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A prismatic approach to crystalline local systems

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Abstract

Let \(X\) be a smooth \(p\)-adic formal scheme. We show that integral crystalline local systems on the generic fiber of \(X\) are equivalent to prismatic \(F\)-crystals over the analytic locus of the prismatic site of \(X\). As an application, we give a prismatic proof of Fontaine’s \(\mathrm {C}_{{\mathrm {crys}}}\)-conjecture, for general coefficients, in the relative setting, and allowing ramified base fields. Along the way, we also establish various foundational results for the cohomology of prismatic \(F\)-crystals, including various comparison theorems, Poincaré duality, and Frobenius isogeny.

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Notes

  1. In [28, Def. 1.1] such objects are called completed prismatic \(F\)-crystals. To see that their notion is compatible with ours, note that by [28, Prop. 4.13] and Beauville–Laszlo gluing, the restriction of a completed prismatic \(F\)-crystal to the open subset \(\operatorname{Spec}(\mathfrak {S})\setminus V(p,E)\) of a Breuil–Kisin prism \((\mathfrak {S},E)\) is a vector bundle. Moreover, the global sections of a vector bundle over \(\operatorname{Spec}(A)\setminus V(p,I)\) are a finitely presented module over \(A\) (cf. Lemma 5.8).

  2. For any multiindex \(\alpha \in \mathbf {Z}^{d}_{\ge 0}\), we use the customary notation , , and .

  3. Here we use that the ramification index of \(V_{0}\) is \(1 \le p-1\); when it is \(\ge p\), the analogous limit \(\lim _{n \to \infty} \lvert \pi ^{n}/n!\rvert = \infty \) for a uniformizer \(\pi \in V_{0}\).

  4. To see this, we recall from Definition 2.23 and Remark 2.24 that there are natural injections \(\operatorname{gr}^{\bullet }\mathbb{B}_{{\mathrm {crys}}}(S,S^{+})\to \operatorname{gr}^{\bullet }\mathbb{B}_{{\mathrm {crys}}, K}(S,S^{+}) \to \operatorname{gr}^{\bullet }\mathbb{B}_{{\mathrm {dR}}}(S,S^{+})\). Thus, the claimed isomorphism follows from the fact that the composed map is a graded isomorphism, thanks to [63, Cor. 2.25].

  5. We warn the reader that our construction of \(\mathcal {O}\mathbb{A}_{{\mathrm {crys}}}\) is ad hoc and different from that of [63], where \(\mathcal {O}_{K}\) is assumed to be unramified.

  6. More conceptually, (10) forms a diagram of objects in an appropriately defined infinitesimal site and there is a natural infinitesimal crystal associated to ℰ that can evaluate on the diagram.

  7. The uniqueness follows from the fact that the set of elements \(\{e\otimes f \mid e\in E(R[1/p]),~f\in \mathbb{B}^{+}_{{\mathrm {dR}}}(S,S^{+}) \}\) is dense in \(\mathcal {O}\mathbb{B}_{{\mathrm {dR}}}^{+}(S,S^{+})\), thanks to the explicit formula in [57, Prop. 6.10].

  8. Let \(d\) be a generator of . Then \(\delta (d)+p^{p-1} \cdot (d/p)^{p}\) is invertible in , so the element \(\varphi (d)=d^{p}+p\delta (d) = p(\delta (d)+p^{p-1} \cdot (d/p)^{p})\) is automatically invertible in .

  9. The fact that the vanishing locus of \(V(\mu )\) in \(\mathcal {Y}_{[0,\infty )}\) is the union \(\bigcup _{n>0} V(\varphi ^{-n}(I))\) follows from the intersection formula \(\bigcap _{r} \frac{\mu}{\varphi ^{-r}(\mu )}\mathrm {A}_{\inf}= \mu \mathrm {A}_{ \inf}\) in [22, Lem. 3.23]: for fixed \(s\in \mathbf {N}\) and \(m \gg 0\), the ideal \(\varphi ^{-m}(I)\) is invertible in \(\mathcal {O}_{\mathcal {Y}_{[0,s]}}\), so the sequence of ideals \(\frac{\mu}{\varphi ^{-r}(\mu )}\cdot \mathcal {O}_{\mathcal {Y}_{[0,s]}}= \varphi ^{-1}(I) \cdots \varphi ^{-r}(I) \cdot \mathcal {O}_{\mathcal {Y}_{[0,s]}}\) eventually stabilizes for large enough \(r\).

  10. To see the injectivity, we note that for the Breuil–Kisin prism \((\mathfrak{S},I=(E(u))\) one has the injections \(\mathfrak{S} \to \mathfrak{S}\langle \varphi ^{n}(I)/p\rangle \to \mathfrak{S}\langle I/p \rangle \). So the case of \(S'\) follows by taking the \(p\)-complete flat base change along and a further localization.

