On local properties of non- Archimedean analytic spaces. Michael Temkin (talk at Muenster Workshop, April 4, 2000) Plan: 1. Introduction. 2. Category bir k. 3. The reduction functor. 4. Main result. 5. Applications. 1
1. Introduction. Let k be a non-archimedean field with nontrivial valuation. All analytic spaces considered are strictly k-analytic. Red : Germs bir k (X x X x ) Main results give criterions 1. for an analytic space X to be good at a point x X; 2. for a morphism Y X to be closed at a point y Y. Applications (W. Lütkebohmert, 1990, for k with discrete valuation): 1. f s : Y s X s is proper f η : Y η X η is proper. 2. X R is a separated morphism to an affinoid space any affinoid domain U X has an affinoid extension V X s.t. U Int(V/R). 2
2. Category bir k. In this section the base field k is arbitrary. Def: For a field K/k, P K = {ν k O ν K}, its topology basis consists of affine sets P K {f 1,..., f n } = {ν f 1,..., f n O ν }. Def: bir k is the category of maps X f X P K, where f X is a local homeomorphism and X is a connected quasi-compact and quasi-separated space. A morphism is a pair, an embedding i : K L and a commutative diagram Y f X h i # X f Y P L P K Def: (i,h) is separated (resp. proper) if Y P L PK X is injective (resp. bijective). 3
Def: V ar k is the category of dominant morphisms Spec(K) X, where X is integral of finite type over k. Morphisms are commutative diagrams Spec(L) i Spec(K) Y X f Def: A morphism (i, f) is 1. separated (resp. proper) if f is separated (resp. proper); 2. birational if i is an isomorphism and f is proper. 4
Construction: V al : V ar k bir k V al(spec(k) X) = {O ν K, Spec(O ν ) X}. Fact: V al(spec(k) X) lim Y, where the limit is taken over all birational morphisms Spec(K) Spec(K) In particular, it is quasi-compact and quasiseparated. Y X Fact: V al induces an equivalence of categories V ar k /{birational morphisms} bir k. 5
3. Reduction functor. Def: The spectrum M(A) of a Banach k- algebra A is the set of all bounded multiplicative seminorms on A. x M(A) χ x : A H(x), where χ x is bounded, H(x) is a non-archimedean field and H(x) = Q(χ x (A)). Analytic spaces are glued from spectra of affinoid algebras. An analytic space is good if any its point has an affinoid neighborhood. Fact: Y X are affinoid, y Y is a point H X (y) = H Y (y). Corollary: For any analytic space X and a point x X the morphism M(H(x)) i x X is well defined. 6
Construction: The reduction functor Red : Germs bir k (X x X x ) is defined as follows: X x Ob(Germs) Y x (with compact Y ) (Y, x Y ) (Spec( H(x)) ĩx Y s ) (Spec( H(x)) ĩx Im(ĩx)) Ob(V ar k) X x Ob(bir k) where Y is an admissible formal k -scheme and Y η Y. If Y is such that Y η Y, then Y proper over both Y and Y and such that Y η Y. Hence the result does not depend on the choice of Y. 7
Fact: Y X is an analytic subdomain, x Y Ỹx X x is an open embedding. Fact: x Y, Z X Y x Z x Ỹx Z x and Y x Z x Ỹx Z x ; if X = M(A) and f A, then X x {f} X x { f}. Th: {equiv. classes of subdomains Y x X x } {open quasi-comp. subsets of X x }. Remark: In Huber s adic spaces, X x x. 8
4. Main result. Th: A germ X x is good iff X x is affine. Proof (sketch): Step 1: We may assume, that X x = Y x Z x, Y x and Z x are good, Ỹx = X x { f} and Z x = X x { f 1 }. Step 2: Given step 1, X x is good. Lemma: Let X = M(B) M(C), M(A) = M(B) M(C), A = B{f 1 } = C{g}, k{t 1,..., T n, S, ps 1 } φ B : T i f i, S f k{t 1,..., T n, q 1 S, S 1 } ψ C : T i g i, S g p 1 q, f i g i A < 1, f g A < 1 and A is induced from k{t i, S, S 1 } X = M(D) and k{t i, q 1 S, ps 1 } D : T i h i, S h. 9
5. Applications. Def: Let A be a k-affinoid algebra and φ : B D a bounded A-homomorphism, where B is A-affinoid and D is A-Banach. Then φ is inner w.r.t. A if ψ : A{T 1,..., T n } B s.t. φ(ψ(t i )) sup < 1. Notation: Let M(B) = Y Y = M(B ) be affinoid spaces over X = M(A) s.t. B B is inner w.r.t. A, then one writes Y X Y. Def (Kiehl): Y f X is locally proper if X = X i and f 1 (X i ) = j Ji Y ij = j Ji Y ij s.t. Y ij Xi Y ij ; if J i may be chosen finite, then f is proper. 10
f Def: Given M(B) = Y X = M(A) and y Y, suppose that B χ y H(y) is inner w.r.t. A, then y Int(Y/X) and f is closed at y. Def: A morphism f : Y X of good spaces is closed if Y = Int(Y/X); f is proper if in addition it is a proper map of topological spaces. Def: f : Y X is closed (resp. proper) if for any morphism X X with good X, the base change morphism f : Y X X X is a closed (resp. proper) morphism of good spaces. 11
Recall, that a germ X x is good iff X x is affine. Corollary: A morphism Y y X x is closed (resp. separated) iff its reduction Ỹy X x is proper (resp. separated). Corollary: Y X is a proper (resp. locally proper) morphism of admissible formal k -schemes Y η X η is proper (resp. closed) in the sense of Berkovich Y rig X rig is proper (resp. locally proper) in the sense of Kiehl. 12
Th: X R, R is affinoid, M(D) = U Int(X/R) U R U = M(D ) and U is a Weierstrass domain in U. Proof (sketch): Step 1: We may assume, that U{f} and U{f 1 } have analogous extensions. Step 2: Reduction to the relative version of the previous lemma. 13
Rigid definition of good spaces: X is good if it is separated and X = I X i = I X i s.t. X i, X i are affinoid and X i X X i, i.e. for any affinoid Y X and i I we have (X i Y ) Y (X i Y ). Conjecture: If X is good, then for any affinoid subdomain Y X there exists a bigger affinoid subdomain Y s.t. Y X Y. 14