Mordell-Lang in positive characteristic

Mordell-Lang in positive characteristic

PAULZIEGLER(*)

ABSTRACT- We give a new proof of the Mordell-Lang conjecture in positive characteristic for finitely generated subgroups. We also make some progress towards the full Mordell- Lang conjecture in positive characteristic.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 14G05; 14K12, 14G17 KEYWORDS. Mordell-Lang, positive characteristic.

1. Introduction

This article is concerned with the full Mordell-Lang conjecture in positive characteristic, which is the following statement:

CONJECTURE1.1. LetLbe an algebraically closed field of characteristicp>0.

LetAbe a semiabelian variety overL, letXAbe an irreducible subvariety and let G A(L) be a subgroup of finite rank. IfX(L)\Gis Zariski dense inX, then there exist a semiabelian varietyBoverF_p, a subvarietyYofBoverF_p, a homomorphism h:BL!A=StabA(X) with finite kernel and an elementa2(A=StabA(X))(L) such thatX=StabA(X)ˆh(Y)‡a.

Here Stab_A(X) denotes the translation stabilizer ofXinA. We call irreducible subvarietiesXsatisfying the conclusion of the above conjecturespecial.

We develop a new approach to Conjecture 1.1. Using this approach we give a new algebro-geometric proof of

THEOREM 1.2 (Hrushovski, [18]). Conjecture1.1 holds in case the groupG is finitely generated.

(*) Indirizzo dell'A.: Department of Mathematics, ETH ZuÈrich, CH-8092 ZuÈrich, Switzerland.

Supported by the Swiss National Science Foundation.

E-mail: [email protected]

This was first proved by Hrushovski in [18] using model-theoretic methods. An algebro-geometric proof of this result has previously been given by RoÈssler [24].

The general case of Conjecture 1.1 is still open. We show that in caseAis an ordinary abelian variety it can be reduced to the following conjecture, which is essentially a special case of Conjecture 1.1 (c.f. Section 6):

CONJECTURE1.3. LetL0be a field which is finitely generated overFpand let L^per₀ be a perfect closure of L0. Let A be a semiabelian variety over L0 and XA_L^per

0 an irreducible subvariety. If X(L^per₀ ) is Zariski dense in X_L^per

0 , then a translate ofXby an element ofA(L^per₀ ) is defined overL₀.

We give an overview of the structure of this article. Since we will studyXAas above through the completionsX^ A^ Spf(R[[x1;. . .;xn]]) along the origin when both X and A are defined over the valuation ring R of a local field of positive characteristic, in Section 2 we provide a number of facts about closed formal sub- schemes of Spf(R[[x₁;. . .;x_n]]). In fact we study more generally closed subschemes of the formal spectrum of the mixed power series ringR[[x₁;. . .;x_n]]hy₁;. . .;y_mi since this seems to be the natural setting for these results.

In Section 3 we collect some facts about special subvarieties and give a criterion for subvarieties to be special (Theorem 3.10).

In Section 4 we set up our method. In Subsection 4.1 we collect some facts about so-called completely slope divisiblep-divisible groups. These arep-divisible groups which possess a filtration such that on each graded piece, a power of the relative Frobenius coincides, up to an isomorphism, with a power of the multiplication-by-p morphism.

In Subsection 4.2, we construct our central tool, a certain Frobenius morphism F: LetRbe the valuation ring of a local fieldKof positive characteristic, letkbe the residue field ofR, letK be an algebraic closure ofK and letR^perK be the perfection of R. LetA be a semiabelian scheme overR. Denote by A^the com- pletion ofAalong the zero section of the special fiber. This is ap-divisible group over the formal spectrum Spf(R) ofR. Assume thatA^is completely slope divisible.

Then the facts from Subsection 4.1 yield a canonical isomorphism (A^_k)_Spf(R^per₎ A^_Spf(R^per₎. By transfering the Frobenius endomorphism ofA^_k with respect to the finite fieldkvia this isomorphism, we obtain an endomorphismF_A^ofA^_Spf(R^per₎. Its significance lies in the following characterization of special subvarieties of AK, where for a subvariety X of A containing the zero section we denote by X^ its completion along the zero section of the special fiber:

THEOREM 1.4 (see Theorem 4.17). Let X AK be an irreducible subvariety.

