A new comparison between overconvergent isocrystals and arithmetic D - PowerPoint PPT Presentation

Introduction Constructible isocrystals The trace map Dual constructibility A new comparison between overconvergent isocrystals and arithmetic D † -modules joint with Tomoyuki Abe Christopher Lazda Warwick Mathematics Institue Padova, 19th September 2019 Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Introduction 1 Constructible isocrystals 2 The trace map 3 Dual constructibility 4 Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility V = complete DVR k = residue field, perfect, char p > 0 K = fraction field K , char 0 X / k variety (= separated scheme of finite type), Isoc † ( X / K ) = F -able overconvergent isocrystals on X If we have an embedding X ֒ → P with P smooth and proper over V , then Isoc † ( X / K ) ֒ → MIC( j † X O ] Y [ ) , where Y is the closure of X inside P k , and H i rig ( X / K , E ) := H i (] Y [ , E ⊗ Ω • ] Y [ ) Good formal properties: finite dimensional, versions with support, excision exact sequences, &c. Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Beyond “smooth” coefficient objects: theory of arithmetic D † -modules (Berthelot/Caro). Locally: take ´ etale co-ordinates x 1 , . . . , x d on P and set  �  �   � � a k ∈ O P Q , ∃ λ > 1 s.t. � � � � a k � λ | k | → 0 D † a k ∂ [ k ] P Q = �   � k where ∂ [ k ] = ∂ k 1 x 1 . . . ∂ k d x d k 1 ! . . . k d ! . Caro defines hol ( D † coh ( D † D b P Q ) ⊂ D b P Q ) “ F -able overholonomic complexes”, stable under: f ! D P f + for f proper, ⊗ † R Γ † Z for Z ⊂ P closed O P by work of Caro/Caro–Tsuzuki. Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Given X ֒ → P , define hol ( X / K ) := � P Q ) supported on X � D b M ∈ D b hol ( D † , these are independent of the embedding, and support a formalism of the 6 functors ( f + , f + ) , ( f ! , f ! ) , ⊗ , D . Comparison of coefficients: ∃ fully faithful functor sp X , + : Isoc † ( X / K ) → D b hol ( X / K ) ⊂ D b hol ( D † P Q ) and can describe the essential image. Defined by Caro when X is smooth, and extended to the non-smooth case by Abe. Example If X is a dense open inside P = P k , and P \ X is a divisor, then sp + = sp ∗ is just pushforward along sp : P K → P . Much more difficult to describe in general! Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility For E ∈ Isoc † ( X / K ), can define its “ D -module cohomology” D ( X / K , E ) := H i − d X ( f + sp + E ) H i where f : X → Spec ( k ) is the structure morphism, inducing f + : D b hol ( X / K ) → D b hol (Spec ( k ) / K ) ∼ = D b ( K ) , and d X = dim X . Concretely, if X ֒ → P then H i D ( X / K , E ) = H i − d X + d P ( P , sp + E ⊗ O P Ω • P / V ) . where d P = dim P . Question Do we always have H i rig ( X / K , E ) ∼ = H i D ( X / K , E )? This is not obvious! Today: describe a new construction of sp + which makes comparison theorems easier to prove. Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Introduction 1 Constructible isocrystals 2 The trace map 3 Dual constructibility 4 Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility P = smooth, proper formal V -scheme, P = special fibre, p 1 , p 2 :] P [ P 2 → P K the two projections. Definition (Berthelot) A convergent stratification on an O P K -module E is an isomorphism ∼ p ∗ → p ∗ 2 E 1 E satisfying the cocycle condition. Definition (Le Stum) E is called constructible if there exists a stratification P = � i P i such that E | ] P i [ is coherent. Isoc † cons ( P / K ) = ( F -able) constructible O P K -modules with convergent stratification. Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Example X ֒ → P locally closed immersion, with closure α : Y ֒ → P , then we have ] α [:] Y [ → P K , and if E ∈ Isoc † ( X / K ) ⊂ MIC( j † X O ] Y [ ), then ] α [ ! E ∈ Isoc † cons ( P / K ) so we have a fully faithful functor ] α [ ! : Isoc † ( X / K ) → Isoc † cons ( P / K ) . Conjecture (Le Stum) R sp ∗ induces an equivalence of categories ∼ Isoc † → Perv( D † cons ( P / K ) P Q ) . This is a theorem when dim P / V = 1. Christopher Lazda

