti des motifs de Voevodsky

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in \mathbf{Z}$, we study the set $\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}-\Sigma_{X}, N_{X}(p)\neq a_{i} [\bmod p]\rbrace$ where $\Sigma_{X}$ is the finite set of primes of bad reduction for $X$. In case $\dim X\leq 3$, we show the set is either empty or has positive lower-density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of $\lbrace p\in\mathcal{P}, p\nmid N_{X}(p)\rbrace$ on average in particular families of hyperelliptic curves.

Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of ${\cal L} X$ in the neighborhood of non-degenerate arcs, we show that a canonical ``basic function'' can be defined on the non-degenerate locus of ${\cal L} X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an ``$L$-monoid''. Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second author.

The purpose of this paper is twofold: to develop a theory of etale cohomological dimension in the context of schemes of finite type over a separably closed field that would be analogous to the well-known theory of quasicoherent cohomological dimension, and to apply our theory to prove new results about the topology of algebraic varieties of small codimension in n-space. The cohomological dimension of a scheme X relative to a closed subscheme Y, denoted by cd(X, Y), is the largest integer r such that the local cohomology group H' (X, F) $& 0 for some abelian torsion sheaf F on Xet, where F consists only of torsion prime to all the residual characteristics of X. The gist of our theory is a technique for proving various upper bounds on cd(X, Y), especially in the case where Y has small codimension in X. Local bounds are the most important as well as the easiest to state. Accordingly, for the purposes of this introduction, let X = Spec A, where A is a strictly Henselian local ring of a finite-type scheme over a separably closed field. Just what kind of bounds on cd(X, Y) one should expect is indicated by the theory of quasicoherent cohomological dimension, which has been developed by several authors; see, for example, [Fa], [Grl], [Hal], [Ha2], [HaSp], [HuLy], [01] and [PesSz]. The quasicoherent cohomological dimension of X relative to Y, denoted by qccd(X, Y), is the largest integer r such that there exists a quasicoherent sheaf F on Xzar with H'(X, F) 54 0. In the above-stated local case, the main results of the theory of quasicoherent cohomological dimension are the following (where n = dim X): (i) qccd(X, Y) < n (see [Gri], 1.12). (ii) qccd(X, Y) < t if Y is set-theoretically defined by t equations (see [Gri], 2.3).

Numerical modelling of two-dimensional morphodynamics with applications to river bars and bifurcations

Serre finiteness and Serre vanishing forBnon-commutative -bundles

Blowing up monomial ideals

Albanese varieties provide a standard tool in algebraic geometry for converting questions about general varieties into questions about Abelian varieties. A result of Serre provides the existence of an Albanese variety for any geometrically connected and geometrically reduced scheme of finite type over a field, and a result of Grothendieck–Conrad establishes that Albanese varieties are stable under base change of field provided the scheme is, in addition, proper. A result of Raynaud shows that base change can fail for Albanese varieties without this properness hypothesis. In this paper we show that Albanese varieties of geometrically connected and geometrically reduced schemes of finite type over a field are stable under separable field extensions. We also show that the failure of base change in general is explained by the L/K-image for purely inseparable extensions L/K.

We extend Weil’s construction of topologies on sets of rational points of schemes over local fields to the case of n-local fields and their rings of integers by using the sequential properties of higher topologies. In order to do this, we endow each scheme of finite type over a sequential ring with a topology in a functorial way, and study the properties of this construction. Finally, we show openness of reduction maps for schemes of finite type over rings of integers of 2-local fields.

Intersection theory in algebraic cobordism

We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups and obtains a bijection with the set of classifying spaces of compact connected Lie groups topologically localised away from the characteristic. We also study the representations of perfectly reductive groups. We establish a highest weight classification of simple modules, the decomposition into blocks, and relate extension groups to those of the underlying abstract group.

Homology of linear groups via cycles in BG× X

Let$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

We show that Bloch's complex of relative zero-cycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields, finite fields, local fields of mixed characteristic, and rings of integers in number rings, generalizing results which so far have only been known for smooth schemes or in low dimensions, and unifying the p-adic and l-adic theory. As an application, we generalize Rojtman's theorem to normal, projective schemes.

Let X be a separated scheme of finite type over an algebraically closed field k and let m be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod m Suslin homology and the first mod m tame étale cohomology of X . We show that the induced homomorphism from the mod m Suslin homology to the abelianized tame fundamental group of X mod m is surjective. It is an isomorphism of finite abelian groups if (m, char(k)) = 1 , and for general m if resolution of singularities holds over k .