相关论文: Explicit computation of the first \'etale cohomolo…

This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the \'etale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three…

We present an explicit expression of the cohomology complex of a constructible sheaf of abelian groups on the small \'etale site of an irreducible curve over an algebraically closed field, when the torsion of the sheaf is invertible in the…

We present an algorithm to compute the torsion component $\mathrm{Pic}^\tau X$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^\tau X$ as a closed subscheme of a projective…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

We present an algorithm which, given a connected smooth projective curve $X$ over an algebraically closed field of characteristic $p>0$ and its Hasse--Witt matrix, as well as a positive integer $n$, computes all \'etale Galois covers of $X$…

Let $k$ be a non-archimedean complete valued field and let X be a smooth Berkovich analytic $k$-curve. Let $F$ be a finite locally constant \'{e}tale sheaf on $k$ whose torsion is prime to the residue characteristic. We denote by $|X|$ the…

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0 they see the…

The purpose of this article is to show how one might compute the \'etale cohomology groups $H^p_{\acute{e}t}(X,G_m)$ in degrees $p=0$, $1$ and $2$ of a toric variety $X$ with coefficients in the sheaf of units. The method is to reduce the…

Given an elliptic fibration $f \colon X \to S$ over the spectrum of a complete discrete valuation ring with algebraically closed residue field, we use a Hochschild--Serre spectral sequence to express the torsion in $R^1f_\ast \mathscr{O}_X$…

We provide a cohomological interpretation of the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a…

We compute the \'etale cohomology groups H^i(X,G_m) in several cases, where X is a smooth tame Deligne-Mumford stack of dimension 1 over an algebraically closed field. We have complete results for orbicurves (and, more generally, for…

Let $X$ be an algebraic scheme over an algebraically closed field and $\ell$ a prime number invertible on $X$. According to classical results (due essentially to A. Grothendieck, M. Artin and P. Deligne), the \'etale cohomology groups…

Let $X$ be a smooth projective curve over a finite field of characteristic $p$. We describe and implement a practical algorithm for computing the $p$-divisible group $Jac(X)[p^\infty]$ via computing its Dieudonn\'{e} module, or equivalently…

For a curve $C$ over a perfect field $k$ of characteristic $p > 0$ we study the tame cohomology of $X = \textit{Spa}(C,k)$ introduced in arXiv:1801.04776. We prove that the tame cohomology groups of $X$ with $p$-torsion coefficients satisfy…

Given a smooth and separated K(pi,1) variety X over a field k, we associate a "cycle class" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute…

Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of…

We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all…

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also…

Let F be a smooth surface in a smooth projective threefold T, and let X=2F be the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve C in X gives rise to two effective divisors on F, namely the curve part P of the…

We compute explicit bases for the de Rham cohomology of cyclic covers of the projective line defined over an algebraically closed field of characteristic $p\geq 0$. For both Kummer and Artin-Schreier extensions, we describe precise…