Poincaré duality, non-lisse direct images, and the non-proper extension in p-adic Hodge theory for rigid-analytic varieties

Scholze's 2013 paper 'p-adic Hodge theory for rigid-analytic varieties' proved finiteness of p-adic étale cohomology and de Rham comparison isomorphisms for proper smooth rigid-analytic varieties with lisse $\mathbb{Z}_p$ -coefficients, and its introduction leaves open: (1) Poincaré duality for this cohomology; (2) removal of the smoothness hypothesis in the finiteness theorem; (3) a relative comparison theorem without the assumption that higher direct images are lisse; and (4) extension of the theory to non-proper rigid-analytic varieties. Of these, (1) and (2) are resolved by later work (Zavyalov; Mann; Lan-Liu-Zhu; Li-Reinecke-Zavyalov; Scholze's survey), and the lisse hypothesis in (3) is automatic at the torsion level (Li-Reinecke-Zavyalov, building on Scholze-Weinstein). The open core is (3) at the $\mathbb{Z}_p$ - and $\mathbb{Q}_p$ -level, where the lisseness of the higher direct images is not known for arbitrary proper smooth analytic families — for general pro-étale $\mathbb{Q}_p$ -local systems the direct images are instead naturally Banach-Colmez-valued, and only the aggregate duality in the solid quasi-coherent framework is known — and (4), whose precise current formulation is the Cst conjecture of Colmez-Nizioł: a non-derived comparison theorem recovering de Rham and Hyodo-Kato cohomology from geometric pro-étale cohomology for general smooth partially proper rigid-analytic varieties, verified so far only for proper varieties, Stein varieties, analytifications of algebraic varieties, complements of small tubes, products of these with finite-dimensional de Rham cohomology, and products of proper and Stein varieties.

Background

A rigid-analytic variety over a p-adic field $k$ (a complete field for a non-archimedean absolute value extending the $p$ -adic one on $\mathbb{Q}_p$ ) is the $p$ -adic analogue of a complex-analytic space: it is glued from affinoids, the common zero loci of families of convergent power series. Unlike the complex case, the étale cohomology of such spaces with $p$ -adic coefficients has no algebraic theory to lean on, and even finite-dimensionality is a theorem rather than a formality. A lisse $\mathbb{Z}_p$ -sheaf (a $p$ -adic local system) on a rigid-analytic variety is the analogue of a local system on a complex manifold: a sheaf locally constant of finite rank; an $\mathbb{F}_p$ -local system is the same with coefficients in the finite field $\mathbb{F}_p$ .

Classical $p$ -adic Hodge theory relates, for a smooth proper variety over $k$ , its $p$ -adic étale cohomology with its de Rham cohomology: the two become canonically isomorphic after tensoring with the de Rham period ring $B_{dR}$ , a topological $\mathbb{Q}_p$ -algebra whose filtration records Hodge-theoretic data. Scholze's 2013 paper extended this theory to the rigid-analytic category, its natural home for analytic families, using two tools: perfectoid spaces (a class of analytic spaces whose $p$ -adic geometry can be studied through an associated 'tilted' space of characteristic $p$ , making the method of almost étale extensions available), and the pro-étale site (a refinement of the étale site permitting certain infinite covers, on which period sheaves such as $\mathbb{B}^+_{dR}$ exist). Theorem 1.1 of that paper proves that for $X$ proper and smooth over a complete algebraically closed extension $K$ of $\mathbb{Q}_p$ and any $\mathbb{F}_p$ -local system $\mathbb{L}$ on $X$ , the groups $H^i(X_{\acute{e}t}, \mathbb{L})$ are finite-dimensional $\mathbb{F}_p$ -vector spaces, vanishing for $i > 2\dim X$ . Theorem 1.6 proves a de Rham comparison for de Rham lisse $\mathbb{Z}_p$ -sheaves (those whose associated $\mathbb{B}^+_{dR}$ -local system comes from a filtered module with integrable connection), and Theorem 1.10 a relative version: for a proper smooth morphism $f : X \to Y$ of smooth rigid-analytic varieties over $k$ and a de Rham lisse $\mathbb{Z}_p$ -sheaf $\mathbb{L}$ on $X$ , if each higher direct image $R^i f_{pro\acute{e}t*}\mathbb{L}$ is a lisse $\mathbb{Z}_p$ -sheaf on $Y$ (automatic when the family is the analytification of an algebraic one), then $R^i f_{pro\acute{e}t*}\mathbb{L}$ is de Rham, with associated filtered module $R^i f_{dR*}(\mathcal{E}, \nabla, \mathrm{Fil}^\bullet)$ .

