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Ofer Gabber

Publications and source records attributed to Ofer Gabber.

18 recordsLinked to original sources
arXiv2026

On endomorphisms of affine spaces and the Jacobian problem

Let $p$ be a prime. We provide examples which show that \'etale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\in\mathbb N\times\mathbb N^{\ast}$. We provide examples of such \'etale endomorphisms whose images have complements of cardinality $l$ and whose geometric degrees are $pm$. Several conjectures are disproved, and in particular we provide an analog over $k$ of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each $k$ are included in dimension $2$ (resp.\ in all dimensions at least $3$); for instance, if $p=2$, then we show for each $m$ there exist surjective \'etale endomorphisms of the affine spaces over $k$ of dimension at least $3$ of geometric degree $m$. If $e:X\rightarrow X$ is an endomorphism of a variety over an algebraically closed field $K$, then we show that there exists $n\in\mathbb N$ such that $\Imm(e^n)=\Imm(e^{n+1})$ provided either (i) $e$ is quasi-finite or (ii) $\dim(X)\le 2$ and $X\setminus\Imm(e)$ is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over $K$ whose images have complements of cardinality $l$ and for all $n\in\mathbb N$ we have $\Imm(e^n)\neq\Imm(e^{n+1})$. We prove that all affine moduli schemes of \'etale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over $K$ are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over $\mathbb C$ and Adjamagbo's analog of it over $k$ hold for \'etale endomorphisms of affine spaces that are composites $g\circ f$, where $f$ is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and $g$ is a projection that omits one coordinate.

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arXiv2026

Infinite transitivity of tame groups of automorphisms of affine spaces

For positive integers $n$ and $m$, we study the actions of the groups of tame automorphisms of the $n$-dimensional affine spaces over finite fields on ordered subsets of $m$ points. Our primary interest lies in constructing tame automorphisms that take one ordered sequence to another and in proving upper and lower bounds on the maximal complexity of such automorphisms. Our preferred measure of complexity of an automorphism is the maximum of the degrees of the polynomials that define it and its inverse. Using methods and results from various branches of mathematics, including the theory of symmetric groups, affine geometry over finite fields, polynomial interpolation, combinatorics of projective spaces over fields, and polynomial automorphisms, we obtain a wide variety of qualitative and quantitative results.

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arXiv2025

Bounding crystalline torsion from \'etale torsion

In this note, we prove that given a smooth proper family over a $p$-adic ring of integers, one gets a control of its crystalline torsion in terms of its \'{e}tale torsion, the cohomological degree, and the ramification. Our technical core result is a boundedness result concerning annihilator ideals of $u^{\infty}$-torsion in Breuil--Kisin prismatic cohomology, which might be of independent interest.

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arXiv2024

Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria

We give sufficient conditions for cohomological flatness (in dimension 0) over discrete valuation rings, generalizing classical results of Raynaud in two different ways. The first is a higher dimensional generalization of Raynaud's numerical criteria, in both the variant for the multiplicity of the special fibre and that for the index of the generic fibre. The second is a logarithmic criterion: we show that, over a log regular base, a proper flat fs log smooth morphism is cohomologically flat in dimension 0. We apply this latter result to curves and torsors under abelian varieties with good reduction, providing necessary and sufficient conditions for the log smoothness of their regular models over arbitrary discrete valuation rings.

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arXiv2024

Algebraization Techniques and Rigid-Analytic Artin-Grothendieck Vanishing

First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean geometry. The result says that we can always algebraize a geometrically reduced affinoid rigid-analytic space in "one direction" in an appropriate sense. As an application of this result, we show the remaining cases of the Artin-Grothendieck Vanishing for affinoid algebras, which were previously conjectured in [BM21, {\S}7]. This allows us to deduce a stronger version of the rigid-analytic Artin-Grothendieck Vanishing Conjecture (see [Han20, Conj. 1.2]) over a field of characteristic 0. Using a completely different set of ideas, we also obtain a weaker version of this conjecture over a field of characteristic p>0.

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arXiv2020

Purity for Barsotti-Tate groups in some mixed characteristic situations

Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of $k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $\text{Frac}(R)$ which extends to every local ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $e\le p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.

