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Bogdan Zavyalov

Publications and source records attributed to Bogdan Zavyalov.

14 recordsLinked to original sources
arXiv2026

Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems

We prove finiteness and Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems on proper $p$-adic rigid-analytic spaces in the framework of Banach--Colmez spaces and establish their optimal cohomological vanishing bounds. As a consequence, we also obtain ordinary finiteness and duality for their arithmetic pro-étale cohomology. We deduce these results from their analogs for perfect complexes on the Fargues--Fontaine curve, which in turn reduce to Poincaré duality for perfect complexes over period rings. We give a simple proof of the latter duality via the same diagrammatic argument as in our previous work for finite coefficients. Along the way, we establish optimal $v$-descent for perfect complexes over certain period rings on affinoid perfectoid spaces.

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arXiv2026

Formality for rigid-analytic spaces satisfying the weight-monodromy conjecture

We prove that étale and de Rham cohomology algebras of a smooth proper rigid-analytic space over a finite extension of $\mathbf{Q}_p$ are formal if the rigid-analytic space satisfies the weight-monodromy conjecture. This is achieved by showing that the underlying $E_\infty$-algebra of a monodromy-pure $E_\infty$-algebra in Weil--Deligne representations is formal. We give examples of smooth proper rigid-analytic surfaces whose cohomology algebras are not formal.

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arXiv2026

Almost coherent modules and almost coherent sheaves

We review the theory of almost coherent modules that was introduced in "Almost Ring Theory" by Gabber and Ramero. Then we globalize it by developing a new theory of almost coherent sheaves on schemes and on a class of "nice" formal schemes. We show that these sheaves satisfy many properties similar to usual coherent sheaves, i.e. the Almost Proper Mapping Theorem, the Formal GAGA, etc. We also construct an almost version of the Grothendieck twisted image functor $f^!$ and verify its properties. Lastly, we study sheaves of $p$-adic nearby cycles on admissible formal models of rigid spaces and show that these sheaves provide examples of almost coherent sheaves. This gives a new proof of the finiteness result for étale cohomology of proper rigid spaces obtained before in the work of Peter Scholze "$p$-adic Hodge Theory For Rigid-Analytic Varieties".

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arXiv2026

Poincaré Duality in abstract 6-functor formalisms

We study Poincaré Duality in the context of abstract 6-functor formalisms. In particular, we give a small and simple list of assumptions that implies Poincaré Duality. As an application, we give new uniform (and essentially formal) proofs of some previously established Poincaré Duality results.

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arXiv2025

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, §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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arXiv2025

Arithmetic Properties Of $\ell$-adic Étale Cohomology and Nearby Cycles of Rigid-Analytic Spaces

We prove a number of results on the étale cohomology of rigid analytic varieties over $p$-adic non-archimedean local fields. Among other things, we establish bounds for Frobenius eigenvalues, show a strong version of Grothendieck's local monodromy theorem, prove mixedness of the nearby cycle sheaf, and show that for any formal model, the IC sheaf on the special fiber is captured by the nearby cycles of the IC sheaf on the generic fiber. We also prove a local version of Deligne's weight-monodromy conjecture, by a novel perfectoid analysis of nearby cycles. Along the way, we develop the theory of "constructible $\ell$-adic complexes on Deligne's topos" (six operations, perverse t-structure, a notion of mixedness, etc.), which is prerequisite to a precise discussion of the Galois action on nearby cycles for algebraic and rigid analytic varieties over non-archimedean fields.

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arXiv2025

Some Foundational Results In Adic Geometry

In this paper, we record some foundational results on adic geometry that seem to be missing in the existing literature. Namely, we develop the Proj construction and a theory of lci closed immersions in the context of locally noetherian analytic adic spaces. In the context of rigid-analytic spaces, these topics have previously been considered in [GL21] and [Con07]. We also develop an etale six functor formalism in the analytic geometry and give a categorical description of lisse and constructible sheaves. All results of this paper are probably well-known to the experts.

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arXiv2024

Relative Poincaré duality in nonarchimedean geometry

We prove a conjecture of Bhatt-Hansen that derived pushforwards along proper morphisms of rigid-analytic spaces commute with Verdier duality on Zariski-constructible complexes. In particular, this yields duality statements for the intersection cohomology of proper rigid-analytic spaces. In our argument, we construct cycle classes in analytic geometry as well as trace maps for morphisms that are either smooth or proper or finite flat, with appropriate coefficients. As an application of our methods, we obtain new, significantly simplified proofs of $p$-adic Poincaré duality and the preservation of $\mathbf{F}_p$-local systems under smooth proper higher direct images.

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arXiv2024

Lefschetz theorems in flat cohomology and applications

We prove a version of the Lefschetz hyperplane theorem for fppf cohomology with coefficients in any finite commutative group scheme over the ground field. As consequences, we establish new Lefschetz results for the Picard scheme.

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arXiv2024

Mod-p Poincaré Duality in p-adic Analytic Geometry

We show Poincaré Duality for $\mathbf{F}_p$-étale cohomology of a smooth proper rigid-analytic space over a non-archimedean field $K$ of mixed characteristic $(0, p)$. It positively answers the question raised by P. Scholze in [Sch13a]. We prove duality via constructing Faltings' trace map relating Poincaré Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce Poincaré Duality for $\mathbf{Z}/p^n\mathbf{Z}$, $\mathbf{Z}_p$, and $\mathbf{Q}_p$-coefficients.

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arXiv2022

Altered Local Uniformization Of Rigid-Analytic Spaces

We prove a version of Temkin's local altered uniformization theorem. We show that for any rig-smooth, quasi-compact and quasi-separated admissible formal $\mathcal{O}_K$-model $\mathfrak{X}$, there is a finite extension $K'/K$ such that $\mathfrak{X}_{\mathcal{O}_{K'}}$ locally admits a rig-étale morphism $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and a rig-isomorphism $h\colon \mathfrak{X}" \to \mathfrak{X}'$ with $\mathfrak{X}'$ being a successive semi-stable curve fibration over $\mathcal{O}_{K'}$ and $\mathfrak{X}"$ being a poly-stable formal $\mathcal{O}_{K'}$-scheme. Moreover, $\mathfrak{X}'$ admits an action of a finite group $G$ such that $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ is $G$-invariant, and the adic generic fiber $\mathfrak{X}'_{K'}$ becomes a $G$-torsor over its quasi-compact open image $U=g_{K'}(\mathfrak{X}'_{K'})$. Also, we study properties of the quotient map $\mathfrak{X}'/G \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and show that it can be obtained as a composition of open immersions and rig-isomorphisms.

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arXiv2016

Chow Groups of Abelian Varieties and Beilinson's Conjecture

In the present paper we introduce the property AA of a subsemigroup of the endomorphism semigroup of an abelian variety, which holds for semigroup of endomorphisms of an abelian variety defined over a number field, and show that the orbit of any cycle under a semigroup with property AA in the Chow group $\otimes\Q$ has finite dimensional span.

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