A prismatic approach to crystalline local systems
Inventiones mathematicae - 2024
Tóm tắt
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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citation_journal_title=Rend. Semin. Mat. Univ. Padova; citation_title=Semistable sheaves and comparison isomorphisms in the semistable case; citation_author=F. Andreatta, A. Iovita; citation_volume=128; citation_publication_date=2012; citation_pages=131-285; citation_id=CR1
citation_journal_title=J. Inst. Math. Jussieu; citation_title=Comparison isomorphisms for smooth formal schemes; citation_author=F. Andreatta, A. Iovita; citation_volume=12; citation_issue=1; citation_publication_date=2013; citation_pages=77-151; citation_id=CR2
citation_journal_title=Forum Math. Pi; citation_title=Prismatic Dieudonné theory; citation_author=J. Anschütz, A.-C. Bras; citation_volume=11; citation_publication_date=2023; citation_id=CR3
citation_journal_title=Duke Math. J.; citation_title=On the Beilinson fiber square; citation_author=B. Antieau, A. Mathew, M. Morrow, T. Nikolaus; citation_volume=171; citation_issue=18; citation_publication_date=2022; citation_pages=3707-3806; citation_id=CR4
citation_journal_title=Camb. J. Math.; citation_title=On the crystalline period map; citation_author=A. Beilinson; citation_volume=1; citation_issue=1; citation_publication_date=2013; citation_pages=1-51; citation_id=CR5
citation_journal_title=Publ. Math. Inst. Hautes Études Sci.; citation_title=Étale cohomology for non-Archimedean analytic spaces; citation_author=V.G. Berkovich; citation_volume=78; citation_publication_date=1994; citation_pages=5-161; citation_id=CR6
citation_title=Cohomologie cristalline des schémas de caractéristique
; citation_publication_date=1974; citation_id=CR7; citation_author=P. Berthelot; citation_publisher=Springer
Berthelot, P.: Cohomologie rigide et cohomologie rigide à supports propres, Première partie (version provisoire 1991), Prépublication IRMAR (1996).
https://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf
citation_title=Notes on Crystalline Cohomology; citation_publication_date=1978; citation_id=CR9; citation_author=P. Berthelot; citation_author=A. Ogus; citation_publisher=Princeton University Press
Bhatt, B.:
$p$
-adic derived de Rham cohomology. Preprint (2012).
https://arxiv.org/abs/1204.6560
citation_journal_title=Camb. J. Math.; citation_title=Algebraization and Tannaka duality; citation_author=B. Bhatt; citation_volume=4; citation_issue=4; citation_publication_date=2016; citation_pages=403-461; citation_id=CR11
citation_title=Specializing varieties and their cohomology from characteristic 0 to characteristic
; citation_inbook_title=Algebraic Geometry: Salt Lake City 2015; citation_publication_date=2018; citation_pages=43-88; citation_id=CR12; citation_author=B. Bhatt; citation_publisher=Am. Math. Soc.
Bhatt, B.: Cohen-Macaulayness of absolute integral closures. Preprint (2020).
https://arxiv.org/abs/2008.08070
Bhatt, B., de Jong, A.J.: Crystalline cohomology and de Rham cohomology. Preprint (2011).
https://arxiv.org/abs/1110.5001
Bhatt, B., Lurie, J.: Absolute prismatic cohomology. Preprint (2022).
https://arxiv.org/abs/2201.06120
Bhatt, B., Lurie, J.: The prismatization of
$p$
-adic formal schemes. Preprint (2022).
