Searcharxiv⌕ Search

arXiv subjects

Cristian D. Gonzalez-Aviles

Publications and source records attributed to Cristian D. Gonzalez-Aviles.

At least 19 recordsLinked to original sources
arXiv2024

Local duality theorems for commutative algebraic groups

If k is an arbitrary field, we construct a category of k-1-motives in which every commutative algebraic k-group G has a dual object $G^{\vee}$. When k is a local field of arbitrary characteristic, we establish Pontryagin duality theorems that relate the fppf cohomology groups of G to the hypercohomology groups of the k-1-motive $G^{\vee}$. We also obtain a duality theorem for the second cohomology group of an arbitrary k-1-motive. These results have applications (to be discussed elsewhere) to certain extensions of Lichtenbaum-van Hamel duality to a class of non-smooth proper k-varieties.

math.NT↗
arXiv2024

On the new theory of 1-motives

This is a letter (not intended for publication in a regular journal) written in response to two referees of my preprint "Local duality theorems for commutative algebraic groups". In it, I discuss possible applications of the new theory of 1-motives introduced in that preprint. One of these is a new approach to the BSD conjecture for abelian varieties over global fields (developed here only partially) which builds on Bloch's well-known volume-theoretic interpretation of the conjecture in the number field case.

math.NT↗
arXiv2022

The algebraic Brauer group of a pinched variety

Added lemma provided by Michel Brion. Other (minor) changes. Submitted version. Let k be any field, let X' be a projective and geometrically integral k-scheme and let Y' be a finite closed subscheme of X'. If f: Y'-> Y is a schematically dominant morphism between finite k-schemes and X is obtained by pinching X' along Y' via f, we describe the kernel (and, in certain cases, the cokernel) of the induced pullback map Br_{1}X -> Br_{1}X' between the corresponding algebraic (cohomological) Brauer groups of X and X'_solely_ in terms of the Brauer groups of the residue fields of Y and Y' and the Amitsur subgroups of X and X' in Br k. As an application, we compute the Brauer group of a projective and geometrically integral k-rational curve (with arbitrary singularities) over any field k. We also extend a well-known theorem of Roquette and Lichtenbaum to a class of singular curves over any local field.

math.AG↗
arXiv2022

The Greenberg functor revisited

The proof, but not the statement, of Proposition 18.2 contained an error which is repaired in this version. See Remark 18.3 in this version. No other changes. We extend Greenberg's original construction to arbitrary (in particular, non-reduced) schemes over (certain types of) local artinian rings. We then establish a number of basic properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor.

math.NT↗
arXiv2020

Brauer groups and Neron class groups

Let K be a global field, let S be a finite set of primes of K containing the archimedean primes and let A be an abelian variety over K. We generalize the duality theorem established in our paper "On Neron class groups of abelian varieties" by removing the hypothesis in [op.cit.] that the Tate-Shafarevich group of A is finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve X over K to a certain finite subquotient of the Brauer group of X. The sequence alluded to above may be regarded as a global analog of an exact sequence of S.Biswas.

math.NT↗
arXiv2019

Cokernels of restriction maps and subgroups of norm one, with applications to quadratic Galois coverings

Let f: S' --> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. For every r>0, let Res_{G}^{(r)}: H^{r}(S_{et},G)---> H^{r}(S'_{et},G) and Cores_{G}^{(r)}: H^{r}(S'_{et},G)---> H^{r}(S_{et},G) be, respectively, the restriction and corestriction maps in etale cohomology induced by f. For certain pairs (f, G), we construct maps α_{r}: Ker Cores_{G}^{(r)}---> Coker Res_{G}^{(r)} and β_{r}: Coker Res_{G}^{(r)}---> Ker Cores_{G}^{(r)} such that α_{r}oβ_{r}=β_{r}oα_{r}=n. In the simplest nontrivial case, namely when f is a quadratic Galois covering, we identify the kernel and cokernel of β_{r} with the kernel and cokernel of another map Coker Cores_{G}^{(r-1)}---> KerRes_{G}^{(r+1)}. We then discuss several applications, for example to the problem of comparing the (cohomological) Brauer group of a scheme S to that of a quadratic Galois cover S' of S.