The paper's introduction flags precisely what its methods do not deliver: 'results like Poincaré duality are not known for the p-adic étale cohomology of rigid-analytic varieties over p-adic fields'; the smoothness assumption in Theorem 1.1 'is probably unnecessary, and an artifact of the proof'; the lisse hypothesis in Theorem 1.10 is not known for general proper smooth rigid-analytic families; and finiteness 'is false if one does not make a restriction to the proper case', so a non-proper theory must handle genuinely infinite cohomology.

The later literature, needed to state the current status, introduced further objects. Poincaré duality was proved for $\mathbb{F}_p$ - and $\mathbb{Z}/p^n$ -coefficients by Zavyalov and, independently with a full formalism, by Mann, and uniformly in the torsion order by Li-Reinecke-Zavyalov; Lan-Liu-Zhu had proved the rational statement (after inverting $p$ ) for spaces Zariski-open in proper smooth ones. For general coefficients a new phenomenon appears: the cohomology of a finite-rank pro-étale $\mathbb{Q}_p$ -local system (a locally free $\hat{\mathbb{Q}}_p$ -module on the pro-étale site, not required to come from a lisse $\mathbb{Z}_p$ -sheaf, i.e. not required to admit a stable $\mathbb{Z}_p$ -lattice) on a proper smooth rigid variety is typically a Banach-Colmez space — an object whose values are mixtures of finite-dimensional $\mathbb{Q}_p$ - and $\mathbb{C}$ -vector spaces, realizable as coherent cohomology on the Fargues-Fontaine curve (a fundamental curve of $p$ -adic Hodge theory whose vector bundles encode $p$ -adic Galois representations) — and can be infinite-dimensional over $\mathbb{Q}_p$ . Anschütz-Le Bras-Mann built a 6-functor formalism (the system of six operations $f^*, f_*, f_!, f^!, \otimes, \underline{Hom}$ on sheaves, here for solid quasi-coherent sheaves — a framework from condensed mathematics in which topological modules admit homological algebra) on the relative Fargues-Fontaine curve, in which duality and finiteness for general pro-étale $\mathbb{Q}_p$ -local systems hold in this upgraded sense. On the non-proper side, a smooth partially proper rigid-analytic variety (one whose affinoid subdomains satisfy a boundary-separatedness condition; every quasi-Stein space, i.e. a countable increasing union of affinoids along open immersions, is partially proper) has a well-behaved geometric pro-étale cohomology valued in topological vector spaces, for which Colmez-Gilles-Nizioł proved Poincaré duality, and Bosco established rational comparison theorems without any properness or smoothness hypothesis. The remaining target for non-proper spaces is the Cst conjecture of Colmez-Nizioł: for a smooth partially proper rigid-analytic variety $Y$ over a finite extension of $\mathbb{Q}_p$ , a non-derived comparison recovering the de Rham cohomology and the Hyodo-Kato cohomology of $Y$ (de Rham cohomology equipped with its Frobenius $\varphi$ , monodromy operator $N$ , and Galois action, obtained from semistable reduction data) from the geometric pro-étale cohomology of $Y$ over an algebraic closure, by taking invariants under the Galois action after tensoring with the semistable period ring $B_{st}$ and its variants. Its derived-level precursor is unconditional.

