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IntroductionÉtale cohomology of rigid spacesRelative perversityRecent developments in étale cohomologyDavid HansenMPIM BonnNovember 10, 2021David HansenRecent developments in étale cohomology1 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityIntroductionRecall: Étale cohomology is the correct analogue of singular cohomologyin algebraic geometry. Invented to prove the Weil conjectures, but now ofcentral importance for many different reasons.David HansenRecent developments in étale cohomology2 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityIntroductionRecall: Étale cohomology is the correct analogue of singular cohomologyin algebraic geometry. Invented to prove the Weil conjectures, but now ofcentral importance for many different reasons.Key points: For any reasonable scheme, have a category Dcb (X , Q )which satisfies a six operations formalism; Lefschetz trace formula; theoryof weights for varieties over finite fields.David HansenRecent developments in étale cohomology2 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA misconceptionIn talking to graduate students, I’ve noticed a common misconceptionthat étale cohomology is a “dead” / “static” / “ancient” subject, withnothing left to be done.David HansenRecent developments in étale cohomology3 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA misconceptionIn talking to graduate students, I’ve noticed a common misconceptionthat étale cohomology is a “dead” / “static” / “ancient” subject, withnothing left to be done.Figure: The creation of étale cohomologyDavid HansenRecent developments in étale cohomology3 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA misconceptionIn talking to graduate students, I’ve noticed a common misconceptionthat étale cohomology is a “dead” / “static” / “ancient” subject, withnothing left to be done.Figure: The creation of étale cohomologyIt is true that many mathematicians can profitably use étale cohomologyas a black box, never looking beyond Freitag-Kiehl or Milne.David HansenRecent developments in étale cohomology3 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA misconceptionIn talking to graduate students, I’ve noticed a common misconceptionthat étale cohomology is a “dead” / “static” / “ancient” subject, withnothing left to be done.Figure: The creation of étale cohomologyIt is true that many mathematicians can profitably use étale cohomologyas a black box, never looking beyond Freitag-Kiehl or Milne. However, itis not a dead subject!David HansenRecent developments in étale cohomology3 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyDavid HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyFirst mentioned (without name) in Grothendieck’s 1958 ICM report.David HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyFirst mentioned (without name) in Grothendieck’s 1958 ICM report.Key foundations laid in SGA4 (M. Artin, Deligne, Grothendieck,Verdier): Basic definitions, construction of the six operations,smooth and proper base change, Poincaré duality, comparison withsingular cohomology for complex varieties, affine vanishing,(conditional) finiteness and biduality theorems in char. 0. All withtorsion coefficients.David HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyFirst mentioned (without name) in Grothendieck’s 1958 ICM report.Key foundations laid in SGA4 (M. Artin, Deligne, Grothendieck,Verdier): Basic definitions, construction of the six operations,smooth and proper base change, Poincaré duality, comparison withsingular cohomology for complex varieties, affine vanishing,(conditional) finiteness and biduality theorems in char. 0. All withtorsion coefficients.SGA 4 1/2 (Deligne): Unconditional finiteness and bidualitytheorems for schemes of finite type over regular bases of dimension 1, complete proof of the Lefschetz trace formula for Frobenius.David HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyFirst mentioned (without name) in Grothendieck’s 1958 ICM report.Key foundations laid in SGA4 (M. Artin, Deligne, Grothendieck,Verdier): Basic definitions, construction of the six operations,smooth and proper base change, Poincaré duality, comparison withsingular cohomology for complex varieties, affine vanishing,(conditional) finiteness and biduality theorems in char. 0. All withtorsion coefficients.SGA 4 1/2 (Deligne): Unconditional finiteness and bidualitytheorems for schemes of finite type over regular bases of dimension 1, complete proof of the Lefschetz trace formula for Frobenius.“This report should allow the user to forget about SGA 5, which canbe considered as a series of digressions, some very interesting.”David HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyFirst mentioned (without name) in Grothendieck’s 1958 ICM report.Key foundations laid in SGA4 (M. Artin, Deligne, Grothendieck,Verdier): Basic definitions, construction of the six operations,smooth and proper base change, Poincaré duality, comparison withsingular cohomology for complex varieties, affine vanishing,(conditional) finiteness and biduality theorems in char. 0. All withtorsion coefficients.SGA 4 1/2 (Deligne): Unconditional finiteness and bidualitytheorems for schemes of finite type over regular bases of dimension 1, complete proof of the Lefschetz trace formula for Frobenius.