https://arxiv.org/abs/2201.06124
citation_journal_title=Duke Math. J.; citation_title=The arc-topology; citation_author=B. Bhatt, A. Mathew; citation_volume=170; citation_issue=9; citation_publication_date=2021; citation_pages=1899-1988; citation_id=CR17
citation_journal_title=Astérisque; citation_title=The pro-étale topology for schemes; citation_author=B. Bhatt, P. Scholze; citation_volume=369; citation_publication_date=2015; citation_pages=99-201; citation_id=CR18
citation_journal_title=Invent. Math.; citation_title=Projectivity of the Witt vector affine Grassmannian; citation_author=B. Bhatt, P. Scholze; citation_volume=209; citation_issue=2; citation_publication_date=2017; citation_pages=329-423; citation_id=CR19
citation_journal_title=Ann. Math. (2); citation_title=Prisms and prismatic cohomology; citation_author=B. Bhatt, P. Scholze; citation_volume=196; citation_issue=3; citation_publication_date=2022; citation_pages=1135-1275; citation_id=CR20
citation_journal_title=Camb. J. Math.; citation_title=Prismatic
-crystals and crystalline Galois representations; citation_author=B. Bhatt, P. Scholze; citation_volume=11; citation_issue=2; citation_publication_date=2023; citation_pages=507-562; citation_id=CR21
citation_journal_title=Publ. Math. Inst. Hautes Études Sci.; citation_title=Integral
-adic Hodge theory; citation_author=B. Bhatt, M. Morrow, P. Scholze; citation_volume=128; citation_publication_date=2018; citation_pages=219-397; citation_id=CR22
citation_journal_title=Publ. Math. Inst. Hautes Études Sci.; citation_title=Topological Hochschild homology and integral
-adic Hodge theory; citation_author=B. Bhatt, M. Morrow, P. Scholze; citation_volume=129; citation_publication_date=2019; citation_pages=199-310; citation_id=CR23
Bogdan, Z.: Mod-
$p$
Poincaré duality in
$p$
-adic analytic geometry. Preprint (2021).
https://arxiv.org/abs/2111.01830
citation_title=Non-Archimedean Analysis; citation_publication_date=1984; citation_id=CR25; citation_author=S. Bosch; citation_author=U. Güntzer; citation_author=R. Remmert; citation_publisher=Springer
Brinon, O.: Représentations
$p$
-adiques cristallines et de Rham dans le cas relatif. Mém. Soc. Math. Fr. (N. S.), no. 112, vi+159 (2008)
citation_journal_title=Ann. Math. (2); citation_title=Purity for flat cohomology; citation_author=K. Česnavičius, P. Scholze; citation_volume=199; citation_issue=1; citation_publication_date=2024; citation_pages=51-180; citation_id=CR27
Du, H., Liu, T., Suk Moon, Y., Shimizu, K.: Completed prismatic
$F$
-crystals and crystalline
$\mathbf{Z}_{p}$
-local systems. Preprint (2022).
https://arxiv.org/abs/2203.03444
citation_title=Crystalline cohomology and -adic Galois-representations; citation_inbook_title=Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988); citation_publication_date=1989; citation_pages=25-80; citation_id=CR29; citation_author=G. Faltings; citation_publisher=Johns Hopkins University
citation_journal_title=Ann. Math. (2); citation_title=Sur certains types de représentations
-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate; citation_author=J.-M. Fontaine; citation_volume=115; citation_issue=3; citation_publication_date=1982; citation_pages=529-577; citation_id=CR30
citation_title=
-Adic periods and -adic étale cohomology; citation_inbook_title=Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985); citation_publication_date=1987; citation_pages=179-207; citation_id=CR31; citation_author=J.-M. Fontaine; citation_author=W. Messing; citation_publisher=Am. Math. Soc.