Then the following are equivalent:

(i) The subvarietyXKis special inAK.

(ii) There existx2X(K) andn51such that the schematic closureXofX x inAsatisfiesF_Aⁿ_^(X)b X.b

This can be considered as a formal analogue of the classification of subvarieties of semiabelian subvarieties which are invariant under an isogeny due to Pink and RoÈssler (c.f. Theorem 3.7). Finally, in Subsection 4.3, we show that given a finitely generated fieldL₀and a semiabelian varietyAoverL₀, one can always embedL₀ into a local field over whichA up to isogeny can be spread out to a semiabelian schemeAas above.

In Section 5, using the methods from Section 4, we prove Theorem 1.2: First we reduce to the situation considered above which allows us to defineF. Then using Theorem 1.4 we show that Theorem 1.2 follows from a variant of the following formal Mordell-Lang result:

THEOREM1.5 (see Theorem 5.1). LetG be a formal group over k which as a formal scheme is isomorphic toSpf(k[[x₁;. . .;x_n]]). LetGG(R)a finitely gen- erated subgroup. Let X G_Spf(R) be a closed formal subscheme. If X is the minimal closed formal subscheme containing X(R)\G, then there exist closed formal subschemesX1;. . .;XmofGSpf(R)defined over a finite field extension of k and elementsg₁;. . .;g_m2G(R)such thatX ˆ [_jX_j‡g_j.

Theorem 1.5 is proven by the same method which was used by Abramovich and Voloch in [2] to prove Theorem 1.2 in the case that the ambient semiabelian variety is defined over a finite field: First one reduces to the case thatX is irreducible in a suitable sense. Then after a suitable translation one may assume thatX(R)\pⁱG is dense inX for alli. Denote byR^pithe subring ofRconsisting of allpⁱ-th powers.

SincepⁱGG(R^pi) it follows thatX is defined overR^pifor alli50. From this it follows thatX is defined overkˆ \_i50R^pi.

In Section 6 we show that in caseAis an ordinary abelian variety Conjecture 1.1 can be reduced to Conjecture 1.3 by combining our method with a reduction due to Ghioca, Moosa and Scanlon. This depends crucially on the fact that in case Ais ordinary, the endomorphismFA^ofA^_Spf(R^per₎described above can already be defined over Spf(R). The argument proceeds similarly to the proof of Theorem 1.2 sket- ched above by reduction to an analogous statement (see Theorem 2.44) for formal group schemes.

NOTATION. We will frequently work over a local fieldKof characteristicp>0.

For such aK, we will always fix an algebraic closureK which we endow with the unique extension of the valuation onK.

We will denote byR(resp.R) the valuation ring of K(resp.K), by m(resp.m) the maximal ideal ofR(resp.R) and by k(resp.k) the residue field ofR(resp.R).

We will denote byK^ the completion ofK. By a complete overfield K⁰K^ ofK we mean a field which is complete with respect to the valuation induced fromK.^ The valuation ring of such aK⁰will be denotedR⁰and the formal scheme associated to R⁰ equipped with the valuation topology will be denoted by Spf(R⁰). We will denote byK^per the perfection ofK insideK and byR^per the valuation ring ofK^per.

Fori50 we letK^pibe the field consisting ofpⁱ-th powers of elements ofKand R^pi the valuation ring ofK^pi.

By a subvariety of a scheme which is of finite type over some field we mean a geometrically reduced closed subscheme.

2. Formal Schemes

LetKbe a local field of characteristicp>0.

By an adic ring we mean the same as in [15, 7.1.9], that is a complete and se- parated topological ring whose topology is defined by an idealJ. We will also call such a ring aJ-adic ring.

ForR⁰the valuation ring of a complete overfieldK⁰K^ andn;m50 we denote by C_n;m⁰ the ringR⁰[[x1;. . .;xn]]hy1;. . .;ymiwhich consists of those power series P

I2(Z⁵⁰)ⁿ;J2(Z⁵⁰)^maIJx^Iy^J with coefficientsaIJ2R⁰ such that for eachI the coef-

ficientsa_IJconverge to zero asJgoes to infinity. We endowC_n;m⁰ with the topology defined by the idealJ⁰_n;mgenerated bymand the variablesx₁;. . .;x_n. This makes C_n;m⁰ into an adic ring. ForR⁰ˆRwe letC_n;m:ˆC⁰_n;mandJ_n;m:ˆJ⁰_n;m.