“This report should allow the user to forget about SGA 5, which canbe considered as a series of digressions, some very interesting.”Deligne’s Weil I paper: Unconditional definition of Dcb (X , Q ) w. sixoperations formalism for X a variety over any finite or alg. closedfield. Enough to prove the Weil conjectures.David HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityA brief and subjective historyFirst mentioned (without name) in Grothendieck’s 1958 ICM report.Key foundations laid in SGA4 (M. Artin, Deligne, Grothendieck,Verdier): Basic definitions, construction of the six operations,smooth and proper base change, Poincaré duality, comparison withsingular cohomology for complex varieties, affine vanishing,(conditional) finiteness and biduality theorems in char. 0. All withtorsion coefficients.SGA 4 1/2 (Deligne): Unconditional finiteness and bidualitytheorems for schemes of finite type over regular bases of dimension 1, complete proof of the Lefschetz trace formula for Frobenius.“This report should allow the user to forget about SGA 5, which canbe considered as a series of digressions, some very interesting.”Deligne’s Weil I paper: Unconditional definition of Dcb (X , Q ) w. sixoperations formalism for X a variety over any finite or alg. closedfield. Enough to prove the Weil conjectures.End of the initial period of development.David HansenRecent developments in étale cohomology4 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:Beilinson-Bernstein-Deligne-Gabber ’83: Perverse sheaves,t-structures, decomposition theorem, purity for intersectioncohomology.David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:Beilinson-Bernstein-Deligne-Gabber ’83: Perverse sheaves,t-structures, decomposition theorem, purity for intersectioncohomology.Thomason ’84, Gabber ’94: Proof of Grothendieck’s absolute purityconjecture, by infusion of ideas from K -theory.David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:Beilinson-Bernstein-Deligne-Gabber ’83: Perverse sheaves,t-structures, decomposition theorem, purity for intersectioncohomology.Thomason ’84, Gabber ’94: Proof of Grothendieck’s absolute purityconjecture, by infusion of ideas from K -theory.Ekedahl ’90, Bhatt-Scholze ’15: Proper development of theformalism with Q -coefficients.David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:Beilinson-Bernstein-Deligne-Gabber ’83: Perverse sheaves,t-structures, decomposition theorem, purity for intersectioncohomology.Thomason ’84, Gabber ’94: Proof of Grothendieck’s absolute purityconjecture, by infusion of ideas from K -theory.Ekedahl ’90, Bhatt-Scholze ’15: Proper development of theformalism with Q -coefficients.Gabber, late ’00s: New proof of absolute purity, optimal finitenessand biduality theorems for excellent schemes. Very sophisticatedarguments.David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:Beilinson-Bernstein-Deligne-Gabber ’83: Perverse sheaves,t-structures, decomposition theorem, purity for intersectioncohomology.Thomason ’84, Gabber ’94: Proof of Grothendieck’s absolute purityconjecture, by infusion of ideas from K -theory.Ekedahl ’90, Bhatt-Scholze ’15: Proper development of theformalism with Q -coefficients.Gabber, late ’00s: New proof of absolute purity, optimal finitenessand biduality theorems for excellent schemes. Very sophisticatedarguments.Laszlo-Olsson ’05-’06, Liu-Zheng, ’12: Flexible six operationsformalism for sheaves on Artin stacks.David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityHistory cont’dSome key later developments:Beilinson-Bernstein-Deligne-Gabber ’83: Perverse sheaves,t-structures, decomposition theorem, purity for intersectioncohomology.Thomason ’84, Gabber ’94: Proof of Grothendieck’s absolute purityconjecture, by infusion of ideas from K -theory.Ekedahl ’90, Bhatt-Scholze ’15: Proper development of theformalism with Q -coefficients.Gabber, late ’00s: New proof of absolute purity, optimal finitenessand biduality theorems for excellent schemes. Very sophisticatedarguments.Laszlo-Olsson ’05-’06, Liu-Zheng, ’12: Flexible six operationsformalism for sheaves on Artin stacks.So much for being a dead subject. Is there still anything left to be done?David HansenRecent developments in étale cohomology5 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityÉtale cohomology of rigid spacesNatural question: Is there a good étale cohomology formalism for rigidanalytic spaces?David HansenRecent developments in étale cohomology6 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityÉtale cohomology of rigid spacesNatural question: Is there a good étale cohomology formalism for rigidanalytic spaces? The answer should obviously be “yes”, but setting upthe formalism presents some new challenges.David HansenRecent developments in étale cohomology6 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityÉtale cohomology of rigid spacesNatural question: Is there a good étale cohomology formalism for rigidanalytic spaces? The answer should obviously be “yes”, but setting upthe formalism presents some new challenges.Foundations laid by Berkovich and Huber in the ’90s: Construction of thesix operations, smooth and proper base change, Poincaré duality, somecomparison and finiteness theorems.David HansenRecent developments in étale cohomology6 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityÉtale cohomology of rigid spacesNatural question: Is there a good étale cohomology formalism for rigidanalytic spaces? The answer should obviously be “yes”, but setting upthe formalism presents some new challenges.Foundations laid by Berkovich and Huber in the ’90s: Construction of thesix operations, smooth and proper base change, Poincaré duality, somecomparison and finiteness theorems.MISSING: A natural class of sheaves (with torsion or Z -coefficients)stable under the six operations, admitting a perverse t-structure,satisfying affine vanishing, etc.David HansenRecent developments in étale cohomology6 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesDavid HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesLet X be a rigid space, Λ a Noetherian ring. Key new definition:David HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesLet X be a rigid space, Λ a Noetherian ring. Key new definition:Definition An étale sheaf F Sh(X , Λ) is Zariski-constructible if there is alocally finite stratification X i I Xi for some Zariski locally closedsubsets Xi X such that F Xi is finite locally constant for all i I .David HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesLet X be a rigid space, Λ a Noetherian ring. Key new definition:Definition An étale sheaf F Sh(X , Λ) is Zariski-constructible if there is alocally finite stratification X i I Xi for some Zariski locally closedsubsets Xi X such that F Xi is finite locally constant for all i I . A complex A D(X , Λ) is Zariski-constructible if all cohomologysheaves are.David HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesLet X be a rigid space, Λ a Noetherian ring. Key new definition:Definition An étale sheaf F Sh(X , Λ) is Zariski-constructible if there is alocally finite stratification X i I Xi for some Zariski locally closedsubsets Xi X such that F Xi is finite locally constant for all i I . A complex A D(X , Λ) is Zariski-constructible if all cohomologysheaves are.(In first part, can replace “locally finite” with “finite” unless dim X .)David HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesLet X be a rigid space, Λ a Noetherian ring. Key new definition:Definition An étale sheaf F Sh(X , Λ) is Zariski-constructible if there is alocally finite stratification X i I Xi for some Zariski locally closedsubsets Xi X such that F Xi is finite locally constant for all i I . A complex A D(X , Λ) is Zariski-constructible if all cohomologysheaves are.(In first part, can replace “locally finite” with “finite” unless dim X .)Key motivation: If X is an algebraic variety, the natural pullbackSh(X , Λ) Sh(X an , Λ) carries constructible sheaves on X toZariski-constructible sheaves on X an .David HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheavesLet X be a rigid space, Λ a Noetherian ring. Key new definition:Definition An étale sheaf F Sh(X , Λ) is Zariski-constructible if there is alocally finite stratification X i I Xi for some Zariski locally closedsubsets Xi X such that F Xi is finite locally constant for all i I . A complex A D(X , Λ) is Zariski-constructible if all cohomologysheaves are.(In first part, can replace “locally finite” with “finite” unless dim X .)Key motivation: If X is an algebraic variety, the natural pullbackSh(X , Λ) Sh(X an , Λ) carries constructible sheaves on X toZariski-constructible sheaves on X an .However, NO interesting properties of these sheaves are obvious!David HansenRecent developments in étale cohomology7 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheaves cont’dUntil further notice: K a nonarchimedean field of characteristic zeroand residue characteristic p 0, Λ Z/nZ.David HansenRecent developments in étale cohomology8 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheaves cont’dUntil further notice: K a nonarchimedean field of characteristic zeroand residue characteristic p 0, Λ Z/nZ.Main theorem (Bhatt-H.)(b)On rigid spaces X /K , Dzc (X , Λ) is stable under the operations f , Rf for proper f , Rf! and Rf on lisse sheaves for Zariski-compactifiable f ,Rf ! if p - n or f is finite, and RH om (under a finite tor-dimensionassumption), and Verdier duality.David HansenRecent developments in étale cohomology8 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheaves cont’dUntil further notice: K a nonarchimedean field of characteristic zeroand residue characteristic p 0, Λ Z/nZ.Main theorem (Bhatt-H.)(b)On rigid spaces X /K , Dzc (X , Λ) is stable under the operations f , Rf for proper f , Rf! and Rf on lisse sheaves for Zariski-compactifiable f ,Rf ! if p - n or f is finite, and RH om (under a finite tor-dimension(b)assumption), and Verdier duality. Moreover, Dzc (X , Λ) carries a naturalperverse t-structure whose abelian heart satisfies all expected properties.David HansenRecent developments in étale cohomology8 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheaves cont’dUntil further notice: K a nonarchimedean field of characteristic zeroand residue characteristic p 0, Λ Z/nZ.Main theorem (Bhatt-H.)(b)On rigid spaces X /K , Dzc (X , Λ) is stable under the operations f , Rf for proper f , Rf! and Rf on lisse sheaves for Zariski-compactifiable f ,Rf ! if p - n or f is finite, and RH om (under a finite tor-dimension(b)assumption), and Verdier duality. Moreover, Dzc (X , Λ) carries a naturalperverse t-structure whose abelian heart satisfies all expected properties.All of these statements are compatible with their schematic counterpartsunder analytification, and with extensions of the base field.David HansenRecent developments in étale cohomology8 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheaves cont’dUntil further notice: K a nonarchimedean field of characteristic zeroand residue characteristic p 0, Λ Z/nZ.Main theorem (Bhatt-H.)(b)On rigid spaces X /K , Dzc (X , Λ) is stable under the operations f , Rf for proper f , Rf! and Rf on lisse sheaves for Zariski-compactifiable f ,Rf ! if p - n or f is finite, and RH om (under a finite tor-dimension(b)assumption), and Verdier duality. Moreover, Dzc (X , Λ) carries a naturalperverse t-structure whose abelian heart satisfies all expected properties.All of these statements are compatible with their schematic counterpartsunder analytification, and with extensions of the base field. Similarresults hold with Z -coefficients.David HansenRecent developments in étale cohomology8 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityZariski-constructible sheaves cont’dUntil further notice: K a nonarchimedean field of characteristic zeroand residue characteristic p 0, Λ Z/nZ.Main theorem (Bhatt-H.)(b)On rigid spaces X /K , Dzc (X , Λ) is stable under the operations f , Rf for proper f , Rf! and Rf on lisse sheaves for Zariski-compactifiable f ,Rf ! if p - n or f is finite, and RH om (under a finite tor-dimension(b)assumption), and Verdier duality. Moreover, Dzc (X , Λ) carries a naturalperverse t-structure whose abelian heart satisfies all expected properties.All of these statements are compatible with their schematic counterpartsunder analytification, and with extensions of the base field. Similarresults hold with Z -coefficients.Proof requires many auxiliary ingredients, possibly of independentinterest.David HansenRecent developments in étale cohomology8 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityFirst main ingredient:David HansenRecent developments in étale cohomology9 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityFirst main ingredient:Algebraization theorem (Bhatt-H.)If A is a K -affinoid ring and X is a scheme of finite type over SpecA, thebnatural functor ( )an : Dcb (X, Λ) Dzc(Xan , Λ) is fully faithful. IfX SpecA is proper, this functor is an equivalence of categories.David HansenRecent developments in étale cohomology9 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityFirst main ingredient:Algebraization theorem (Bhatt-H.)If A is a K -affinoid ring and X is a scheme of finite type over SpecA, thebnatural functor ( )an : Dcb (X, Λ) Dzc(Xan , Λ) is fully faithful. IfX SpecA is proper, this functor is an equivalence of categories.Most useful case: X SpecA.David HansenRecent developments in étale cohomology9 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityFirst main ingredient:Algebraization theorem (Bhatt-H.)If A is a K -affinoid ring and X is a scheme of finite type over SpecA, thebnatural functor ( )an : Dcb (X, Λ) Dzc(Xan , Λ) is fully faithful. IfX SpecA is proper, this functor is an equivalence of categories.Most useful case: X SpecA. This is a key input into the proof of thefollowing result.David HansenRecent developments in étale cohomology9 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityFirst main ingredient:Algebraization theorem (Bhatt-H.)If A is a K -affinoid ring and X is a scheme of finite type over SpecA, thebnatural functor ( )an : Dcb (X, Λ) Dzc(Xan , Λ) is fully faithful. IfX SpecA is proper, this functor is an equivalence of categories.Most useful case: X SpecA. This is a key input into the proof of thefollowing result.Locality theorem (Bhatt-H.)Zariski-constructibility is an étale-local property.David HansenRecent developments in étale cohomology9 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityFirst main ingredient:Algebraization theorem (Bhatt-H.)If A is a K -affinoid ring and X is a scheme of finite type over SpecA, thebnatural functor ( )an : Dcb (X, Λ) Dzc(Xan , Λ) is fully faithful. IfX SpecA is proper, this functor is an equivalence of categories.Most useful case: X SpecA. This is a key input into the proof of thefollowing result.Locality theorem (Bhatt-H.)Zariski-constructibility is an étale-local property.Upshot: In the proof of the main theorem, all claims can be checkedlocally in the analytic topology.David HansenRecent developments in étale cohomology9 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.2) For any F D (SpecA, Λ), the natural mapRΓ(SpecA, F ) RΓ(SpaA, F an ) is an isomorphism.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.2) For any F D (SpecA, Λ), the natural mapRΓ(SpecA, F ) RΓ(SpaA, F an ) is an isomorphism.1) proved by Berkovich and Huber if f proper or Y SpecK . Generalcase uses Nagata compactification, Gabber’s finiteness theorems, andresolution of singularities (many times) to reduce to a purity theoremproved by Huber.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.2) For any F D (SpecA, Λ), the natural mapRΓ(SpecA, F ) RΓ(SpaA, F an ) is an isomorphism.1) proved by Berkovich and Huber if f proper or Y SpecK . Generalcase uses Nagata compactification, Gabber’s finiteness theorems, andresolution of singularities (many times) to reduce to a purity theoremproved by Huber.2) proved by Huber if F has constant cohomology sheaves. General casefollows from a trick.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.2) For any F D (SpecA, Λ), the natural mapRΓ(SpecA, F ) RΓ(SpaA, F an ) is an isomorphism.1) proved by Berkovich and Huber if f proper or Y SpecK . Generalcase uses Nagata compactification, Gabber’s finiteness theorems, andresolution of singularities (many times) to reduce to a purity theoremproved by Huber.2) proved by Huber if F has constant cohomology sheaves. General casefollows from a trick.1) 2) immediately give full faithfulness in the algebraization theorem.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.2) For any F D (SpecA, Λ), the natural mapRΓ(SpecA, F ) RΓ(SpaA, F an ) is an isomorphism.1) proved by Berkovich and Huber if f proper or Y SpecK . Generalcase uses Nagata compactification, Gabber’s finiteness theorems, andresolution of singularities (many times) to reduce to a purity theoremproved by Huber.2) proved by Huber if F has constant cohomology sheaves. General casefollows from a trick.1) 2) immediately give full faithfulness in the algebraization theorem.Essential surjectivity can then be checked on hearts.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityProof of the algebraization theorem relies, in turn, on two separateingredients.Comparison theorems (H.)Let A be a K -affinoid ring.1) Let f : X Y be any finite type map of locally finite typeSpecA-schemes. Then for any F Dc (X, Λ), the natural map(Rf F )an Rf an F an is an isomorphism.2) For any F D (SpecA, Λ), the natural mapRΓ(SpecA, F ) RΓ(SpaA, F an ) is an isomorphism.1) proved by Berkovich and Huber if f proper or Y SpecK . Generalcase uses Nagata compactification, Gabber’s finiteness theorems, andresolution of singularities (many times) to reduce to a purity theoremproved by Huber.2) proved by Huber if F has constant cohomology sheaves. General casefollows from a trick.1) 2) immediately give full faithfulness in the algebraization theorem.Essential surjectivity can then be checked on hearts. After stratifying,reduce to proving that lisse sheaves algebraize.David HansenRecent developments in étale cohomology10 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityWe now use the second ingredient:David HansenRecent developments in étale cohomology11 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityWe now use the second ingredient:Extension theorem (H.)Let X be a normal rigid space, U X the complement of anowhere-dense closed analytic subset. Then any finite étale map V Uextends uniquely to a finite map V 0 X .David HansenRecent developments in étale cohomology11 / 24

IntroductionÉtale cohomology of rigid spacesRelative perversityWe now use the second ingredient:Extension theorem (H.)Let X be a normal rigid space, U X the complement of anowhere-dense closed analytic subset. Then any finite étale map V Uextends uniquely to a finite map V 0 X .Essential case of X smooth and X U an snc divisor treated byLütkebohmert. General case can be deduced by resolution of singularities.If X is a scheme of finite type over Spec

Recent developments in etale cohomology David Hansen MPIM Bonn November 10, 2021 David Hansen Recent developments in etale cohomology 1/24. . History cont'd Some key later developments: Beilinson-Bernstein-Deligne-Gabber '83: Perverse sheaves, t-structures, decomposition theorem, purity for intersection

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