Several forms of Drinfeldβs lemma Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/ Recent Advances in Modern π -Adic Geometry virtual seminar November 12, 2020 Supported by NSF (grant DMS-1802161) and UC San Diego (Warschawski Professorship). Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 1 / 30
Drinfeldβs lemma for schemes Contents 1 Drinfeldβs lemma for schemes 2 Drinfeldβs lemma for perfectoid spaces (and diamonds) 3 Drinfeldβs lemma for πΊ -isocrystals Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 2 / 30
Drinfeldβs lemma for schemes References for this section Eike Lau, On generalised π -shtukas, PhD thesis (Bonn, 2004), pdf. KSK, Sheaves, stacks, and shtukas, Arizona Winter School 2017 (pdf). Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 3 / 30
Drinfeldβs lemma for schemes Setup: a formal quotient by Frobenius π = a scheme over πΎ π π = an algebraically closed fjeld of characteristic π We will consider β π π /π π β is a formal quotient: an object of some type isomorphism with its π π -pullback. Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 4 / 30 π π = π Γ πΎ π π π π = the pullback to π π of the absolute Frobenius on Spec π over π π /π π is an object of the same type over π π equipped with an
Drinfeldβs lemma for schemes Coherent sheaves Theorem (Drinfeld, Lau) (coherent sheaves on π ) β (coherent sheaves on π π /π π ) is an equivalence of categories and preserves cohomology. Idea of proof: reduce to π projective, trivialize π π -action on πΌ 0 (π, β°(π)) . Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 5 / 30 For π/πΎ π of fjnite type, the base extension functor
Drinfeldβs lemma for schemes Finite Γ©tale covers and profjnite fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeldβs lemma Kiran S. Kedlaya Warning: in general π 0 (π π ) β π 0 (π) . For example, if π = Spec β is a 1 1 π¦ β π π , π prof Corollary For any π , FEt (π) β FEt (π π /π π ) is an equivalence. Corollary 6 / 30 For π connected, π π /π π is connected and for any geometric point (π π /π π , π¦) β π prof (π, π¦) . geometric point, π 0 (π π ) β Μ β€ indexed by identifjcations of the copies of πΎ π in π and β ; but π π acts on π 0 (π π ) by translation by β€ .
Drinfeldβs lemma for schemes Finite Γ©tale covers and profjnite fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeldβs lemma Kiran S. Kedlaya Warning: in general π 0 (π π ) β π 0 (π) . For example, if π = Spec β is a 1 1 π¦ β π π , π prof Corollary For any π , FEt (π) β FEt (π π /π π ) is an equivalence. Corollary 6 / 30 For π connected, π π /π π is connected and for any geometric point (π π /π π , π¦) β π prof (π, π¦) . geometric point, π 0 (π π ) β Μ β€ indexed by identifjcations of the copies of πΎ π in π and β ; but π π acts on π 0 (π π ) by translation by β€ .
Drinfeldβs lemma for schemes Products of two (or more) fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeldβs lemma Kiran S. Kedlaya 2 1 7 / 30 1 π prof qcqs, and for any geometric point π¦ β π , Corollary For π 1 , π 2 two connected qcqs πΎ π -schemes, put π = π 1 Γ πΎ π π 2 and let π 1 , π 2 βΆ π β π be the partial Frobenius maps. Then π/π 2 is connected (π/π 2 , π¦) β π prof (π 1 , π¦) Γ π prof (π 2 , π¦).