By formal schemes, we mean the same as in [15, Section 10]. In this section, we are concerned with affine formal schemesX over Spf(R) defined by the following class of rings:

DEFINITION2.1 (c.f. [19, Section 2.1] and [4, Section 1]). A topologicalR-algebra C is of formally finite type if it is adic and if for some ideal of definitionJ the quotientsC=Jⁱare of finite type overRfor alli50.

DEFINITION2.2. We denote by AFS_Rthe full subcategory of of the category of formal schemes over Spf(R) whose objects are the formal schemes of the form Spf(C) forCa topologicalR-algebra of formally finite type.

LEMMA 2.3 ([4, Lemma 1.2]). For a J-adic R-algebra C the following are equivalent:

(i) The ringCis of formally finite type.

(ii) The ringC=J²is finitely generated overR.

(iii) The ringC is topologically isomorphic over Rto a quotient of Cn;m for somen;m50.

REMARK 2.4. Let X ˆSpf(C)2AFSR. By the remark after [15, Definition 10.14.2] closed formal subschemes ofX correspond to ideals ofC. Thus by Lemma 2.3 a formal scheme over Spf(R) is in AFS_R if and only if it admits a closed em- bedding into Spf(C_n;m) for somen;m50.

DEFINITION2.5 (c.f. [5, Def. III.2.8.1]). A continuous homomorphismh:G!G⁰ of topological groups isstrictif the quotient topology on the image ofhinduced from G coincides with the subspace topology induced from G⁰. A continuous homo- morphism of topological rings or modules is strict if the underlying homomorphism of topological groups is strict.

The following summarizes properties of topological R-algebras of formally fi- nite type:

PROPOSITION 2.6. Let C and C⁰ be topological R-algebras of formally finite type.

(i) The Jacobson radical ofCis an ideal of definition, in fact it is the largest ideal of definition. In particular there is a unique topology on the ringC which makesCinto a topologicalR-algebra of formally finite type.

(ii) Every homomorphismC!C⁰is continuous.

(iii) Every homomorphismC!C⁰is strict.

(iv) Each ideal ofCis closed.

(v) The ringC_RKis Jacobson.

(vi) For each maximal idealnofC_RK the quotient(C_RK)=nis a finite field extension ofK.

PROOF. For (i) and (ii) see [19, Lemma 2.1]. For (iii) and (iv) see [4, Lemma 1.1].

For (v) see [19, Proposition 2.16] and for (vi) see [19, Lemma 2.3]. p We will also have to work with formal schemesX_Spf(R⁰₎forX 2AFS_R andR⁰ the valuation ring of a complete overfield K⁰K^ and with closed formal sub- schemes of such formal schemes. However, in [15] the notion of a formal subscheme is only defined for locally Noetherian formal schemes, and valuation rings R⁰ as above are in general not Noetherian. Thus we make the following ad hoc definition:

DEFINITION 2.7. A morphism Spf(C)!Spf(C⁰) of affine formal schemes is a closed embedding if the corresponding homomorphism C⁰!C is surjective and strict. In this case we will say that Spf(C) is aclosed formal subschemeof Spf(C⁰).

Thus closed formal subschemes of Spf(C⁰) correspond to closed ideals ofC⁰. In caseC⁰is Noetherian, this definition coincides with the one from [15] by the remark after [15, Definition 10.14.2].

DEFINITION2.8. LetR⁰be the valuation ring of a complete overfieldK⁰K. We^ denote by AFS_R⁰ the full subcategory of the category of formal schemes over Spf(R⁰) whose objects are those affine formal schemes which admit a closed em- bedding into Spf(C⁰_n;m) for somen;m50.

LEMMA 2.9. Let C be a Noetherian J-adic ring and C⁰ a J⁰-adic ring. Let C!C⁰be a ring homomorphism such that JC⁰J⁰and such that for each i50the induced homomorphism C=Jⁱ!C⁰=(J⁰)ⁱis faithfully flat.

Let 0!M⁰!M!M⁰⁰!0 a sequence of finitely generated C-modules.

