1 Existence of the Néron model


 Magnus Rich
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1 Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable Sscheme of finite type X with X K = A. The nicest possible model over S is an Sabelian scheme (proper, smooth). Not every abelian variety admits such a model: if K is a padic field and S = O K, then A admitsanabelianschememodelovers iffthegaloisactionofγ K onh 1 ét (A,Q l)isunramified. Definition 1 A Neron model X of A over S is a smooth model that satisfies the Néron Mapping Property : For every smooth Sscheme Z and every map u K : Z K A,!v : Z X with v K = u K. Definition 2 A weak Néron model is a smooth model that satisfies the mapping property for every étale scheme Z. Remark 3 If the model exists, it is unique up to isomorphism by the universal property. The group structure of X K can be lifted to X by the mapping property. Proposition 4 If X is an Sabelian scheme, then X is the Néron model of its generic fiber. Proof. Let Z be any smooth Sscheme and u K : Z K X K be any map. By definition u is a rational map u : Z > X. Let V Z be the domain of definition of u. Clearly V contains Z K. Claim. s S closed point, V s is dense in Z s. Let ζ be a generic point of Z s. Then O Z,s is a DVR. By the valuative criterion for properness, there is a map Spec(O Z,ζ ) X. There exists U, an open neighborhood of ζ, such that u is defined on U. This implies the claim. The claim implies that the rational map u is well defined on a subscheme of codimension at most 1 (in fact, if dimx K = n, then dimx = n + 1 and u is well defined on a dense subscheme of Z K ). By a theorem of Weil, u is a morphism: Theorem 5 (Weil, BLR 4.4) Let S be a (not necessarily reduced) noetherian base scheme, u Z G an Srational map from a smooth Sscheme Z to a smooth separable Sgroup scheme G. If u is well defined on a subscheme of codimension at most 1, then u is a morphism. As for the uniqueness, one has Theorem 6 (EGA 1, ) If we have two morphisms f 1,f 2 of Sschemes Z X, Z reduced, X separated. If there exists an open and dense subscheme U of Z s.t. f 1 and f 2 agree on U, then f 1 f 2. 1
2 1 Existence of the Néron model Let S be a Dedekind ring, K the field of fractions, A/K an abelian variety. We want to show that A admits a Néron model. Now A is a closed subscheme of a certain P n K, which in turn (canonically) embeds as an open subset of P n S. Take X to be the schematic closure of the inclusion A P n S. Then X is a proper model of A over S. Moreover, as X K Spec(K) is smooth, there exists an open subscheme S of S such that X S Spec(S ) is smooth. Now if for every point s S\S one can construct a Néron model of A over the local ring O S,s then we can just glue them together to get the global Néron model. Theorem 7 (BLR 1.2.4) For any model X/S of A, the following two conditions are equivalent: X is a Néron model; s S, X S O S,s is a Néron model. We are thus reduced to the local case. From now on, S = Spec(R), where R is a DVR, K is the fraction field, and k is the residue field. Let k s be the separable closure of k. Let R sh be the strict henselianization of R, i.e. 1.1 Plan of proof We want to constuct a sequence of models R sh = lim R R /R étale k /k k s /k X 0 X 1 X 2 X 3 X 4 where X 0 is a proper model of A, X 1 is obtained from X 0 by blowing up some components of the special fiber, X 2 is the smooth locus of X 1 (and is a weak Néron model), X 3 is obtained from X 2 by removing some components (and has a birational group structure), and X 4 is obtained from X 3 by Weil s theorem that extends the birational group structure to a group structure. Now X 0 is proper, and X 0,k is proper and geometrically connected (EGA IV ). However, X 0,k is in general neither smooth nor reduced. Aim: recover smoothness dropping properness Definition 8 Let X be a flat Rscheme of finite type with smooth generic fiber. A smoothening of X is a proper map X X such that: (1) X K = X K ; (2) for every étale extension R /R, X smooth (R ) X(R ) is surjective; or equivalently, (2 ) X smooth (Rsh ) X(R sh ). Theorem 9 X as above. A smoothening always exists. 2
