Related papers: A decisive Theorem (Un th\'eor\`eme d\'ecisif)

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an…

We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a…

We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where…

A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…

Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (\'etale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Th\'el\`ene and Voisin. If k is separably closed, finite or…

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…

We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with…

A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition.…

We define E-theory for separable C*-algebras over second countable topological spaces and establish its basic properties. This includes an approximation theorem that relates the E-theory over a general space to the E-theories over finite…

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…

In this paper we study extension problems for torsors in positive characteristic. Let $F$ be a field of characteristic $p>0$ and $U/F$ be a unipotent algebraic group. As our first main result, we prove that every $U$-torsor defined over the…

We prove that under suitable graded and local hypothesis, a formally unramified algebra over a field must be reduced. We detail examples, including one due to Gabber, to show that it is not possible to generalize these results further.

Let $T$ be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that $T$ has continuum-many countable models. The proof is purely first-order, but raises the question of Borel completeness of…

An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation…

We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our…

Brehm's extension theorem states that a non-expansive map on a finite subset of a Euclidean space can be extended to a piecewise-linear map on the entire space. In this note, it is verified that the proof of the theorem is constructive…

In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…

We prove that a finite torsion-free conformal Lie algebra with a splitting solvable radical has a finite faithful conformal representation.

Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange…

A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras.…