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1. Étale structures and the Joyal–Tierney representation theorem in countable model theory

   https://doi.org/10.1017/bsl.2025.10086  

   Chen, Ruiyuan (July 2025, The Bulletin of Symbolic Logic)

   An étale structure over a topological space X is a continuous family of structures (in some first-order language) indexed over X. We give an exposition of this fundamental concept from sheaf theory and its relevance to countable model theory and invariant descriptive set theory. We show that many classical aspects of spaces of countable models can be naturally framed and generalized in the context of étale structures, including the Lopez-Escobar theorem on invariant Borel sets, an omitting types theorem, and various characterizations of Scott rank. We also present and prove the countable version of the Joyal–Tierney representation theorem, which states that the isomorphism groupoid of an étale structure determines its theory up to bi-interpretability; and we explain how special cases of this theorem recover several recent results in the literature on groupoids of models and functors between them. 

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2. Cohomological Invariants in Positive Characteristic

   https://doi.org/10.1093/imrn/rnaa321  

   Totaro, Burt (January 2021, International Mathematics Research Notices)

   Abstract We determine the mod $$p$$ cohomological invariants for several affine group schemes $$G$$ in characteristic $$p$$. These are invariants of $$G$$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $$p$$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $$p$$ Milnor K-theory of fields. 

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3. Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts

   Alper, Jarod; Halpern-Leistner, Daniel; Hall, Jack; Rydh, David (February 2024, Forum of Mathematics, Sigma)

   We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth, or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin-Tyomkin. 

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4. Syntomic complexes and p -adic étale Tate twists

   https://doi.org/10.1017/fmp.2022.21  

   Bhatt, Bhargav; Mathew, Akhil (January 2023, Forum of Mathematics, Pi)

   Abstract The primary goal of this paper is to identify syntomic complexes with the p -adic étale Tate twists of Geisser–Sato–Schneider on regular p -torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘ F -smooth’. The F -smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes. 

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5. Integral Points on Varieties With Infinite Étale Fundamental Group

   https://doi.org/10.1093/imrn/rnad147  

   Achenjang, Niven T; Morrow, Jackson S (July 2023, International Mathematics Research Notices)

   Abstract We study integral points on varieties with infinite étale fundamental groups. More precisely, for a number field $$F$$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $$\varphi \colon Y \to X$$ of degree at least $$2\dim (X)^{2}$$, there exists an ample line bundle $$\mathscr{L}$$ on $$Y$$ such that for a general member $$D$$ of the complete linear system $$|\mathscr{L}|$$, $$D$$ is geometrically irreducible and any set of $$\varphi (D)$$-integral points on $$X$$ is finite. We apply this result to varieties with infinite étale fundamental group to give new examples of irreducible, ample divisors on varieties for which finiteness of integral points is provable. 

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