Guo, H.: Crystalline cohomology of rigid analytic spaces. Preprint (2021).
https://arxiv.org/abs/2112.14304
Guo, H., Petrov, A., Yang, Z.: Pointwise criteria of
$p$
-adic local systems (2023, in preparation)
citation_journal_title=Proc. Am. Math. Soc.; citation_title=Some results on coherent rings; citation_author=M.E. Harris; citation_volume=17; citation_publication_date=1966; citation_pages=474-479; citation_id=CR34
citation_journal_title=Isr. J. Math.; citation_title=A homotopy theory for stacks; citation_author=S. Hollander; citation_volume=163; citation_publication_date=2008; citation_pages=93-124; citation_id=CR35
citation_journal_title=Math. Z.; citation_title=A generalization of formal schemes and rigid analytic varieties; citation_author=R. Huber; citation_volume=217; citation_issue=4; citation_publication_date=1994; citation_pages=513-551; citation_id=CR36
citation_title=Étale cohomology of rigid analytic varieties and adic spaces; citation_publication_date=1996; citation_id=CR37; citation_author=R. Huber; citation_publisher=Friedr. Vieweg & Sohn
citation_journal_title=J. Reine Angew. Math.; citation_title=Arc-descent for the perfect loop functor and
-adic Deligne-Lusztig spaces; citation_author=A.B. Ivanov; citation_volume=794; citation_publication_date=2023; citation_pages=1-54; citation_id=CR38
citation_journal_title=Tohoku Math. J. (2); citation_title=Syntomic cohomology and
-adic étale cohomology; citation_author=K. Kato, W. Messing; citation_volume=44; citation_issue=1; citation_publication_date=1992; citation_pages=1-9; citation_id=CR39
citation_title=
-Adic properties of modular schemes and modular forms; citation_inbook_title=Modular Functions of One Variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972); citation_publication_date=1973; citation_pages=69-190; citation_id=CR40; citation_author=N.M. Katz; citation_publisher=Springer
citation_journal_title=Ann. Math. (2); citation_title=A
-adic local monodromy theorem; citation_author=K.S. Kedlaya; citation_volume=160; citation_issue=1; citation_publication_date=2004; citation_pages=93-184; citation_id=CR41
citation_title=Some ring-theoretic properties of , -adic Hodge theory; citation_inbook_title=Simons Symp.; citation_publication_date=2020; citation_pages=129-141; citation_id=CR42; citation_author=K.S. Kedlaya; citation_publisher=Springer
citation_journal_title=Astérisque; citation_title=Relative
-adic Hodge theory: foundations; citation_author=K.S. Kedlaya, R. Liu; citation_volume=371; citation_publication_date=2015; citation_pages=239; citation_id=CR43
citation_journal_title=Invent. Math.; citation_title=Rigidity and a Riemann-Hilbert correspondence for
-adic local systems; citation_author=R. Liu, X. Zhu; citation_volume=207; citation_issue=1; citation_publication_date=2017; citation_pages=291-343; citation_id=CR44
citation_title=Higher Topos Theory; citation_publication_date=2009; citation_id=CR45; citation_author=J. Lurie; citation_publisher=Princeton University Press
citation_title=Higher Algebra; citation_publication_date=2017; citation_id=CR46; citation_author=J. Lurie
Mann, L.: A
$p$
-adic 6-functor formalism in rigid-analytic geometry. Preprint (2022).
https://arxiv.org/abs/2206.02022
citation_journal_title=Adv. Math.; citation_title=The Galois group of a stable homotopy theory; citation_author=A. Mathew; citation_volume=291; citation_publication_date=2016; citation_pages=403-541; citation_id=CR48
citation_journal_title=J. Pure Appl. Algebra; citation_title=Faithfully flat descent of almost perfect complexes in rigid geometry; citation_author=A. Mathew; citation_volume=226; citation_issue=5; citation_publication_date=2022; citation_id=CR49
Min, Y., Wang, Y.: Relative
$(\varphi ,\Gamma )$
-modules and prismatic
$F$
-crystals. Preprint (2021).
https://arxiv.org/abs/2110.06076
Morrow, M., Tsuji, T.: Generalised representations as q-connections in integral
$p$
-adic Hodge theory. Preprint (2020)
https://arxiv.org/abs/2010.04059
Nie, T.: Absolute Hodge cycles in prismatic cohomology. Ph.D. thesis (2021).