math.AG↗
arXiv2019

The norm map and the capitulation kernel

Let f: S'--> S be a finite and faithfully flat morphism of locally noetherian schemes of constant rank n > 1 and let G be a smooth, commutative and quasi-projective S-group scheme with connected fibers. Under certain restrictions on f and G, we relate the kernel of the restriction map in degree r>0 étale cohomology Res_{G}^{(r)}: H^{r}(S_{\et},G)--> H^{r}(S'_{\et},G) to a certain quotient of the kernel of the mod n corestriction map in degree r-1, namely Cores_{G}^{(r-1)}/n: H^{r-1}(S'_{\et},G)/n\to H^{r-1}\lbe(S_{\et},G)/n. When r=1 and f is a Galois covering with Galois group D, our main theorem relates Ker Res_{G}^{(1)}=H^{1}(D,G(S')) to the subgroup of G(S') of those sections whose S'/S-norm lies in G(S)^{n}. We also include applications to the capitulation problem for Neron-Raynaud class groups of invertible tori and Tate-Shafarevich groups of abelian varieties.

math.NT↗
arXiv2018

Cech cohomology and the Capitulation kernel

We discuss the capitulation kernel associated to a degree n covering using Cech cohomology and the Kummer sequence. The main result is a five-term exact sequence that relates the capitulation kernel to the Cech cohomology of the n-th roots of unity and a certain subquotient of the group of units modulo n-th powers.

math.NT↗
arXiv2017

The units-Picard complex of a reductive group scheme

Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To do so, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.

math.AG↗
arXiv2017

On the group of units and the Picard group of a product

Let S be a reduced scheme and let f: X--> S and g: Y-->S be faithfully flat morphisms locally of finite presentation with geometrically connected and geometrically reduced maximal fibers. We discuss the canonical maps G_{m,S}(X)+G_{m,S}(Y)-->G_{m,S}(X x_{S} Y) and Pic X+Pic Y-->Pic(X x_{S} Y) induced by f and g (both sums are direct). Under certain additional conditions on X,Y,S, f and g, we describe the kernel and cokernel of the preceding maps, thereby extending known results when S is the spectrum of a field.

math.NT↗
arXiv2017

The relative units-Picard complex and the Brauer group of a product

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors. Under certain hypotheses, we obtain a five-term exact sequence involving the preceding groups which enables us to solve the indicated problem in the case of, e.g., (certain types of) ruled varieties over a field of characteristic zero.

math.AG↗
arXiv2017

Some remarks on descent data with applications to Galois descent

In this expository paper we present a proof of the equivalence of the standard definition of descent data on schemes with another one mentioned in the literature that involves certain cartesian diagrams. Using this equivalence, we discuss the Galois descent of both schemes and morphisms of schemes.

math.AG↗
arXiv2016

On the perfection of schemes

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors.

math.AG↗
arXiv2016

On Tate-Shafarevich groups of 1-motives

We establish the finiteness of the kernel and cokernel of the restriction map III^{i}(F,M) ---> III^{i}(K,M)^{G} for i=1 and 2, where M is a (Deligne) 1-motive over a global field F and K/F is a finite Galois extension of global fields with Galois group G.

math.NT↗
arXiv2016

Groups of components and Weil restriction

We determine the behavior under Weil restriction of the group of connected components of the special fiber of an arbitrary smooth group scheme (whose Weil restriction exists) over an arbitrary (commutative and unital) local ring. Applications to Néron models are given.

math.NT↗
arXiv2016

On Tate-Shafarevich groups of 1-motives over Galois extensions

Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any finiteness hypotheses. We show that these groups are finite and obtain, in particular, formulas for the corresponding quotient of group orders.

math.NT↗
arXiv2013

On the cohomology of tori over local fields with perfect residue field

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension <= 1, our results have a particularly simple form and yield, more generally, isomorphisms between the abelian K-cohomology of a reductive group G over K and the k-cohomology of a certain quotient of the algebraic fundamental group of G.

math.NT↗