3 Let X 0 be a proper model of A, X 1 X 0 a smoothening of X 0. Then we can take X 2 to be the smooth locus of X 1. Proposition 10 X 2 is a weak Néron model of A. Proof. By definition of a smoothening (specifically, property 2), every étale map lifts to X 2. Remark 11 A weak Néron model is not unique. 1.2 Existence of smoothenings The idea is to blow up some components of X K. Definition 12 (measure for defect of smoothness) X as above. For any a : Spec(R sh ) X we set δ(a) to be the length of the torsion submodule of a Ω 1 X/R. For any subset E of X(R sh ) we denote δ(x,e) = max{δ(a),a E} Theorem 13 (BLR 3.3 prop 3) δ(a) is bounded as a varies in X(R sh ) Lemma 14 δ(a) = 0 iff a factors through the smooth locus of X. Proof. : if a is smooth, then locally around a the sheaf Ω 1 X/R is free of finite type, hence a Ω 1 X/R is free : if a Ω 1 X/R = 0, d k denotes the dimension of X k, and d K denotes the dimension of X K, then by semicontinuity d k d K. As R sh is a DVR, the pullback a Ω 1 X/R is free (since it is torsionfree of finite type), so it is generated by d K elements. But then a k Ω1 X k /k is generated by d K elements, and d K d k, which implies that  locally around a k  X k /k is smooth of dimension d K. Since X/R is flat and locally X K /K and X k /k are smooth, X/R is smooth Canonical partition Given X as above and E X(R sh ), we will define a collection of geometrically reduced ksubschemes Y 1,U 1,Y 2,U 2,...,Y t,u t of X k and a partition E = E 1 E 2... E t. Define π to be the specialization map on R sh, i.e. π : X(R sh ) X k (k s ). 1. Let Y 1 be the Zariski closure of π(e) in X k 2. Let U 1 be the largest subscheme of Y such that U 1 is smooth; Ω 1 X/K Y 1 is locally free 3. E 1 := π 1 (U 1 (k s )) E. Note that Y 1 is geometrically reduced and that U 1 is dense in Y 1. Now suppose we have defined E 1,...,E i for i 1. Apply this construction to E \ (E 1... E i ) to get a closed subscheme Y i+1, an open dense U i+1, and a subset E i+1. This process will terminate in finitely many steps as X k is Noetherian. We denote t = t(x,e) the length of this partition ( length of partition of E in X ). 3
4 1.2.2 End of proof Lemma 15 (BLR 3.3.5) X,E as above. Let a E be such that a k = π(a) is a singular point of X k. Assume that a E i (i.e. a U i (k s )). Let X X be the blowup along Y i and a be the unique lift of a to X (existence and uniqueness of the lift follow from the valuation criterion for properness and the fact that X X is proper). Then δ(a ) {0,δ(a) 1}. Let X,E as above. Denote E k = π(e). Definition 16 Let Y be a closed subscheme of X k. We say that it is Epermissible if Y is geometrically reduced and F = Y(k s ) E k satisfies: 1. F lies in Y smooth 2. F lies in the largest subscheme of Y over which Ω 1 X/R is locally free 3. F is dense in Y In case Y is permissible, we say that a blowup X X along Y is Epermissible. Theorem 17 X,E as above. There is a X X that is a finite sequence of Epermissible blowups such that for any a E, there exists a unique lift of a to a point a in X sm(r sh ). Remark 18 The smoothening theorem is the case E = R sh Proof. By induction on δ(x,e)+t(x,e) 1. If δ(x,e) = 0, then all the points of X are smooth, and there is nothing to do. So we can assume δ(x,e) 1. Consider the canonical partition E = E 1... E t and let X X be the Epermissible blowup along Y t. Recall that by construction we have π(e t ) U t (k s ) Y t (k s ), and that π(e t ) are dense in Y t. By the lemma, for every a in E t the unique lift a of a satisfies δ(a ) < δ(a), and therefore δ(x,e t ) < δ(x,e t ). By induction, there is a finite chain of blowups such that E t lifts uniquely to X smooth. If t = 1 we are done. If t > 1, consider E = E \E t on X ; E has a canonical partition which is again given by E 1,...,E t 1, and t(x,e ) < t(x,e). Then induct. 4
