Rigid Geometry and Applications II. Kazuhiro Fujiwara & Fumiharu Kato

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1 Rigid Geometry and Applications II Kazuhiro Fujiwara & Fumiharu Kato

2 Birational Geometry from Zariski s viewpoint S U D S : coherent (= quasi-compact and quasi-separated) (analog. compact Hausdorff) U = S \ D S quasicompact open immersion (U: dense in S ). I I D = defining ideal of D

3 Basic Question: Extension problem P: a property of morphisms (e.g. P = flat ) f U : X U U: finitely presented /U with property P. Assume f : X S : an extension of f U. Can one find such an f with P? NO in general. Need to allow birational changes of S preserving U.

4 Modifications MD (S,U) : Category of U-admissible modifications: dense Objects: U S proper S Morphisms: S S S The category MD (X,U) is cofiltered.

5 BL (S,U) : Full subcategory of U-admissible blowups: Objects: S S : a blow-up with the center S \ U (set-theoretically). Example: S = Spec A, D = V(I) S = Proj n 0 Jn, I k J for k.

6 Strict Transform S S : U-admissible modification. f : X S : a morphism. X X S = X S S S S f S f X S X : dividing out I D -torsions. X

7 Modified Extension Problem f U : X U U: finitely presented /U with the property P. Assume f : X S : an extension of f U. Can one find a U-admissible modification (resp. blow-up) S S such that the strict transform f : X S has P?

8 Flattening Theorem Case: P=flat Theorem (Gruson-Raynaud, 1970). f U : X U U: flat, finitely presented = S S : U-admissible blow-up such that f : X S : flat, finitely presented.

9 Corollary. BL (S,U) is cofinal in MD (S,U). Comments on Flattening Theorem Clear if S = Spec V, V: DVR. If V: valuation ring, flatness is clear, while finite presentation is rather difficult.

10 Proof - Revival of Zariski s idea Zariski-Riemann space (Zariski, 1939) U cpt = lim S S BL (S,U) Projective limit taken in the category of local ringed spaces (1) Generalization of abstract Riemann surface: S : regular curve = U cpt = S (Dedekind-Weber) (2) Zariski s motivation: resolution of singularities

11 Points and local rings x U cpt U x U = O U cpt,x = O U,x. x U = V x : valuation ring (of height 1) with 1 α: Spec V x S x = {p S } S BL (S,U) : compatible system of points. (p S : the image of the closed points of Spec V x.) S

12 O U cpt : I D -valuative ring A composite of local rings of U and valuation rings. Notice: The valuation rings are not necessarily of height 1 (even when S : algebraic variety /k). One has to consider valuation rings of higher height.

13 Example of valuation rings S :algebraic surface ht. rat. rk. 0 0 trivial valuation 1 1 divisorial non-divisorial non-divisorial composite of two divisorial valuations

14 V: Valuation ring Intuitive Description Spec V = Long Curve S U D T D/S U cpt = U T D/S : The set of long curves passing through D: Tubular neighborhood

15 Fundamental Property Theorem (Zariski, 1944). U cpt is quasi-compact. Flattening theorem: consequence of the quasicompactness. Proofs: (1) lim coherent topoi = coherent topos (cf. lim cpt Hausdorff spaces = cpt Hausdorff space) + Deligne s theorem (existence of points) (2) Stone s representation theorem for distributive lattices.

16 Proof of flattening theorem Idea: Reduction to long curves using the quasi-compactness (1) True for long curves, i.e. S=Spec V. (2) True for S=Spec local ring of a point of U cpt. (Follows from (1) and true on U by patching.) (1), (2), and P: locally finitely presented = Claim is true locally on U cpt, i.e., x U cpt, U x Claim is true on U x. (3) Quasi-compactness = Done by birational patching S S U-Admissible blow-up

17 Other applications Embedding theorem for algebraic spaces (Nagata (1963): for Noetherian schemes). Resolution of singularities of quasi-excellent surfaces (Zariski).

18 Contents of the Book (Vol. I) 1. Preliminaries 2. Formal geometry 3. Basics on rigid geometry 4. Formal flattening theorem 5. Enlargement theorem

19 To be discussed (Vol. II)? 1. Etale topology 2. Lefschetz trace formula 3. Relationship with logarithmic geometry & ramification theory

20 Why de we do this? 1. Arithmetic geometry of Shimura varieties (local models, p-adic modular forms). 2. Cohomology theory of schemes (l-adic Lefschetz trace formula, rigid cohomology of Berthelot). 3. Mirror symmetry (expected construction of Mirror partner (Kontsevich, Fukaya,...)).

21 Recall: Our approach Birational geometry of formal spaces = Rigid geometry Raynaud s viewpoint of rigid geometry Zariski s viewpoint of birational geometry Geometry of models Rigid analytic geometry Zariski-Riemann space Geometry of Formal Schemes

22 Birational approach to Rigid Geometry Birational Geometry p-adic Rigid Geometry S U D D: p = 0 S Spf Z p

23 Adequate Formal schemes Defined ring theoretically. Noetherian outside the ideal of definition. Objects contain: Spf V, V: a-adically complete valuation ring. Noetherian formal schemes. Functoriality: Closed under finite type extensions. Base change by finite type morphisms.

