arXiv subjects

Sally Gilles

Publications and source records attributed to Sally Gilles.

8 recordsLinked to original sources

arXiv2026

$\mathbb{G}_m$-cohomology of $p$-adic Stein spaces

We compute the \'etale $\mathbb{G}_m$-cohomology of some $p$-adic rigid analytic Stein spaces. The computation is done by considering the filtration induced by the subgroup of principal units $U=1+ \mathfrak{m} \mathcal{O}^+$ of $\mathbb{G}_m$. We then determine the $U$-cohomology via methods from $p$-adic Hodge theory (passage to the pro-\'etale site, comparison theorems with $p$-adic cohomologies), while the $\mathbb{G}_m/U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space.

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arXiv2025

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

Let $\mathbf{B}$ be the ring of analytic functions on the Fargues-Fontaine curve $Y_{\rm FF}$. We show that adding $p$-adic analogs of $\log p$ and $\log 2\pi i$ kills its Galois cohomology in degrees~$\geq 1$. The analogous result for $\mathbf{B}^+_{\rm dR}$ is folklore. This makes it possible to formulate $C_{\rm dR}$ and $C_{\rm st}$-type conjectures for compact support cohomology of $p$-adic analytic varieties.

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arXiv2025

A pro-\'etale-to-de Rham comparison theorem for curves

We state a conjecture relating de Rham cohomology of a smooth rigid analytic variety to its compactly supported pro-\'etale cohomology. We prove the conjecture in the cases where the variety is a Stein curve of dimension one or a Stein space of higher dimension with low Frobenius slopes. The proof uses computation of the Galois cohomology of the almost de Rham period ring BdR[log(t)].

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arXiv2025

Compactly supported $p$-adic pro-\'etale cohomology of analytic varieties

We study properties of compactly supported $p$-adic pro-\'etale 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

Duality for $p$-adic geometric pro-\'etale cohomology

We prove that $p$-adic geometric pro-\'etale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincar\'e duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizio{\l}, from a Poincar\'e 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 Poincar\'e 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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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

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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arXiv2022

Morphismes de périodes et cohomologie syntomique

We give the geometric version of a construction of Colmez-Niziol which establishes a comparison theorem between arithmetic p-adic nearby cycles and syntomic sheaves. The local construction of the period isomorphism uses $(ϕ,Γ)$-modules theory and is obtained by reducing the period isomorphism to a comparison theorem between cohomologies of Lie algebras. By applying the method of "more general coordinates" used by Bhatt-Morrow-Scholze, we construct a global isomorphism. In particular, we deduce the semi-stable conjecture of Fontaine-Jannsen. This result was also proved by (among others) Tsuji, via the Fontaine-Messing map, and by Cesnavicius and Koshikawa, which generalized the proof of the crystalline conjecture by Bhatt, Morrow and Scholze. We use the previous map to show that the period morphism of Tsuji and the one of Cesnavicius-Koshikawa are the same.

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