We endow these modules with the J-adic topology. Then the sequence 0!M⁰!M!M⁰⁰!0 is exact if and only if 0!M⁰^_CC⁰!M^_CC⁰! M⁰⁰^_CC⁰!0 is exact. If this holds, the homomorphism M^_CC⁰!M⁰⁰^_CC⁰ is strict.

PROOF. Assume that 0!M⁰!M!M⁰⁰!0 is exact.

For i50 letM⁰_i:ˆJⁱM\M⁰. SinceCis Noetherian, by [6, Theorem III.3.2.2]

the topology onM⁰defined by theM⁰_iis theJ-adic topology. This together with the fact that JC⁰J⁰ implies that (M_iC(J⁰)ⁱC⁰)_i50 is a fundamental system of neighborhoods of the identity inM⁰_CC⁰. ThusM⁰^_CC⁰can be written as

M⁰^_CC⁰ˆlim

i

M⁰=M⁰_iCC⁰=(J⁰)ⁱ:

Fori50 there is an exact sequence 0!M⁰=M_i⁰!M=JⁱM!M⁰⁰=JⁱM⁰⁰!0 of C=Jⁱ-modules. SinceC=Jⁱ!C⁰=(J⁰)ⁱis flat, this induces an exact sequence 0!M⁰=M⁰_iC=JiC⁰=(J⁰)ⁱ!M=JⁱM_C=JiC⁰=(J⁰)ⁱ!M⁰⁰=JⁱM⁰⁰_C=JiC⁰=(J⁰)ⁱ!0:

The transition morphisms M⁰=M⁰_iCC⁰=(J⁰)ⁱ!M⁰=M⁰_{i 1C}C⁰=(J⁰)^{i 1} are sur- jective. By [3, Proposition 10.2] this surjectivity implies that the sequence 0!M⁰^_CC⁰!M^_CC⁰!M⁰⁰^_CC⁰!0 obtained by taking the inverse limit of the above exact sequences is again exact.

To prove the other direction of the claim, by a direct verification it suffices to show that if M is a non-zero finitely generated C-module endowed with theJ-adic topology, the ringM^_CC⁰is non-zero. As above we can writeM^_CC⁰ as lim

i

M=JⁱM_C=JⁱC⁰=(J⁰)ⁱ with surjective transition homomorphisms M=JⁱM_C=JⁱC⁰=(J⁰)ⁱ!M=J^{i 1}M_C=J^{i 1}C⁰=(J⁰)^{i 1}. As it is finitely generated over the complete Noetherian ring C, the module M is complete. Hence the modulesM=JⁱMare non-zero. Thus by the faithful flatness of C=Jⁱ!C⁰=(J⁰)ⁱ the modules M=JⁱM_C=JiC⁰=(J⁰)ⁱ are non-zero. Hence the surjectivity of the transition morphisms implies that M^_CC⁰ is non-zero.

It remains to prove the claim about strictness. SinceJC⁰J⁰, the topology on M^_CC⁰ and M⁰^_CC⁰ is the J⁰-adic topology. Hence if the homomorphism M^_CC⁰!M⁰^_CC⁰ is surjective, its strictness follows from the fact that p((J⁰)ⁱ(M^CC⁰))ˆ(J⁰)ⁱ(M⁰^_CC⁰). p LEMMA 2.10. Let R⁰ be the valuation ring of a completely valued overfield K⁰K of K. For n;^ m;i50 the ring homomorphism Cn;m=Jⁱ_n;m!C_n;m⁰ =(J⁰_n;m)ⁱ induced by the inclusion Cn;m,!C_n;m⁰ is faithfully flat.

PROOF. The homomorphism in question is

(R=mⁱ)[x₁;. . .;x_n;y₁;. . .;y_n]=(x₁;. . .;x_n)ⁱ

!(R⁰=(mR⁰)ⁱ)[x1;. . .;xn;y1;. . .;yn]=(x1;. . .;xn)ⁱ

R⁰R((R=mⁱ)[x1;. . .;xn;y1;. . .;yn]=(x1;. . .;xn)ⁱ):

Thus the claim follows from the faithful flatness ofR!R⁰. p LEMMA 2.11. Let X ˆSpf(C)2AFSR and X⁰ˆSpf(C⁰) a closed formal subscheme of X defined by an ideal I of C. Let R⁰ be the valuation ring of a complete overfield K⁰K of K. Then^ X⁰_Spf(R0) is the closed formal subscheme of X_Spf(R⁰₎defined by the ideal I(C^_RR⁰)of C^_RR⁰. This ideal is equal to I^_RR⁰.