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arXiv2019

Sur le produit de variétés localement factorielles ou Q-factorielles

We show that the factorial and Q-factorial loci of algebraic varieties defined over an algebraically closed field are open, that products of locally factorial varieties are still locally factorial, and that this property remains true for Q-factorial varieties if the ground field is not the algebraic closure of a finite field.

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arXiv2018

Foundations for almost ring theory -- Release 7.5

This is release 7.5 of our project, aiming to provide a complete treatment of the foundations of almost ring theory, following and extending Faltings's method of "almost etale extensions". The central result is the "almost purity theorem", for whose proof we adapt Scholze's method, based on his perfectoid spaces. This release provides the foundations for our generalization of Scholze's perfectoid spaces, and reduces the proof of the almost purity theorem to a general assertion concerning the étale topology of adic spaces, whose proof uses previous work by the first author. As usual, this new release is a mix of corrections and various improvements, with a final chapter dedicated to applications; notably, we include a generalization of Y.André's "perfectoid Abhyankar's lemma" which we use to give a proof of a generalization of the "direct summand conjecture", extending André's recent work.

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arXiv2017

Finiteness of étale fundamental groups by reduction modulo $p$

We introduce a spreading out technique to deduce finiteness results for étale fundamental groups of complex varieties by characteristic $p$ methods, and apply this to recover a finiteness result proven recently for local fundamental groups in characteristic $0$ using birational geometry.

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arXiv2014

Points in algebraic geometry

We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, étale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show direct image along a closed immersion is exact for all these topologies except qfh. The methods are transportable to other categories of sheaves as well.

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arXiv2014

Fibrés principaux sur les corps valués henséliens

Let (K,v) be a valued field, Y a K-variety, G an algebraic group over K (not necessarily smooth), and f: X->Y a G-torsor over Y. We consider the induced map X(K)-->Y(K), which is continuous for the topologies deduced from the valuation. Let I denote the image of this map. We prove that if (K,v) is henselian and its completion is a separable extension, then: - I is locally closed in Y(K); - the induced surjection X(K)-->I is a principal bundle with group G(K) (also topologized by the valuation).

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arXiv2014

Hypersurfaces in projective schemes and a moving lemma

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F. Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if R is the ring of integers of a number field, or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1-cycles on a regular scheme X quasi-projective and flat over S. We also show the existence of a finite surjective S-morphism to the projective space P_S^d for any scheme X projective over S when X/S has all its fibers of a fixed dimension d.

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arXiv2012

The index of an algebraic variety

Let K be the field of fractions of a Henselian discrete valuation ring O_K. Let X_K/K be a smooth proper geometrically connected scheme admitting a regular model X/O_K. We show that the index δ(X_K/K) of X_K/K can be explicitly computed using data pertaining only to the special fiber X_k/k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an FA-scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant γ(A) of a singular local ring (A, \m): the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all \m-primary ideals Q in \m. We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of Spec(A), and we give a new way of computing the index of a smooth subvariety X_K/K of P^n_K over any field K, using the invariant γof the local ring at the vertex of a cone over X.

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arXiv2012

Dimensions of group schemes of automorphisms of truncated Barsotti--Tate groups

Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $γ_D(i):=\dim(\pmb{Aut}(D[p^i]))$. We show that $n_D\le cd$, $(γ_D(i+1)-γ_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb N$, for $cd>0$ we have $γ_D(1)<γ_D(2)<...<γ_D(n_D)$, and for $m\in\{1,...,n_D-1\}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.

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arXiv2007

Sur la p-dimension des corps

Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already proved by Kazuya KATO, we prove the following formula: cd_p(K) = dim(A) + p-rank(k), if k is separably closed and K of characteristic zero. A similar statement is valid without those assumptions on k and K.

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arXiv2002

Almost ring theory - sixth release

We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to p-adic analytic geometry. You should really have a look at the introductions (each chapter has one).

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arXiv2000

Almost ring theory

The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct the almost version of the cotangent complex and we use it to generalise some results of Faltings on the lifting of almost etale morphisms and almost etale algebras over nilpotent extensions. We also study the "almost trace" of an almost flat and almost finitely presented morphism, in particular we show that the almost trace is (almost) perfect if and only if the morphism is almost etale. Finally we study some cases of non-flat descent for almost rings, and establish the invariance of almost etale morphisms under Frobenius.

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