Problem Statement

Scholze's paper proves, for proper smooth rigid-analytic varieties over $p$ -adic fields, finiteness of $p$ -adic étale cohomology with $\mathbb{F}_p$ -local system coefficients (Theorem 1.1), a de Rham comparison for de Rham lisse $\mathbb{Z}_p$ -sheaves (Theorem 1.6), and a relative comparison for proper smooth morphisms $f : X \to Y$ , conditional on each higher direct image $R^i f_{pro\acute{e}t*}\mathbb{L}$ being a lisse $\mathbb{Z}_p$ -sheaf on $Y$ (Theorem 1.10). The completion program that the paper's introduction explicitly leaves open, in its original generality, is: (1) prove Poincaré duality for the $p$ -adic étale cohomology of proper smooth rigid-analytic varieties; (2) remove the smoothness hypothesis in the finiteness theorem; (3) establish the relative de Rham comparison without assuming that the higher direct images $R^i f_{pro\acute{e}t*}\mathbb{L}$ are lisse $\mathbb{Z}_p$ -sheaves, i.e. for smooth proper rigid-analytic families that do not arise by analytification of algebraic families; and (4) extend the comparison, finiteness, and duality theory to non-proper rigid-analytic varieties, where finiteness of $p$ -adic étale cohomology fails and the correct replacement statements must be identified and proved. By the results cited in the References, components (1) and (2) are resolved and the lisse hypothesis in (3) is automatic at the $\mathbb{Z}/p^n$ -torsion level. The remaining open targets, which are independent of one another, are: (3*) For a proper smooth morphism $f : X \to Y$ of smooth rigid-analytic varieties over $k$ and a lisse $\mathbb{Z}_p$ -sheaf $\mathbb{L}$ on $X$ (in particular a de Rham lisse one), establish that each higher direct image $R^i f_{pro\acute{e}t*}\mathbb{L}$ is again a lisse $\mathbb{Z}_p$ -sheaf on $Y$ for arbitrary proper smooth analytic families — without the algebrizability that makes this automatic and without assuming lisseness of the direct images — so that for de Rham lisse $\mathbb{L}$ the relative de Rham comparison of Theorem 1.10 holds without its lisseness hypothesis, with filtered module $R^i f_{dR*}(\mathcal{E}, \nabla, \mathrm{Fil}^\bullet)$ ; this lisse-preservation is a theorem at the $\mathbb{Z}/p^n$ -torsion level (Li-Reinecke-Zavyalov, building on Scholze-Weinstein), so the open content is the $\mathbb{Z}_p$ - and $\mathbb{Q}_p$ -level. (4*) For non-proper spaces, the Cst conjecture of Colmez-Nizioł (Conjecture 4.1 of their survey, with the compactly supported variant Conjecture 4.10, formulated in Colmez-Gilles-Nizioł arXiv:2511.14675): for every smooth partially proper rigid-analytic variety $Y$ over a finite extension of $\mathbb{Q}_p$ , the non-derived comparison recovering the de Rham and Hyodo-Kato cohomologies of $Y$ from the geometric pro-étale cohomology of $Y$ over an algebraic closure via period-ring invariants — currently verified only for proper varieties, Stein varieties, analytifications of algebraic varieties, complements of small tubes, products of these with finite-dimensional de Rham cohomology, and products of proper and Stein varieties (Theorem 4.3(ii) of the survey). The model-compactification Question 4.5 of that survey (compactify the model of a quasi-compact $Y$ so that the complement of the special fiber has codimension at least 1), which would imply Cst for quasi-compact dagger (overconvergent) varieties, is an independent sub-target attached to (4*).

