arXiv subjects

Jinhe Ye

Publications and source records attributed to Jinhe Ye.

24 records · Page 2Linked to original sources

arXiv2021

The étale-open topology and the stable fields conjecture

For an arbitrary field $K$ and $K$-variety $V$, we introduce the étale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably closed, real closed, or $p$-adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of $K$. For example, the étale-open topology on $\mathbb{A}^1(K)$ is not discrete if and only if $K$ is large. As an application, we show that a large stable field is separably closed.

math.LO↗

arXiv2021

Classifying spaces and the Lascar group

We show that the Lascar group $\operatorname{Gal}_L(T)$ of a first order theory $T$ is naturally isomorphic to the fundamental group $π_1(|\operatorname{Mod}(T)|)$ of the classifying space of the category of models of $T$ and elementary embeddings.

math.LO↗

arXiv2021

Peterzil-Steinhorn subgroups and $μ$-stabilizers in ACF

We consider $G$, a linear group defined over $k$, an algebraically closed field. By considering $k$ as an embedded residue field of an algebraically closed valued field $K$, we can associate to it a compact $G$-space $S^μ_G(k)$, consisting of $μ$-types on $G$. We showed that for each $p_μ\in S^μ_G(k)$, $\text{Stab}^μ(p)=\text{Stab}(p_μ)$ is a solvable infinite algebraic group when $p_μ$ is centered at infinity and residually algebraic. Moreover we give a description of the dimension $\text{Stab}(p_μ)$ in terms of dimension of $p$.

math.LO↗

arXiv2021

Cohomology of algebraic varieties over non-archimedean fields

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $Γ_\infty$, where $Γ$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $Γ_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

math.AG↗

arXiv2020

Pro-definability of spaces of definable types

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists in showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loeser. Uniform definability of definable types is finally achieved by studying classes of stably embedded pairs.

math.LO↗

arXiv2020

Homotopy Types of Abstract Elementary Classes

We prove that for any homotopy type $X$, there is an abstract elementary class $\mathcal{C}$, with joint embedding, almagamation and no maximal models such that the classifying space realizes the homotopy type $X$. We provide a few explicit examples.

math.LO↗