5 2 Part 2  from weak models to strong ones Recall that we are working with S = Spec(R), R a DVR, and K the field of fractions of R. Let X be a weak Néron model of A. 2.1 Invariant differential forms and minimal components Let S be a scheme, G/S a smooth Sgroup scheme of relative dimension d. The sheaf Ω d G/S := Λd Ω 1 G/S is invertible, generated by a (left)invariant differential form. Choose such a form ω for A/K. As A/K is proper, hence commutative, ω is also rightinvariant. Let X/R be a smooth model of A, of relative dimension d. The sheaf Ω d X/R is again invertible, and satisfies Ω d X/R R K = Ω d A/K. Now for every irreducible component W of X k let η be the generic point and note that O X,η is a DVR, with the same uniformizer π as R. Hence there exists an r Z such that Ω d X/R is generated by π r ω on O X,η. We set ord W (ω) = r and ρ := min W ord W(ω). Replacing ω by π ρ ω we can suppose ρ = 0. We say that a component of X is minimal if ord W (ω) = 0. We take X 3 (our next step in constructing a Néron model) to be X minus the nonminimal components. Lemma 19 In this case (i.e. ρ = 0), ω extends to a section of Ω d X/R, and it is a generator of Ω d X 3 /R. Proof. 1. As X is normal, if ω is welldefined in codimension at most 1 it extends to a global section. 2. The zero locus of a section of a line bundle on an integral scheme is of pure codimension one. 3. ω X3 does not vanish on codimension 1, so ω X3 is everywhere nonvanishing (hence a generator). 2.2 Properties of the weak Néron model Definition 20 Let S be a scheme and X,Y be flat schemes of finite type over S. 1. We say that an open set U X is Sdense if s S the fiber U s is schematically dense in X s. 5
6 2. An Srational map f : X > Y is an equivalence class of maps U Y where U is an Sdense subscheme of X. The equivalence relation is given by (f : U Y) (g : V Y) if and only if W dense in U V such that f W = g W. Remark 21 If Y is separated and X is irreducible there exists a maximal Sdense open subscheme of X such that f is welldefined on U. This U is called the domain of definition of f. Theorem 22 Let X be a weak Néron model of A. Then X satisfies the Rrational Néron mapping property, namely smooth Rscheme Z, u K : Z K > A,!R rational map f : Z > X such that f K = u K. Proof. Preliminary reductions: 1. We can suppose that Z is of finite type over R (at any rate, the property is local). 2. We can reduce to the case when Z has only one irreducible component in the special fiber: as Z k is smooth over k, Z k = irreducible components. 3. Let W Z be the domain of definition of u K. Take Y the schematic closure of Z k \W in Z. Then we only have to work over Z\Y, i.e. we can suppose that u K is a morphism. Note that Z \Y is Rdense in Z. So (without loss of generality) we have a morphism u K : Z K A. Let Γ K be the closed subscheme of Z K A given by the graph of u K, and take Γ to be the schematic closure of Γ K in Z R X. Let p : Γ Z be the projection. It is enough to prove that p admits a section on an Rdense, open subscheme of Z. Claim: p k : Γ k (k s ) Z k (k s ) is surjective. Proof. Any point z k Z k (k s ) lifts to a z Z(R sh ) since Z/R is smooth and R sh is Henselian. Let z K be the generic point of Z. Consider the map u K : z K Z K A. By the weak Néron property, u K extends to a map u : Spec(R sh ) X. Now u K (Z K ) = X K is a generic point of u(z) = X. We deduce from this a point (Z,X) of Γ, such that (Z k,x k ) projects to z k via p k. Now p k (Γ k ) is a constructible subset of X k. The projection pi k (Γ k ) contains all the k s  points, so it contains a dense open subscheme of Z K. On the other hand, pi k is dominated: for every η Z k, ζ Γ such that π(ζ) = η. Consider the map of DVRs ϕ : O Z,η O Γ,ζ. The map ϕ K is an isomorphism, since p K : Γ K Z K has a section. Hence ϕ is an isomorphism, and by descent theory ϕ induces an isomorphism U V, where U and V are open neighborhoods of η,ζ respectively. This implies that p : Γ Z has a section on U, and as U contains the generic point of Z it is also Rdense in it. 6
