Hodge--Tate crystals on the logarithmic prismatic sites of semi-stable formal schemes

Yu, Miao; Wang, Yupeng

doi:10.48550/arxiv.2205.08895

Cited by 2 publications

(5 citation statements)

References 16 publications

(35 reference statements)

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“…The equivalences in Thm. 5.2 then recover the known results about rational Hodge-Tate crystals proved in [MW21,MW22].…”

Section: De Rham Crystals As Log Connectionssupporting

confidence: 78%

“…Remark 5.5 (Compatibility with Hodge-Tate case). When m = 1, a B + dR,1 -crystal is precisely a (rationa) Hodge-Tate crystal treated in [MW21,MW22]. A log-A(u)-connection is simply a K-vector space with a K-linear endomorphism.…”

Section: De Rham Crystals As Log Connectionsmentioning

confidence: 99%

“…When m = 1, i.e., the case with nearly Hodge-Tate resp. log-nearly Hodge-Tate representations, these are proved in [Gao22] and [MW22] respectively. We now prove the general (torsion) case by a dévissage argument inspired by [Fon04,Thm.…”

mentioning

confidence: 94%

“…The relative case of Thm. 1.17 is established in [MW,MW22] for Hodge-Tate crystals, and will be established in full generality in the upcoming [GMW] (which in addition will generalize Thm. 1.18 to the relative case).…”

Section: Introductionmentioning

confidence: 99%

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de Rham crystals on the prismatic site of $\mathcal{O}_K$

Gao¹,

Yu²,

Wang³

2022

Preprint

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Let O K be a mixed characteristic complete discrete valuation ring with perfect residue field. We classify de Rham crystals over the (log-) prismatic site of O K using certain log connections. By establishing a Sen-Fontaine theory for B + dR -representations over a Kummer tower, we further classify these crystals by (log-) nearly de Rham representations. In addition, we compare (log-) prismatic cohomology of these crystals with the corresponding de Rham cohomology and Galois cohomology. Contents1. Introduction 1 2. de Rham crystals and stratifications 7 3. Stratifications I: structure of cosimplicial rings 9 4. Stratifications II: axiomatic computation of log connections 12 5. de Rham crystals as log connections 19 6. Cohomology of de Rham crystals I: vs. de Rham cohomology 21 7. de Rham crystals and B + dR -representations 23 8. Locally analytic vectors: axiomatic computations 25 9. Locally analytic vectors in rings 27 10. Sen-Fontaine theory for B + dR -representations over Kummer tower 31 11. Cohomology of de Rham crystals II: vs. Galois cohomology 35 12. Log-nearly de Rham representations 36 References 39

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“…The equivalences in Thm. 5.2 then recover the known results about rational Hodge-Tate crystals proved in [MW21,MW22].…”

Section: De Rham Crystals As Log Connectionssupporting

confidence: 78%

Section: De Rham Crystals As Log Connectionsmentioning

confidence: 99%

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

de Rham crystals on the prismatic site of $\mathcal{O}_K$

Gao¹,

Yu²,

Wang³

2022

Preprint

Self Cite

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“…The cohomological comparison Theorem 1.4 was previously known for the small correspondence under various additional hypothesis: These include the algebraic settings of Abbes-Gros and Tsuji [AGT16][AG22], the case of good reduction [Wan21, Thm. 1.1][AHLB23], and arithmetic settings of Galois-equivariant pro-étale vector bundles, namely for Q p -local systems due to Liu-Zhu [LZ17], and more generally by Min-Wang [MW22a]. For curves, Faltings deduced it from the small case.…”

mentioning

confidence: 99%

The p-adic Corlette–Simpson correspondence for abeloids

2022

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For an abeloid variety A over a complete algebraically closed field extension K of $$\mathbb {Q}_p$$ Q p , we construct a p-adic Corlette–Simpson correspondence, namely an equivalence between finite-dimensional continuous K-linear representations of the Tate module and a certain subcategory of the Higgs bundles on A. To do so, our central object of study is the category of vector bundles for the v-topology on the diamond associated to A. We prove that any pro-finite-étale v-vector bundle can be built from pro-finite-étale v-line bundles and unipotent v-bundles. To describe the latter, we extend the theory of universal vector extensions to the v-topology and use this to generalise a result of Brion by relating unipotent v-bundles on abeloids to representations of vector groups.

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Hodge--Tate crystals on the logarithmic prismatic sites of semi-stable formal schemes

Cited by 2 publications

References 16 publications

de Rham crystals on the prismatic site of $\mathcal{O}_K$

de Rham crystals on the prismatic site of $\mathcal{O}_K$

The p-adic Corlette–Simpson correspondence for abeloids

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