https://dash.harvard.edu/handle/1/37370189
citation_journal_title=Ann. Sci. Éc. Norm. Supér. (4); citation_title=Crystalline conjecture via
-theory; citation_author=W. Nizioł; citation_volume=31; citation_issue=5; citation_publication_date=1998; citation_pages=659-681; citation_id=CR53
citation_journal_title=Duke Math. J.; citation_title=
-Isocrystals and de Rham cohomology. II. Convergent isocrystals; citation_author=A. Ogus; citation_volume=51; citation_issue=4; citation_publication_date=1984; citation_pages=765-850; citation_id=CR54
citation_title=The convergent topos in characteristic
; citation_inbook_title=The Grothendieck Festschrift, Vol. III; citation_publication_date=1990; citation_pages=133-162; citation_id=CR55; citation_author=A. Ogus; citation_publisher=Birkhäuser
Ogus, A.: Crystalline prisms: reflections on the present and past. Preprint (2023).
https://arxiv.org/abs/2204.06621
citation_journal_title=Forum Math. Pi; citation_title=
-Adic Hodge theory for rigid-analytic varieties; citation_author=P. Scholze; citation_volume=1; citation_publication_date=2013; citation_id=CR57
citation_journal_title=Forum Math. Pi; citation_title=
-Adic Hodge theory for rigid-analytic varieties—corrigendum; citation_author=P. Scholze; citation_volume=4; citation_publication_date=2016; citation_id=CR58
Scholze, P.: Étale cohomology of diamonds. Preprint (2017).
https://arxiv.org/abs/1709.07343
citation_title=Berkeley Lectures on -Adic Geometry; citation_publication_date=2020; citation_id=CR60; citation_author=P. Scholze; citation_author=J. Weinstein; citation_publisher=Princeton University Press
Shiho, A.: Relative log convergent cohomology and relative rigid cohomology I. Preprint (2007).
https://arxiv.org/abs/0707.1742
citation_journal_title=Compos. Math.; citation_title=A
-adic monodromy theorem for de Rham local systems; citation_author=K. Shimizu; citation_volume=158; citation_issue=12; citation_publication_date=2022; citation_pages=2157-2205; citation_id=CR62
citation_journal_title=Algebra Number Theory; citation_title=Crystalline comparison isomorphisms in
-adic Hodge theory: the absolutely unramified case; citation_author=F. Tan, J. Tong; citation_volume=13; citation_issue=7; citation_publication_date=2019; citation_pages=1509-1581; citation_id=CR63
Tang, L.: Syntomic cycle classes and prismatic Poincaré duality. Preprint (2022).
https://arxiv.org/abs/2210.14279
The Stacks Project Authors: Stacks Project (2022).
https://stacks.math.columbia.edu
citation_journal_title=J. Reine Angew. Math.; citation_title=Finiteness and duality for the cohomology of prismatic crystals; citation_author=Y. Tian; citation_volume=800; citation_publication_date=2023; citation_pages=217-257; citation_id=CR66
citation_journal_title=Invent. Math.; citation_title=
-Adic étale cohomology and crystalline cohomology in the semi-stable reduction case; citation_author=T. Tsuji; citation_volume=137; citation_issue=2; citation_publication_date=1999; citation_pages=233-411; citation_id=CR67
citation_journal_title=Trans. Am. Math. Soc.; citation_title=On finitely generated flat modules; citation_author=W.V. Vasconcelos; citation_volume=138; citation_publication_date=1969; citation_pages=505-512; citation_id=CR68
citation_journal_title=Compos. Math.; citation_title=On higher direct images of convergent isocrystals; citation_author=D. Xu; citation_volume=155; citation_issue=11; citation_publication_date=2019; citation_pages=2180-2213; citation_id=CR69
citation_journal_title=Ann. Math. (2); citation_title=Affine Grassmannians and the geometric Satake in mixed characteristic; citation_author=X. Zhu; citation_volume=185; citation_issue=2; citation_publication_date=2017; citation_pages=403-492; citation_id=CR70