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Several forms of Drinfelds lemma Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/ Recent Advances in Modern -Adic Geometry virtual seminar November 12, 2020

  1. Several forms of Drinfeld’s lemma Kiran S. Kedlaya Department of Mathematics, University of California, San Diego kedlaya@ucsd.edu http://kskedlaya.org/slides/ Recent Advances in Modern π‘ž -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 Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 1 / 30

  2. Drinfeld’s lemma for schemes Contents 1 Drinfeld’s lemma for schemes 2 Drinfeld’s lemma for perfectoid spaces (and diamonds) 3 Drinfeld’s lemma for 𝐺 -isocrystals Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 2 / 30

  3. Drinfeld’s lemma for schemes References for this section Eike Lau, On generalised 𝒠 -shtukas, PhD thesis (Bonn, 2004), pdf. KSK, Sheaves, stacks, and shtukas, Arizona Winter School 2017 (pdf). Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 3 / 30

  4. Drinfeld’s lemma for schemes Setup: a formal quotient by Frobenius π‘Œ = a scheme over 𝔾 π‘ž 𝑙 = an algebraically closed fjeld of characteristic π‘ž We will consider β€œ π‘Œ 𝑙 /πœ’ 𝑙 ” is a formal quotient: an object of some type isomorphism with its πœ’ 𝑙 -pullback. Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 4 / 30 π‘Œ 𝑙 = π‘Œ Γ— 𝔾 π‘ž 𝑙 πœ’ 𝑙 = the pullback to π‘Œ 𝑙 of the absolute Frobenius on Spec 𝑙 over π‘Œ 𝑙 /πœ’ 𝑙 is an object of the same type over π‘Œ 𝑙 equipped with an

  5. Drinfeld’s lemma for schemes Coherent sheaves Theorem (Drinfeld, Lau) (coherent sheaves on π‘Œ ) β†’ (coherent sheaves on π‘Œ 𝑙 /πœ’ 𝑙 ) is an equivalence of categories and preserves cohomology. Idea of proof: reduce to π‘Œ projective, trivialize πœ’ 𝑙 -action on 𝐼 0 (π‘Œ, β„°(π‘œ)) . Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 5 / 30 For π‘Œ/𝔾 π‘ž of fjnite type, the base extension functor

  6. Drinfeld’s lemma for schemes Finite Γ©tale covers and profjnite fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeld’s lemma Kiran S. Kedlaya Warning: in general 𝜌 0 (π‘Œ 𝑙 ) β‰  𝜌 0 (π‘Œ) . For example, if π‘Œ = Spec β„“ is a 1 1 𝑦 β†’ π‘Œ 𝑙 , 𝜌 prof Corollary For any π‘Œ , FEt (π‘Œ) β†’ FEt (π‘Œ 𝑙 /πœ’ 𝑙 ) is an equivalence. Corollary 6 / 30 For π‘Œ connected, π‘Œ 𝑙 /πœ’ 𝑙 is connected and for any geometric point (π‘Œ 𝑙 /πœ’ 𝑙 , 𝑦) β‰… 𝜌 prof (π‘Œ, 𝑦) . geometric point, 𝜌 0 (π‘Œ 𝑙 ) β‰… Μ‚ β„€ indexed by identifjcations of the copies of 𝔾 π‘ž in 𝑙 and β„“ ; but πœ’ 𝑙 acts on 𝜌 0 (π‘Œ 𝑙 ) by translation by β„€ .

  7. Drinfeld’s lemma for schemes Finite Γ©tale covers and profjnite fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeld’s lemma Kiran S. Kedlaya Warning: in general 𝜌 0 (π‘Œ 𝑙 ) β‰  𝜌 0 (π‘Œ) . For example, if π‘Œ = Spec β„“ is a 1 1 𝑦 β†’ π‘Œ 𝑙 , 𝜌 prof Corollary For any π‘Œ , FEt (π‘Œ) β†’ FEt (π‘Œ 𝑙 /πœ’ 𝑙 ) is an equivalence. Corollary 6 / 30 For π‘Œ connected, π‘Œ 𝑙 /πœ’ 𝑙 is connected and for any geometric point (π‘Œ 𝑙 /πœ’ 𝑙 , 𝑦) β‰… 𝜌 prof (π‘Œ, 𝑦) . geometric point, 𝜌 0 (π‘Œ 𝑙 ) β‰… Μ‚ β„€ indexed by identifjcations of the copies of 𝔾 π‘ž in 𝑙 and β„“ ; but πœ’ 𝑙 acts on 𝜌 0 (π‘Œ 𝑙 ) by translation by β„€ .