  11. This is a Breuil–Kisin prism in the sense of [28, Ex. 3.4].

  12. More precisely, given a divided power thickening \((A,J)\) of \(R\), we can lift the composition \(P \twoheadrightarrow R \to A/J\) to a map \(\mathbf {Z}_{p}[ x_{s}]\to A\) by the freeness of \(P\). The base change of \((A,J)\) along the quasi-syntomic cover \(A\to (A\otimes _{\mathbf {Z}_{p} [ x_{s}]} \mathbf {Z}_{p} [x_{s}^{1/p^{\infty}}])^{ \wedge}_{p}\) forms a divided power thickening of \(R\otimes _{P} P^{0}\), and thus admits a map from \(\mathrm {A}_{{\mathrm {crys}}}(R\otimes _{P} P^{0})\).

  13. This is possible by the \(p\)-completeness of \(B\) and the observation that any lift of a unit of \(B/(J,p^{n})\) is a unit in \(B/p^{n}\), since \(J\) is a divided power ideal.

  14. To reduce to the setting of [61, Cor. 2.37], it suffices to notice that each cohomology sheaf \(H^{i}(\mathcal {E}'[1/p])\) is an isocrystal in vector bundles, which produces a finite filtration on \(Rf_{s,{\mathrm {crys}},*} \mathcal {E}'[1/p]\) such that each graded piece is a perfect complex.

  15. Using the perspective on symmetric monoidal \(\infty \)-categories as certain coCartesian fibrations \(\mathcal {C}^{\otimes} \to \mathrm {Fin}_{*}\) (see e.g. [46, Def. 2.0.0.7]), recall that a functor \(F^{\otimes} \colon \mathcal {C}^{\otimes} \to \mathcal {D}^{\otimes}\) over \(\mathrm {Fin}_{*}\) is lax symmetric monoidal if it preserves locally coCartesian lifts of inert maps ([46, Def. 2.1.1.8]) and symmetric monoidal if it preserves all locally coCartesian lifts. It is an equivalence if it is additionally an equivalence on the underlying \(\infty \)-categories ([46, § 2.1.3]). One can check easily that if \(F\) is lax symmetric monoidal and has a homotopy inverse \((F^{\otimes})^{-1}\) which is a symmetric monoidal equivalence, then \(F\) must also preserve all locally coCartesian lifts and hence be a symmetric monoidal equivalence.

  16. The statement in [28] contains the assumption that the \(\varphi \)-module is torsionfree, which is not used in the proof.

  17. The tensor product is automatically complete because is perfect (Proposition 5.11).

  18. Let \(S=\overline{A}/p\) be the quasiregular semiperfect ring. As in [15, Thm. 4.6.1, Ex. 4.6.9], the surjection from \(\mathrm {A}_{{\mathrm {crys}}}(\overline{A}/p) \to \overline{A}/p\) can be understood via the diagram of prismatic cohomology:

  19. By the triviality of the Kähler differential, we identify this de Rham complex with a Koszul complex.

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Acknowledgements

The influence of the work of Bhatt–Scholze [20, 21] on our article is obvious, and we thank them heartily for their pioneering work. We initiated our project during the fall of 2021 after several conversations with Peter Scholze. We are grateful for his support and his many patient explanations throughout the writing of this project, especially his suggestion of the gluing construction in Theorem 4.15 using both the local system and its associated \(F\)-isocrystal. We are indebted to Bhargav Bhatt for many illuminating explanations and suggestions throughout the genesis of this paper. We are also thankful to the anonymous referee for countless remarks and corrections, which substantially enhanced the quality of our manuscript. Further thanks go to Sasha Petrov for suggesting that we should be able to prove the Frobenius isogeny property via Poincaré duality and to Guido Bosco, David Hansen, Hiroki Kato, Shizhang Li, Samuel Marks, Akhil Mathew, Yuchen Wu and Bogdan Zavyalov for helpful discussions and correspondence. We gratefully acknowledge funding through the Max Planck Institute for Mathematics in Bonn, Germany, during the preparation of this work. Additional support came from the University of Chicago (H.G.) and the National Science Foundation under Grant No. DMS-1926686 and the IAS School of Mathematics (E.R.) in the revision stage of the project.

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  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Haoyang Guo

  2. Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Emanuel Reinecke

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Guo, H., Reinecke, E. A prismatic approach to crystalline local systems. Invent. math. 236, 17–164 (2024). https://doi.org/10.1007/s00222-024-01238-4

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