PROOF. Pick a strict continuous surjectionCn;m!C for somen;m50. Note that M^_RR⁰ˆM^_Cn;mC_n;m⁰ for any topological C-moduleM. By Proposition 2.6 the topology on C⁰ is the same as the topology defined by J_n;mC⁰. Hence by Lemma 2.9 applied toCn;m!C_n;m⁰ , which is possible by Lemma 2.10, there is an exact sequence 0!I^RR⁰!C^RR⁰!(C=I)^RR⁰!0 and the homomorphism C^_RR⁰!(C=I)^_RR⁰is strict. ThusX⁰is the closed formal subscheme ofX_Spf(R⁰₎ defined by the idealI^_RR⁰inC^_RR⁰. SinceCis Noetherian, there is a surjective homomorphism of C-modules C^k !Ifor somek50. Again by Lemma 2.9 this induces a surjection (C^_RR⁰)^kC^k^_RR⁰!I^_RR⁰ which implies I^_RR⁰ˆ

I(C^RR⁰). p

DEFINITION 2.12. Let X ˆSpf(C) be an affine formal scheme and X1;. . .;Xm be closed formal subschemes of X defined by closed ideals I1;. . .;ImofC. We say thatX is the union of the formal subschemesX_iif the intersection of the ideals I_i is the zero ideal of C.

2.1 ±Points overR

DEFINITION2.13. Let X ˆSpf(C)2AFS_R. We defineX(R) to be the set of homomorphismsC!R ofR-algebras.

LEMMA2.14. Let C be a topological R-algebra of formally finite type and let h:C!R be a homomorphism of R-algebras.

(i) The homomorphismhfactors through the valuation ringR⁰of a finite field extensionK⁰KofK.

(ii) The homomorphismhis continous.

PROOF. (i): The homomorphism h induces a homomorphism C_RK! R _RK !K with the last homomorphism given by multiplication. Its kernel is

a maximal ideal n of CRK. By Proposition 2.6 the quotient (CRK)=n is a finite field extension of K. This implies (i).

(ii): By (i) it suffices to show that if R⁰ is the valuation ring of a finite field extensionK⁰K ofKany homomorphismh:C!R⁰ofR-algebras is continuous.

SinceR⁰is of formally finite type this is a special case of Proposition 2.6 (ii). p Caution: The ring R is not complete with respect to the valuation topology.

Thus there is no formal scheme Spf(R) and X(R) cannot be considered as X(Spf(R)). The set X(R) is also not the same as X(Spf(R)).^

Our interest in the setX(R) comes from the following situation, to which we will later apply the results of this section: Let Abe a semiabelian scheme of di- mensiongoverRandX Aa closed subscheme containing the zero section. Lets be the closed point of the zero section ofA. LetA^andX^ be the formal schemes associated to the completions of the local ringsO_A;sandO_X;swith respect to their maximal ideals. Then A ^ Spf(R[[x₁;. . .;x_g]]) so thatX^2AFS_R and X(^ R) is naturally identified with the set points inX(R) which map to 0 in the special fiber (c.f. Subsection 2.6).

Note that the formation ofX(R) is functorial in X.

For a finite field extensionK⁰K ofK with valuation ringR⁰andX 2AFS_R we denote byX(R⁰) the set of homomorphismsX !Spf(R⁰) over Spf(R). There is a natural inclusion X(R⁰),!X(R) and Lemma 2.14 (i) implies:

LEMMA 2.15. Let X 2AFSR. The set X(R) is the unionX(R) ˆ [K⁰X(R⁰) where K⁰ varies over all finite field extensions of K contained inK.

REMARK 2.16. For X 2AFS_R, the set X(R) can be described more con- cretely as follows: For anyr₁;. . .;r_n 2m ands₁;. . .;s_n2R there exists a unique continous homomorphism of R-algebras Cn;m!R which sends thexi to ri and the yitosi and each homomorphismCn;m!R is of this form. Thus associating to an elementh2Spf(Cn;m)(R) the images of the x_iand they_igives a bijection Spf(C_n;m)(R) ! mