Algebraic varieties Analytic varieties On p -adic comparison theorems for analytic spaces Wies� lawa Nizio� l, joint with Pierre Colmez CNRS, Sorbonne University July 27, 2020
Algebraic varieties Analytic varieties Algebraic comparison theorem Notation: K / Q p - finite, G K = Gal( K / K ), C = � K , K ⊃ O K → k , F = W ( k ). Theorem (Algebraic comparison theorem) X / K – algebraic variety. There exists a natural B st -linear, G K -equivariant period isomorphism ( r ≥ 0) H r et ( X K , Q p ) ⊗ Q p B st ≃ H r α pst : HK ( X K ) ⊗ F nr B st , ( ϕ, N , G K ) , ´ H r et ( X K , Q p ) ⊗ Q p B dR ≃ H r α dR : dR ( X K ) ⊗ K B dR , Fil , ´ where α dR = α pst ⊗ B dR .
Algebraic varieties Analytic varieties Algebraic comparison theorem Notation: K / Q p - finite, G K = Gal( K / K ), C = � K , K ⊃ O K → k , F = W ( k ). Theorem (Algebraic comparison theorem) X / K – algebraic variety. There exists a natural B st -linear, G K -equivariant period isomorphism ( r ≥ 0) H r et ( X K , Q p ) ⊗ Q p B st ≃ H r α pst : HK ( X K ) ⊗ F nr B st , ( ϕ, N , G K ) , ´ H r et ( X K , Q p ) ⊗ Q p B dR ≃ H r α dR : dR ( X K ) ⊗ K B dR , Fil , ´ where α dR = α pst ⊗ B dR . Here: (1) H r dR ( X K ) – Deligne de Rham cohomology (uses resolution of singularities) (2) H r HK ( X K ) – Beilinson Hyodo-Kato cohomology (uses de Jong’s alterations)
� � � � � Algebraic varieties Analytic varieties Hyodo-Kato cohomology (i) locally : in h -topology alterations allow U � � U sstable h − map Spec O L X finite Spec O K Then we have R Γ cr ( U 0 / O 0 H ∗ - finite rank / F L , ( a ) F L ) , ( ϕ, N ) , ι HK : R Γ cr ( U 0 / O 0 F L ) ⊗ L ( b ) F L L ≃ R Γ dR ( U ) .
� � � � � Algebraic varieties Analytic varieties Hyodo-Kato cohomology (i) locally : in h -topology alterations allow U � � U sstable h − map Spec O L X finite Spec O K Then we have R Γ cr ( U 0 / O 0 H ∗ - finite rank / F L , ( a ) F L ) , ( ϕ, N ) , ι HK : R Γ cr ( U 0 / O 0 F L ) ⊗ L ( b ) F L L ≃ R Γ dR ( U ) . (ii) globalization : make (i) geometric and glue in h -topology. Get H ∗ - finite rank / F nr , R Γ HK ( X K ) , ( ϕ, N , G K ) , ι HK : R Γ HK ( X K ) ⊗ F nr K ≃ R Γ dR ( X K )
Algebraic varieties Analytic varieties Restated algebraic comparison theorem (i) de Rham-to-´ etale comparison : H r et ( X K , Q p ) ≃ ( H r HK ( X K ) ⊗ F nr B st ) ϕ =1 , N =0 ∩ F 0 ( H r dR ( X ) ⊗ K B dR ) , G K , ´
� � � � � � Algebraic varieties Analytic varieties Restated algebraic comparison theorem (i) de Rham-to-´ etale comparison : H r et ( X K , Q p ) ≃ ( H r HK ( X K ) ⊗ F nr B st ) ϕ =1 , N =0 ∩ F 0 ( H r dR ( X ) ⊗ K B dR ) , G K , ´ or: we have a bicartesian diagram ( r ≥ 0) HK ( X K ) ⊗ F nr B + st ) ϕ = p r , N =0 H r ( H r et ( X K , Q p ( r )) ´ dR ( X ) ⊗ K B + � H r dR ( X ) ⊗ K B + F r ( H r dR ) dR We will write it as (upper index refers to cohomology degree) HK r H r r ´ et , r � DR r H r ( F r )
Algebraic varieties Analytic varieties or: there exists an exact sequence r → DR r → 0 0 → H r et , r → H r ( F r ) ⊕ HK r ´
Algebraic varieties Analytic varieties or: there exists an exact sequence r → DR r → 0 0 → H r et , r → H r ( F r ) ⊕ HK r ´ (ii) ´ etale-to-de Rham comparison : et ( X K , Q p ) , B st ) G K − sm ≃ H r Hom( H r HK ( X K ) ∗ , ( ϕ, N , G K ) , ´ Hom G K ( H r et ( X K , Q p ) , B dR ) ≃ H r dR ( X K ) ∗ , Fil ´
Algebraic varieties Analytic varieties Analytic varieties X / K - smooth rigid analytic variety Case 1 : X proper, (A) Scholze: (i) H r et ( X C , Q p ) is finite rank over Q p : ´ • Artin-Schreier to pass to coherent cohomology • Cartier-Serre argument for finitness of coherent cohomology (ii) Hodge-de Rham spectral sequence degenerates ⇒ get de Rham comparison isomorphism : H r et ( X C , Q p ) ⊗ Q p B dR ≃ H r α dR : dR ( X ) ⊗ K B dR , Fil , ´
Algebraic varieties Analytic varieties (B) Colmez-Nizio� l: Algebraic comparison theorem holds (HK-cohomology is defined using Hartl and Temkin alterations instead of de Jong’s) H r et ( X C , Q p ) ⊗ Q p B st ≃ H r α pst : HK ( X C ) ⊗ F nr B st , ( ϕ, N , G K ) , ´ H r et ( X C , Q p ) ⊗ Q p B dR ≃ H r α dR : dR ( X ) ⊗ K B dR , Fil . ´
Algebraic varieties Analytic varieties (B) Colmez-Nizio� l: Algebraic comparison theorem holds (HK-cohomology is defined using Hartl and Temkin alterations instead of de Jong’s) H r et ( X C , Q p ) ⊗ Q p B st ≃ H r α pst : HK ( X C ) ⊗ F nr B st , ( ϕ, N , G K ) , ´ H r et ( X C , Q p ) ⊗ Q p B dR ≃ H r α dR : dR ( X ) ⊗ K B dR , Fil . ´ (i) Tsuji, Kato, CN: p -adic nearby cycles = syntomic cohomology ( τ ≤ r ) ⇒ DR r − 1 f r − 1 f r → H r et , r → H r ( F r ) ⊕ HK r r → DR r → H r +1 − − − − ´ syn , r
Algebraic varieties Analytic varieties (B) Colmez-Nizio� l: Algebraic comparison theorem holds (HK-cohomology is defined using Hartl and Temkin alterations instead of de Jong’s) H r et ( X C , Q p ) ⊗ Q p B st ≃ H r α pst : HK ( X C ) ⊗ F nr B st , ( ϕ, N , G K ) , ´ H r et ( X C , Q p ) ⊗ Q p B dR ≃ H r α dR : dR ( X ) ⊗ K B dR , Fil . ´ (i) Tsuji, Kato, CN: p -adic nearby cycles = syntomic cohomology ( τ ≤ r ) ⇒ DR r − 1 f r − 1 f r → H r et , r → H r ( F r ) ⊕ HK r r → DR r → H r +1 − − − − ´ syn , r (ii) Lift the sequence to the category of Banach-Colmez ( BC ) spaces Suffices : f r − 1 , f r = 0. For f r − 1 , have DR i / H i ( F r ) − Dim = ( d , 0) , H i et − Dim = (0 , h ) , h ≥ 0 . ´ But in BC category there is no map between such spaces. For f r : bring to this situation by twisting.
Algebraic varieties Analytic varieties Digression: Banach-Colmez spaces What structure can we put on st ) N =0 ,ϕ =1 ≃ ( H n HK r r = ( H n HK ( X C ) ⊗ F nr B + HK ( X C ) ⊗ F nr B + cr ) ϕ =1 ?
Algebraic varieties Analytic varieties Digression: Banach-Colmez spaces What structure can we put on st ) N =0 ,ϕ =1 ≃ ( H n HK r r = ( H n HK ( X C ) ⊗ F nr B + HK ( X C ) ⊗ F nr B + cr ) ϕ =1 ? Example 0 → Q p t → B + ,ϕ = p → C → 0 cr So B + ,ϕ = p ∼ C ⊕ Q p . cr
Algebraic varieties Analytic varieties Digression: Banach-Colmez spaces What structure can we put on st ) N =0 ,ϕ =1 ≃ ( H n HK r r = ( H n HK ( X C ) ⊗ F nr B + HK ( X C ) ⊗ F nr B + cr ) ϕ =1 ? Example 0 → Q p t → B + ,ϕ = p → C → 0 cr So B + ,ϕ = p ∼ C ⊕ Q p . cr More generally, we have Fundamental exact sequence : 0 → Q p t m → B + ,ϕ = p m → B + dR / t m B + dR → 0 cr So: B + ,ϕ = p m ∼ C m ⊕ Q p . But In which category ? cr
Algebraic varieties Analytic varieties Digression: Banach-Colmez spaces What structure can we put on st ) N =0 ,ϕ =1 ≃ ( H n HK r r = ( H n HK ( X C ) ⊗ F nr B + HK ( X C ) ⊗ F nr B + cr ) ϕ =1 ? Example 0 → Q p t → B + ,ϕ = p → C → 0 cr So B + ,ϕ = p ∼ C ⊕ Q p . cr More generally, we have Fundamental exact sequence : 0 → Q p t m → B + ,ϕ = p m → B + dR / t m B + dR → 0 cr So: B + ,ϕ = p m ∼ C m ⊕ Q p . But In which category ? cr Remark The category of topological vector spaces is not good: C ⊕ Q p ≃ C !
Algebraic varieties Analytic varieties Theorem (Colmez, Fontaine) There exists an abelian category of Banach-Colmez vector spaces W which are finite dimensional C -vector spaces ± finite dimensional Q p -vector spaces. We have 1. Dim ( W ) := (dim C W , dim Q p W ); set ht W := dim Q p W 2. Dim ( W ) is additive on short exact sequences.
Algebraic varieties Analytic varieties Theorem (Colmez, Fontaine) There exists an abelian category of Banach-Colmez vector spaces W which are finite dimensional C -vector spaces ± finite dimensional Q p -vector spaces. We have 1. Dim ( W ) := (dim C W , dim Q p W ); set ht W := dim Q p W 2. Dim ( W ) is additive on short exact sequences. Example dR / t m is B m with Dim ( B m ) = ( m , 0) . 1. B + 2. B + ,ϕ a = p b is U a , b with Dim ( U a , b ) = ( b , a ) . cr 3. C / Q p is V 1 / Q p with Dim = (1 , − 1) .
Algebraic varieties Analytic varieties Case 2: X / K Stein: 1. there exists an admissible covering by affinoids · · · ⋐ U n ⋐ U n +1 ⋐ · · · 2. H i ( X , F ) = 0, F -coherent, i > 0 3. R Γ pro´ et ( X C , Q p ) ≃ holim n R Γ ´ et ( U n , C , Q p )
Algebraic varieties Analytic varieties Case 2: X / K Stein: 1. there exists an admissible covering by affinoids · · · ⋐ U n ⋐ U n +1 ⋐ · · · 2. H i ( X , F ) = 0, F -coherent, i > 0 3. R Γ pro´ et ( X C , Q p ) ≃ holim n R Γ ´ et ( U n , C , Q p ) Examples (1) X = A K , r > 0 : H r et ( X C , Q p ( r )) ≃ Ω r − 1 ( A C ) / ker d , pro´ H 1 et ( X C , Q p (1)) ≃ O ( A C ) / C pro´
Algebraic varieties Analytic varieties Case 2: X / K Stein: 1. there exists an admissible covering by affinoids · · · ⋐ U n ⋐ U n +1 ⋐ · · · 2. H i ( X , F ) = 0, F -coherent, i > 0 3. R Γ pro´ et ( X C , Q p ) ≃ holim n R Γ ´ et ( U n , C , Q p ) Examples (1) X = A K , r > 0 : H r et ( X C , Q p ( r )) ≃ Ω r − 1 ( A C ) / ker d , pro´ H 1 et ( X C , Q p (1)) ≃ O ( A C ) / C pro´ (2) X = G m , K , there exists an exact sequence 0 → O ( G m , C ) / C → H 1 et ( G m , C , Q p (1)) → Q p < dlog z > → 0 pro´ trivial G K -action on Q p < dlog z >
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