Drinfeldβs lemma for schemes Open subschemes and Γ©tale sheaves Corollary Corollary For π any πΎ π -scheme and β β π prime, are equivalences of categories and preserve cohomology. (And so on.) Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 8 / 30 For any π , quasicompact open subschemes of π and π π /π π are the same. (lisse β β -sheaves on π ) β (lisse β β -sheaves on π π /π π ) (constructible β β -sheaves on π ) β (constructible β β -sheaves on π π /π π )
Drinfeldβs lemma for schemes Context: shtukas and excursion operators These constructions are used to describe excursion operators on moduli stacks of shtukas, in order to describe the Langlands correspondence per V. Lafgorgue. (See last weekβs seminar!) Similarly, other forms of Drinfeldβs lemma are needed to do likewise for local Langlands in mixed characteristic, or for π -adic coeffjcients in positive characteristic. Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 9 / 30
Drinfeldβs lemma for perfectoid spaces (and diamonds) Contents 1 Drinfeldβs lemma for schemes 2 Drinfeldβs lemma for perfectoid spaces (and diamonds) 3 Drinfeldβs lemma for πΊ -isocrystals Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 10 / 30
Drinfeldβs lemma for perfectoid spaces (and diamonds) References for this section CarterβKSKβZΓ‘brΓ‘di, Drinfeldβs lemma for perfectoid spaces and (2020). KSK, Sheaves, stacks, and shtukas, Arizona Winter School 2017 (pdf). (2018). ScholzeβWeinstein, Berkeley Lectures on π -adic Geometry (pdf). Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 11 / 30 overconvergence of multivariate (π, Ξ) -modules, arXiv:1808.03964v2 KSK, Simple connectivity of Fargues-Fontaine curves, arXiv:1806.11528v3
Drinfeldβs lemma for perfectoid spaces (and diamonds) Absolute products of perfectoid spaces Let Pfd be the category of perfectoid spaces in characteristic π . This category admits absolute products. (π’,π£) [π’ β1 π£ β1 ] βΆ π€(π’), π€(π£) < 1}, which is not quasicompact! Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 12 / 30 For example, if π 1 = Spa πΎ π ((π’ π ββ )) , π 2 = Spa πΎ π ((π£ π ββ )) , then π 1 Γ π 2 = {π€ β Spa πΎ π οΏ½ π’, π£ οΏ½ [π’ βπ β , π£ π ββ ] β¨
Drinfeldβs lemma for perfectoid spaces (and diamonds) Quotients by partial Frobenius Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 13 / 30 For π 1 , π 2 β Pfd , put π = π 1 Γ π 2 . This space admits partial Frobenius operators π 1 , π 2 . Unlike for schemes, however, π/π 2 is an object of Pfd ! Moreover, if π 1 , π 2 are quasicompact, then so is π/π 2 .
Drinfeldβs lemma for perfectoid spaces (and diamonds) Product with a geometric point Theorem Fargues-Fontaine curve for π 1 . Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 14 / 30 For π 2 a geometric point, FEt (π 1 ) β FEt (π/π 2 ) is an equivalence. This reduces to the case where π 1 is itself a geometric point. When π 2 = Spa β β π , this can be proved by interpreting π/π 2 in terms of the
Drinfeldβs lemma for perfectoid spaces (and diamonds) Product with a geometric point Theorem completion of πΏ β² (π’) . A direct calculation rules out abelian covers; one then uses π -adic difgerential equations to construct a βramifjcation fjltrationβ to reduce to the abelian case. Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 15 / 30 For π 2 a geometric point, FEt (π 1 ) β FEt (π/π 2 ) is an equivalence. For general π 2 = Spa πΏ , we reduce from πΏ to πΏ β² where πΏ is a
Drinfeldβs lemma for perfectoid spaces (and diamonds) Products of two (or more) fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeldβs lemma Kiran S. Kedlaya multivariate (π, Ξ) -modules (see CarterβKSKβZΓ‘brΓ‘di). 2 1 π -adic representations of π prof A similar statement holds for diamonds. This can be used to describe (π 2 , π¦). 2 (π 1 , π¦) Γ π prof 1 (π/π 2 , π¦) β π prof 1 π prof geometric point, Corollary 16 / 30 For π 1 , π 2 β Pfd connected qcqs, π/π 2 is connected. For π¦ β π a (π 1 , π¦) Γ π prof (π 2 , π¦) in terms of
Drinfeldβs lemma for perfectoid spaces (and diamonds) Products of two (or more) fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeldβs lemma Kiran S. Kedlaya FarguesβFontaine curve? And how to classify the latter? related to vector bundles on the (relative) square of the relative (π 2 , π¦) 2 1 (π 2 , π¦). 2 (π 1 , π¦) Γ π prof 1 (π/π 2 , π¦) β π prof 1 π prof geometric point, Corollary 17 / 30 For π 1 , π 2 β Pfd connected qcqs, π/π 2 is connected qcqs. For π¦ β π a When π 1 = π 2 , are π -adic representations of π prof (π 1 , π¦) Γ π prof
Drinfeldβs lemma for perfectoid spaces (and diamonds) More questions Is there a version for constructible sheaves? (See FarguesβScholze?) Does this build towards an β β = π β Langlands correspondence for β π ? Kiran S. Kedlaya Several forms of Drinfeldβs lemma RAMpAGe, Nov. 12, 2020 18 / 30
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