References

Peter Scholze, p-adic Hodge theory for rigid-analytic varieties, Forum of Mathematics, Pi 1 (2013), e1, DOI 10.1017/fmp.2013.1, https://doi.org/10.1017/fmp.2013.1
Peter Scholze, Perfectoid spaces: A survey, Current Developments in Mathematics 2012, International Press (2013), pp. 193-227, DOI 10.4310/cdm.2012.v2012.n1.a4, https://doi.org/10.4310/cdm.2012.v2012.n1.a4
Kai-Wen Lan, Ruochuan Liu, Xinwen Zhu, De Rham comparison and Poincaré duality for rigid varieties, Peking Mathematical Journal 6 (2022), no. 1, 143-216, DOI 10.1007/s42543-020-00031-5, arXiv:1912.13030, https://doi.org/10.1007/s42543-020-00031-5
Kiran S. Kedlaya, Ruochuan Liu, Finiteness of cohomology of local systems on rigid analytic spaces, arXiv:1611.06930 (2016), https://arxiv.org/abs/1611.06930
Bhargav Bhatt, David Hansen, The six functors for Zariski-constructible sheaves in rigid geometry, Compositio Mathematica 158 (2022), no. 2, 437-482, DOI 10.1112/s0010437x22007291, https://doi.org/10.1112/s0010437x22007291
Peter Scholze, Jared Weinstein, Berkeley Lectures on p-adic Geometry, Princeton University Press (2020), DOI 10.23943/princeton/9780691202082.001.0001, https://doi.org/10.23943/princeton/9780691202082.001.0001
Bogdan Zavyalov, Mod- $p$ Poincaré Duality in $p$ -adic analytic geometry, Annals of Mathematics 201 (2025), no. 3, DOI 10.4007/annals.2025.201.3.2, arXiv:2111.01830, https://doi.org/10.4007/annals.2025.201.3.2
Lucas Mann, A $p$ -Adic 6-Functor Formalism in Rigid-Analytic Geometry, arXiv:2206.02022 (2022), https://arxiv.org/abs/2206.02022
Shizhang Li, Emanuel Reinecke, Bogdan Zavyalov, Relative Poincaré duality in nonarchimedean geometry, arXiv:2410.08200 (2024), https://arxiv.org/abs/2410.08200
Ben Heuer, The Primitive Comparison Theorem in characteristic $p$ , Mathematische Zeitschrift 308 (2024), no. 3, article 55, DOI 10.1007/s00209-024-03613-6, arXiv:2402.00836, https://doi.org/10.1007/s00209-024-03613-6
Guido Bosco, Rational $p$ -adic Hodge theory for rigid-analytic varieties, arXiv:2306.06100 (2023), https://arxiv.org/abs/2306.06100
Pierre Colmez, Sally Gilles, Wiesława Nizioł, Duality for $p$ -adic geometric pro-étale cohomology, arXiv:2411.12163 (2024), https://arxiv.org/abs/2411.12163
Johannes Anschütz, Arthur-César Le Bras, Lucas Mann, A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve, arXiv:2412.20968 (2024), https://arxiv.org/abs/2412.20968
Pierre Colmez, Wiesława Nizioł, Hodge Theory of $p$ -adic analytic varieties: a survey, arXiv:2601.16557 (2026), https://arxiv.org/abs/2601.16557
Bogdan Zavyalov, Almost coherent modules and almost coherent sheaves, arXiv:2110.10773 (2021), https://arxiv.org/abs/2110.10773
Pierre Colmez, Sally Gilles, Wiesława Nizioł, Une conjecture $C_{\rm st}$ pour la cohomologie à support compact, arXiv:2511.14675 (2025), https://arxiv.org/abs/2511.14675
Piotr Achinger, Sally Gilles, Wiesława Nizioł, Compactly supported $p$ -adic pro-étale cohomology of analytic varieties, arXiv:2501.13651 (2025), https://arxiv.org/abs/2501.13651
Xinyu Shao, Hyodo-Kato cohomology in rigid geometry: some foundational results, arXiv:2502.13315 (2025), https://arxiv.org/abs/2502.13315