7 2.3 Properties of X 3 Theorem 23 m K : A A A, the group law on A, can be extended to an Rbirational group law m : X 3 X 3 > X 3 Remark 24 An Rbirational group law is a map that satisfies is Rbirational; is Rbirational. Φ : X 3 X 3 > X 3 X 3 (x, y) (x, m(x, y)) Ψ : X 3 X 3 > X 3 X 3 (x, y) (m(x, y), y) Proof. The properties of the weak Néron model (previous section) imply the existence of a Rrational lift m : X 3 X 3 > X of m K. Let D X 3 R X 3 be the domain of definition of m, and let ϕ : D X 3 R X (x,y) (x,m(x,y)). Let ω be an invariant differential form on Ω d X/R and ω = pr2 ω Ωd X 3 X/R. We have ϕ ω = ω since this is true on the generic fiber. Let ξ = (α,γ) be a generic point of X 3 X 3. We want to show that η = ϕ(ξ) is in X 3 X 3, so that we can consider ϕ as a birational map with image in X 3 X 3. Let r = ord γ (ω) = ord η (ω) 0. The form π r ω is a generator of Ω d D/X 3 in O D,η. On the other hand, ϕ (π r ω ) = bω is also a generator (b O D,ξ ). By comparing orders, r = 0 and b invertible, so (by definition of a minimal component) γ X 3,k and likewise η X 3,k X 3,k since ord η (ω ) = 0. Therefore ϕ maps (X 3 X 3 ) k to (X 3 X 3 ) k. Let U = D (X 3 X 3 ). Then ϕ induces a morphism U X 3 X 3, and we can define m = pr 2 ϕ : U X 3 X 3 X 3. We can also define Ψ in a similar way. To show that Φ is birational, one needs to show that ϕ induces an isomorphism of U onto an open subscheme of X 3 X 3. To do so, simply notice that ϕ is an isomorphism between Ω d X 3 X 3 /X 3 and Ω d U/X 3, hence it is also an isomorphism between the Ω 1 s. It follows from the diagram U ϕ X 3 X 3 smooth X 3 smooth that ϕ is étale (since the schemes are smooth over the base and ϕ is an isomorphism on Ω 1 ). By Zariski s main theorem, α factors through an open immersion U Y and a finite map Y X 3 X 3. But since the finite map is an isomorphism on the generic fiber, it is an isomorphism of Y onto an open subscheme of X 3 X 3, as desired. 7
8 Theorem 25 (BLR 5, 6) Given X 3,A there exists X 4 (smooth separable Rgroup scheme of finite type) containing X 3 as an Rdense open subscheme with the following properties: (X 4 ) K = A; the group law of X 4 extends the Rbirational group law on X 3. X 4 is unique, and is a Néron model. 2.4 Néron models of elliptic curves K a local field, R its ring of integers, k the residue field. Definition 26 A Kcurve is a proper, geometrically connected scheme over K of dimension 1 An Rcurve is a proper flat Rscheme whose geometric fibers are connected Theorem 27 (Resolution of singularities) If C is a smooth Kcurve there exists a regular Rmodel of C. Proof. By blowing up. A proof can be found in [?]. Definition 28 Given a smooth Kcurve C of genus at least 1, a minimal regular Rmodel of C is a regular Rmodel X of C such that for any other regular Rmodel Y of C, there exists a morphism Y X which induces the identity on the generic fibers. Theorem 29 [?, XXX] C as above. There exists a (unique) minimal regular Rmodel of C. Let now E/K be an elliptic curve and E be its minimal regular model. Consider the smooth locus of E. Since (E sm ) K = E admits a map (the identity) towards E, by the Néron mapping property there exists a (canonical) map E sm Néron(E). Theorem 30 (BLR 1.5 proposition 1) This canonical map is an isomorphism. 2.5 Néron model of a Jacobian Let S be any scheme and X/S be an Sscheme. We can define the relative Picard functor by P X/S : S Sch Set T Pic(X S T). The functor Pic X/S is the sheafification of P X/S for the fppf topology on S Sch. We have Pic X/S (T) = H 0 (T,R 1 f T, G m ), where f is the structure map X T T. If C is a smooth curve over K, then Pic C/K is a smooth commutative group scheme over K. Its connected component is an abelian variety over K (the Jacobian of C). if X is an Rmodel of C that is also semistable (i.e. X k is reduced with at most ordinary double points as singularities), then Pic X/S is representable by an Rgroup scheme, whose connected component Pic 0 X/S is a semistable model of Jac(C). By the Néron mapping property, there is a map Pic 0 X/S Néron(Jac(C)), and we have 8
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