  8. Drinfeld’s lemma for schemes Products of two (or more) fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeld’s lemma Kiran S. Kedlaya 2 1 7 / 30 1 𝜌 prof qcqs, and for any geometric point 𝑦 β†’ π‘Œ , Corollary For π‘Œ 1 , π‘Œ 2 two connected qcqs 𝔾 π‘ž -schemes, put π‘Œ = π‘Œ 1 Γ— 𝔾 π‘ž π‘Œ 2 and let πœ’ 1 , πœ’ 2 ∢ π‘Œ β†’ π‘Œ be the partial Frobenius maps. Then π‘Œ/πœ’ 2 is connected (π‘Œ/πœ’ 2 , 𝑦) β‰… 𝜌 prof (π‘Œ 1 , 𝑦) Γ— 𝜌 prof (π‘Œ 2 , 𝑦).

  9. Drinfeld’s lemma for schemes Open subschemes and Γ©tale sheaves Corollary Corollary For π‘Œ any 𝔾 π‘ž -scheme and β„“ β‰  π‘ž prime, are equivalences of categories and preserve cohomology. (And so on.) Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 8 / 30 For any π‘Œ , quasicompact open subschemes of π‘Œ and π‘Œ 𝑙 /πœ’ 𝑙 are the same. (lisse β„š β„“ -sheaves on π‘Œ ) β†’ (lisse β„š β„“ -sheaves on π‘Œ 𝑙 /πœ’ 𝑙 ) (constructible β„š β„“ -sheaves on π‘Œ ) β†’ (constructible β„š β„“ -sheaves on π‘Œ 𝑙 /πœ’ 𝑙 )

  10. Drinfeld’s 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 week’s seminar!) Similarly, other forms of Drinfeld’s lemma are needed to do likewise for local Langlands in mixed characteristic, or for π‘ž -adic coeffjcients in positive characteristic. Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 9 / 30

  11. Drinfeld’s lemma for perfectoid spaces (and diamonds) Contents 1 Drinfeld’s lemma for schemes 2 Drinfeld’s lemma for perfectoid spaces (and diamonds) 3 Drinfeld’s lemma for 𝐺 -isocrystals Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 10 / 30

  12. Drinfeld’s lemma for perfectoid spaces (and diamonds) References for this section Carter–KSK–ZΓ‘brΓ‘di, Drinfeld’s lemma for perfectoid spaces and (2020). KSK, Sheaves, stacks, and shtukas, Arizona Winter School 2017 (pdf). (2018). Scholze–Weinstein, Berkeley Lectures on π‘ž -adic Geometry (pdf). Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 11 / 30 overconvergence of multivariate (πœ’, Ξ“) -modules, arXiv:1808.03964v2 KSK, Simple connectivity of Fargues-Fontaine curves, arXiv:1806.11528v3

  13. Drinfeld’s lemma for perfectoid spaces (and diamonds) Absolute products of perfectoid spaces Let Pfd be the category of perfectoid spaces in characteristic π‘ž . This category admits absolute products. (𝑒,𝑣) [𝑒 βˆ’1 𝑣 βˆ’1 ] ∢ 𝑀(𝑒), 𝑀(𝑣) < 1}, which is not quasicompact! Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 12 / 30 For example, if π‘Œ 1 = Spa 𝔾 π‘ž ((𝑒 π‘ž βˆ’βˆž )) , π‘Œ 2 = Spa 𝔾 π‘ž ((𝑣 π‘ž βˆ’βˆž )) , then π‘Œ 1 Γ— π‘Œ 2 = {𝑀 ∈ Spa 𝔾 π‘ž οΏ½ 𝑒, 𝑣 οΏ½ [𝑒 βˆ’π‘ž ∞ , 𝑣 π‘ž βˆ’βˆž ] ∨

  14. Drinfeld’s lemma for perfectoid spaces (and diamonds) Quotients by partial Frobenius Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 13 / 30 For π‘Œ 1 , π‘Œ 2 ∈ Pfd , put π‘Œ = π‘Œ 1 Γ— π‘Œ 2 . This space admits partial Frobenius operators πœ’ 1 , πœ’ 2 . Unlike for schemes, however, π‘Œ/πœ’ 2 is an object of Pfd ! Moreover, if π‘Œ 1 , π‘Œ 2 are quasicompact, then so is π‘Œ/πœ’ 2 .

