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Wiesława Nizioł

Publications and source records attributed to Wiesława Nizioł.

At least 19 recordsLinked to original sources
arXiv2026

Une conjecture $C_{\rm st}$ pour la cohomologie à support compact

Let $\mathbf{B}$ be the ring of analytic functions on the Fargues-Fontaine curve $Y_{\rm FF}$. We shaw that, by adding sufficiently iterated periods, one can kill its Galois cohomology in degrees $\geq 1$. For $\mathbf{B}_{\rm dR}^+$, the corresponding result is obtained by adjoining just $\log t$ and is folklore. This allows to formulate $C_{\rm dR}$ and $C_{\rm st}$ conjectures for the compactly supported cohomology of $p$-adic analytic varieties.

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arXiv2026

Hodge Theory of $p$-adic analytic varieties: a survey

Hodge Theory of $p$-adic analytic varieties was initiated by Tate in his 1967 paper on $p$-divisible groups, where he conjectured the existence of a Hodge-like decomposition for the $p$-adic étale cohomology of proper analytic varieties. Tate's conjecture was refined by Fontaine who gave the theory its definite shape. A lot of work has been done for algebraic varieties and a number of proofs of Fontaine's conjectures have been obtained between years 1985 and 2011. But the study of Hodge Theory of $p$-adic analytic varieties started really only in 2011 with Scholze's proof of Tate's conjecture using perfectoid methods. Methods that opened the way to an avalanche of results. In this paper, we survey our results and conjectures (comparison theorems and their geometrization, dualities, etc.), focusing on the case of nonproper analytic varieties, where a number of new phenomena occur. We also describe the new objects that appeared along the way.

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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

Topological Vector Spaces

Motivated by applications to duality theorems for $p$-adic pro-étale cohomology of rigid analytic spaces, we study the category of Topological Vector Spaces in the setting of condensed mathematics. We prove that it contains, as full subcategories, both the category of (topologically) bounded algebraic Vector Spaces and the category of perfect complexes on the Fargues-Fontaine curve. Vector Spaces coming from $p$-adic pro-étale cohomology of smooth partially proper rigid analytic varieties are examples of sheaves belonging to the former category.

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arXiv2026

On the de Rham flip-flopping in dual towers

We prove a version of de Rham and Hyodo-Kato flip-flopping for dual towers of rigid analytic spaces including those coming from dual basic local Shimura varieties. The main tool are comparison theorems expressing the two cohomologies as pro-étale cohomology of corresponding relative period sheaves that, by definition, satisfy pro-étale descent. As an application, we show that de Rham and Hyodo-Kato cohomologies of finite level coverings of the Drinfeld space of any dimension $d$ over $K$ are admissible as representations of $\mathbb{GL}_{d+1}(K)$.

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arXiv2025

Duality for $p$-adic geometric pro-étale cohomology

We prove that $p$-adic geometric pro-étale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincaré duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizioł, from a Poincaré duality for solid quasi-coherent sheaves on the Fargues-Fontaine curve representing this cohomology. The latter duality is proved by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincaré duality for ${\mathbf B}^+_{\rm st}$-twisted Hyodo-Kato and filtered $\mathbf{B}^+_{\rm dr}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result.

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arXiv2025

Compactly supported $p$-adic pro-étale cohomology of analytic varieties

We study properties of compactly supported $p$-adic pro-étale cohomology of smooth partially proper rigid analytic varieties. In particular, we prove a comparison theorem, in a stable range, with compactly supported syntomic cohomology, which is built from compactly supported Hyodo-Kato and ${\mathcal B}^+_{\rm dr}$-cohomologies. We derive from that a (limited version of a) fundamental diagram.

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arXiv2024

On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture

Long ago, Fontaine formulated conjectures (now theorems) relating étale and de Rham cohomologies of algebraic varieties over $p$-adic fields. In an earlier work we have shown that pro-étale and de Rham cohomologies of analytic varieties in the two extreme cases: proper and Stein, are also related. In the proper case, the comparison theorems are similar to those for algebraic varieties, but for Stein varieties they are quite different. In this paper, we state analogs of Fontaine's conjectures for general smooth dagger varieties, that interpolate between these two extreme cases, and we prove them for many ``small'' varieties (cases we deal with include products of overconvergent affinoids and proper varieties, analytifications of algebraic varieties, or "almost proper" dagger varieties). The proofs use a ``geometrization'' of all involved cohomologies in terms of quasi-Banach-Colmez spaces (qBC's for short, quasi- because we relax the finiteness conditions). The heart of the proof relies on delicate properties of BC's and qBC's. These properties should be of independent interest and we have devoted a large part of the paper to them.

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arXiv2024

On the $v$-Picard group of Stein spaces

We study the image of the Hodge-Tate logarithm map (in any cohomological degree), defined by Heuer, in the case of smooth Stein varieties. Heuer, motivated by the computations for the affine space of any dimension, raised the question whether this image is always equal to the group of closed differential forms. We show that it indeed always contains such forms but the quotient can be non-trivial: it contains a slightly mysterious $Z_p$-module that maps, via the Bloch-Kato exponential map, to integral classes in the pro-étale cohomology. This quotient is already non-trivial for open unit discs of dimension strictly greater than $1$.

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arXiv2023

On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem

The purpose of this paper is to prove a basic $p$-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a $p$-adic field: $p$-adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over $\bf{B}^+_{\rm dR}$). The key computation is the passage from absolute crystalline cohomology to Hyodo-Kato cohomology and the construction of the related Hyodo-Kato isomorphism. We also "geometrize" our comparison theorem by turning $p$-adic pro-étale and syntomic cohomologies into sheaves on the category ${\rm Perf}_C$ of perfectoid spaces over $C$ (this geometrization will be crucial in our proof of the $C_{\rm st}$-conjecture in the sequel to this paper).

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arXiv2023

Arithmetic duality for $p$-adic pro-étale cohomology of analytic curves

We prove a Poincaré duality for arithmetic $p$-adic pro-étale cohomology of smooth dagger curves over finite extensions of ${\mathbf Q}_p$. We deduce it, via the Hochschild-Serre spectral sequence, from geometric comparison theorems combined with Tate and Serre dualities. The compatibility of all the products involved is checked via reduction to the ghost circle, for which we also prove a Poincaré duality (showing that it behaves like a proper smooth analytic variety of dimension $1/2$). Along the way we study functional analytic properties of arithmetic $p$-adic pro-étale cohomology and prove that the usual cohomology is nuclear Fréchet and the compactly supported one -- of compact type.

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arXiv2023

Factorization de la cohomologie étale p-adique de la tour de Drinfeld

For a finite extension $F$ of ${\mathbf Q}_p$, Drinfeld defined a tower of coverings of ${\mathbb P}^1\setminus {\mathbb P}^1(F)$ (the Drinfeld half-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the $p$-adic geometric étale cohomology of this tower analogous to Emerton's decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finitness theorem for the arithmetic étale cohomology modulo $p$ which is shown by first proving, via a computation of nearby cycles, that this cohomology has finite presentation. This last result holds for all $F$; for $F\neq {\mathbf Q}_p$, it implies that the representations of ${\rm GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$.

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arXiv2023

Correspondance de Langlands locale $p$-adique et anneaux de Kisin

We use a ${\mathcal B}$-adic completion and the $p$-adic local Langlands correspondence for ${\mathrm {GL}}_2({\mathbf Q}_p )$ to give a construction of Kisin's rings and the attached universal Galois representations (in dimension 2 and for ${\mathbf Q}_p$) directly from the classical Langlands correspondence. This gives, in particular, a uniform proof of the geometric Breuil-Mézard conjecture in the supercuspidal case.

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arXiv2022

Cohomologie des courbes analytiques $p$-adiques

Cohomology of affinoids does not behave well; often, this can be remedied by making affinoids overconvergent. In this paper, we focus on dimension 1 and compute, using analogs of pants decompositions of Riemann surfaces, various cohomologies of affinoids. To give a meaning to these decompositions we modify slightly the notion of $p$-adic formal scheme, which gives rise to the adoc (an interpolation between adic and ad hoc) geometry. It turns out that cohomology of affinoids (in dimension 1) is not that pathological. From this we deduce a computation of cohomologies of curves without boundary (like the Drinfeld half-plane and its coverings). In particular, we obtain a description of their $p$-adic pro-étale cohomology in terms of de the Rham complex and the Hyodo-Kato cohomology, the later having properties similar to the ones of $\ell$-adic pro-étale cohomology, for $\ell\neq p$.

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arXiv2021

p-adic etale cohomology of period domains

We compute the p-torsion and p-adic etale cohomologies with compact support of period domains over local fields in the case of basic isocrystals for quasi-split reductive groups. For the p-torsion case, we follow the method used by Orlik in his computations of the l-torsion etale cohomology using as a key new ingredient the computation of Ext groups between mod p generalized Steinberg representations of p-adic groups. For the p-adic case, we don't use Huber's definition of etale cohomology with compact support as Orlik did since it seems to give spaces that are much too big; instead we use continuous etale cohomology with compact support.

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arXiv2019

On uniqueness of $p$-adic period morphisms, II

We prove equality of the various $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterium we had found for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined. We do it by lifting the syntomic and almost étale period morphisms to Voyevodsky triangulated category of motives, where their equality with the $h$-cohomology period morphism can be checked directly using the Beilinson Poincaré Lemma.

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arXiv2019

On p-adic comparison theorems for rigid analytic varieties, I

We compute, in a stable range, the arithmetic p-adic etale cohomology of smooth rigid analytic and dagger varieties (without any assumption on the existence of a nice integral model) in terms of differential forms using syntomic methods. The main technical input is a construction of a Hyodo-Kato cohomology and a Hyodo-Kato isomorphism with de Rham cohomology.

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