� � � � � Introduction Constructible isocrystals The trace map Dual constructibility Not clear how to even define R sp ∗ in general: want a lifting D b ( D † P Q ) forget Isoc † D b ( D P Q ) cons ( P / K ) R sp ∗ but sp − 1 D † P Q doesn’t act on constructible isocrysals (even if they are coherent on all α of P K ). What we did: given X ֒ → Y ֒ → P , construct D b ( D † P Q ) forget � Isoc † � D b ( D P Q ) Isoc † ( X / K ) cons ( P / K ) ] α [ ! R sp ∗ This immediately gives H i rig ( P , ] α [ ! E ) ∼ = H i D ( P , R sp ∗ ] α [ ! E ) . Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Basic example: P = � P 2 V with co-ordinates x 0 , x 1 , x 2 , Y = P 1 k = V ( x 2 ) ⊂ P , and X = A 1 k = D ( x 0 ) ⊂ Y , so we have j α X ֒ → Y ֒ → P and ] α [:] Y [ → P K . We take E = j † X O ] Y [ P . Set U = P \ Y , so for any sheaf F on P K we have the localisation exact sequence 0 → Γ † Y F → F → j † U F → 0 , note that Γ † Y =] α [ ! ] α [ − 1 . We apply this to F = R ] α [ ∗ j † X O ] Y [ =] α [ ∗ j † X O ] Y [ to obtain 0 → ] α [ ! j † X O ] Y [ → ] α [ ∗ j † X O ] Y [ → j † U ] α [ ∗ j † X O ] Y [ → 0 which gives a 2-term resolution of ] α [ ! j † X O ] Y [ . Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Lemma The sheaves ] α [ ∗ j † X O ] Y [ and j † U ] α [ ∗ j † X O ] Y [ are sp ∗ -acyclic. So we have � � R sp ∗ ] α [ ! j † X O ] Y [ ∼ sp ∗ ] α [ ∗ j † X O ] Y [ → sp ∗ j † U ] α [ ∗ j † = X O ] Y [ . If we set u = x 1 / x 0 and v = x 2 / x 0 , and look at global sections, then the first term consists of series f ( u , v ) ∈ K � u , v � such that: ∀ η < 1 ∃ λ > 1 s.t. f ( u , v ) converges for | v | ≤ η and | u | ≤ λ . Can describe the second term similarly, as series f ( u , v ) ∈ K � u , v , v − 1 � such that: there exists ρ < 1 such that ∀ ρ < η < 1 ∃ λ > 1 s.t. f ( u , v ) converges for ρ ≤ | v | ≤ η and | u | ≤ λ . Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Explicitly, the second is Kedlaya’s relative Robba ring R v K � u � † , and the first is it’s plus part R v , + K � u � † consisting of series with terms in non-negative powers of v . ⇒ can see directly that R sp ∗ ] α [ ! j † X O ] Y [ ∼ = v − 1 K � u , v − 1 � † [ − 1] and the D P Q -module structure extends to a D † P Q -module structure. j α Want to generalise this to an arbitrary frame ( X ֒ → Y ֒ → P ) with P smooth and proper over V , and E ∈ Isoc † ( X / K ). Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Three complications: The complement Y \ X might not be a divisor. So we take a suitable open cover 1 X = ∪ a X a and replace E by 0 → ⊕ a j † X a E → ⊕ a 1 , a 2 j † X a 1 ∩ X a 2 E → . . . → j † ∩ a X a E → 0 . We don’t know in general that the j † X a E are ] α [ ∗ -acyclic. So we take the 2 immersions α η : [ Y ] η → ] Y [ of quasi-compact tubes, and replace j † X a E by �� � � res − id α η n ∗ j † α η n ∗ j † lim X a E | [ Y ] η n − → X a E | [ Y ] η n − → n 0 n ≥ n 0 n ≥ n 0 for η n → 1 − . Y might not be a divisor in P , so we need to pick divisors D b such that 3 Y = ∩ b D b and replace the short exact sequence 0 → Γ † Y F → F → j † U F → 0 by the long exact sequence 0 → Γ † Y F → F → ⊕ b j † U \ D b F → . . . → j † U \∪ b D b F → 0 . Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Proposition Given suitable choices X = ∪ a X a and Y = ∩ b D b as above there exists a resolution RC † ( E ) of ] α [ ! E such that the D P Q -module structure on sp ∗ RC † ( E ) extends canonically to a D † P Q -module structure. Changing the X a or the D b results in canonically quasi-isomorphic complexes of D † P Q -modules. Corollary There exists a canonical lifting of ( R sp ∗ ◦ ] α [ ! )[ d P ] to a functor R sp P , ! : Isoc † ( X / K ) → D b ( D † P Q ) such that H i rig ( P , ] α [ ! E ) = H i − 2 d P ( u + R sp P , ! E ) . Example If Y = P and Y \ X is a divisor, then R sp P , ! E = sp ∗ E [ d P ] = sp + E [ d P ]. Christopher Lazda

Introduction Constructible isocrystals The trace map Dual constructibility Introduction 1 Constructible isocrystals 2 The trace map 3 Dual constructibility 4 Christopher Lazda