24 Fs adq Fs Noe Fs fin /DVR Noe Fs fin /V Notice: The height of V can be arbitrary. Includes formal models of Tate s affinoid algebras. Nice Point: Can generalize Theorems in EGA III (Finitudes, GFGA Comparison, GFGA Existence) in derived categorical language.

25 Coherent Rigid Spaces CFs adq = the category of coherent (= quasi-compact and quasi-separated) adequate formal schemes. Notion of admissible blow-ups. Definition. The category of coherent rigid spaces: CRf = CFs adq /{admissible blow-ups}. Quotient functor CFs adq CRf X X rig.

26 Admissible Topology U X: (coherent) open immersion formal model U X: open immersion. {U α X}: (finite) covering formal models U α X such that X = U α. Proposition. Any representable presheaf on CRf ad is a sheaf. (Proof: By birational patching of formal models.)

27 General Rigid Spaces Definition. General Rigid Space = a sheaf F of sets on CRf ad such that (1) surjective map of sheaves ( Y) i F, ( ) i I where Y i : coherent rigid spaces. (2) For i, j I, Y i Y j Y i : ( ) an increasing sequence of open immersions. Rf = the category of general rigid spaces.

28 Relationship with algebraic spaces Theorem. { GAGA functor} Algebraic spaces of finite type /Q p Rf. Ingredients for the proof: Embedding theorem for algebraic spaces. Equivalence Theorem:

29 { } Equivalence Theorem. S : coherent adequate formal scheme X/S : formal algebraic space of {adm. blow-ups} / finite type { } / X/S : formal scheme {adm. blow-ups} of finite type Follows from: Theorem. S : as above, X/S : formal algebraic space of finite type. = π: X X: admissible blow-up such that X : formal scheme.

30 Visualization The ring used yesterday X = X rig : coherent rigid space X = lim X X X (projective limit of all admissible blow-ups). Specialization map sp X : X X. O int (Integral Structure Sheaf): limit of O X. X O Rigid Structure Sheaf: O X = lim Hom O int(i n, O int X ), X n 0 where I: ideal of definition. X In p-adic situation: O X = O int X X O O X [1 p ].

31 I O int X is the canonical integral model of O X. Zariski-Riemann triple ZR(X) = ( X, O int X, O X). X X Defined similarly for general rigid spaces Remark. Notice the analogy: (E, ) : Hermitian bundle O X E with integral model OE X int. Proposition. Topos associated to X admissible topos X ad.

32 Relation with other theories ( ( ) ) Tate s rigid ( ) spaces Affinoid in Rf = ( Rf affinoids affinoids. ( ( ) (1) Tate s rigid spaces are naturally objects of Rf (via Raynaud s theorem & patching). Coherent rigid space of the form (Spf A) rig. (2) Zariski-Riemann triples are regarded as Huber s adic spaces.

33 ( Tate s rigid spaces ) Diagram Rf f.f. ZR Not f.f. f.f. ( ) Huber s adic spaces Birational Geometry f.f. = fully faithful f.f. ( ) Berkovich spaces Spectral Geometry

34 Formal Flattening Theorem Theorem (Bosch-Raynaud, Fujiwara). f : X S : finite type between coherent adequate formal schemes. = the following conditions are equivalent: (1) f rig : X rig S rig : flat ( def f rig : flat as a morphism of local ringed spaces). (2) S S : admissible blow-up such that the strict transform f : X S is flat. Proof: Similar to the algebraic case.

35 Corollaries Corollary. Admissible blow-ups are cofinal in the category of formal modifications. Corollary (Gerritzen-Grauert). In Tate s rigid analytic geometry, sep. étale inj. open imm.

36 Properness in Rigid Geometry Three different definitions 1. Universally closed, separated, of finite type. 2. Raynaud properness; i.e., having proper formal model. 3. Kiehl properness; i.e., existence of affinoid enlargements of coverings for each relatively compact affinoid open subset. Difficult: 2. = 3.

37 For f : X Spf V: of finite type, Showing 2. = 3. Theorem. U X: affine open such that U: proper = π: X X: admissible blow-up and W X : open such that π 1 (U) W, W Spf A: contraction to an affine formal scheme. Remark. Proved by Lütkebohmert, when V = DVR (1990).

38 Formal Geometry and Rigid Geometry General Principle e.g. GFGA theorems GAGA theorems

39 Finiteness Theorem Formal Geometry Y: quasi-compact adequate and amenable f : X Y: proper of finite presentation = R f maps D coh (X) to D coh (Y) for =, +,, b. Rigid Geometry Y: quasi-compact ϕ: X Y: proper = Rϕ maps D coh (X) to D coh (Y) for =, +,, b.

40 Comparison Theorem Formal Geometry (Y, Z): universal adhesive pair, amenable f : X Y: proper of finite presentation = D (X) for coh D ( X) Rigid Geometry (B, I): complete t.u.a., U = Spec B \ V(I). f : X Y: proper U- morphisms. = D coh (X) rig D coh (Xan ) R f D coh (Y) for R f D (Ŷ) R f D coh (Y) rig R f an D (Y an )

41 Existence Theorem Formal Geometry (B, I): complete t.u.a. pair, univ. top. coh. f : X Y = Spec B: proper of finite presentation = D b coh (X) D b coh ( X) Rigid Geometry (B, I): complete t.u.a., U = Spec B \ V(I). f : X Y: proper U- morphisms. = D b coh (X) D b coh (X an )