  15. Drinfeld’s lemma for perfectoid spaces (and diamonds) Product with a geometric point Theorem Fargues-Fontaine curve for π‘Œ 1 . Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 14 / 30 For π‘Œ 2 a geometric point, FEt (π‘Œ 1 ) β†’ FEt (π‘Œ/πœ’ 2 ) is an equivalence. This reduces to the case where π‘Œ 1 is itself a geometric point. When π‘Œ 2 = Spa β„‚ β™­ π‘ž , this can be proved by interpreting π‘Œ/πœ’ 2 in terms of the

  16. Drinfeld’s lemma for perfectoid spaces (and diamonds) Product with a geometric point Theorem completion of 𝐿 β€² (𝑒) . A direct calculation rules out abelian covers; one then uses π‘ž -adic difgerential equations to construct a β€œramifjcation fjltration” to reduce to the abelian case. Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 15 / 30 For π‘Œ 2 a geometric point, FEt (π‘Œ 1 ) β†’ FEt (π‘Œ/πœ’ 2 ) is an equivalence. For general π‘Œ 2 = Spa 𝐿 , we reduce from 𝐿 to 𝐿 β€² where 𝐿 is a

  17. Drinfeld’s lemma for perfectoid spaces (and diamonds) Products of two (or more) fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeld’s lemma Kiran S. Kedlaya multivariate (πœ’, Ξ“) -modules (see Carter–KSK–ZΓ‘brΓ‘di). 2 1 π‘ž -adic representations of 𝜌 prof A similar statement holds for diamonds. This can be used to describe (π‘Œ 2 , 𝑦). 2 (π‘Œ 1 , 𝑦) Γ— 𝜌 prof 1 (π‘Œ/πœ’ 2 , 𝑦) β‰… 𝜌 prof 1 𝜌 prof geometric point, Corollary 16 / 30 For π‘Œ 1 , π‘Œ 2 ∈ Pfd connected qcqs, π‘Œ/πœ’ 2 is connected. For 𝑦 β†’ π‘Œ a (π‘Œ 1 , 𝑦) Γ— 𝜌 prof (π‘Œ 2 , 𝑦) in terms of

  18. Drinfeld’s lemma for perfectoid spaces (and diamonds) Products of two (or more) fundamental groups RAMpAGe, Nov. 12, 2020 Several forms of Drinfeld’s lemma Kiran S. Kedlaya Fargues–Fontaine curve? And how to classify the latter? related to vector bundles on the (relative) square of the relative (π‘Œ 2 , 𝑦) 2 1 (π‘Œ 2 , 𝑦). 2 (π‘Œ 1 , 𝑦) Γ— 𝜌 prof 1 (π‘Œ/πœ’ 2 , 𝑦) β‰… 𝜌 prof 1 𝜌 prof geometric point, Corollary 17 / 30 For π‘Œ 1 , π‘Œ 2 ∈ Pfd connected qcqs, π‘Œ/πœ’ 2 is connected qcqs. For 𝑦 β†’ π‘Œ a When π‘Œ 1 = π‘Œ 2 , are π‘ž -adic representations of 𝜌 prof (π‘Œ 1 , 𝑦) Γ— 𝜌 prof

  19. Drinfeld’s lemma for perfectoid spaces (and diamonds) More questions Is there a version for constructible sheaves? (See Fargues–Scholze?) Does this build towards an β€œ β„“ = π‘ž ” Langlands correspondence for β„š π‘ž ? Kiran S. Kedlaya Several forms of Drinfeld’s lemma RAMpAGe, Nov. 12, 2020 18 / 30

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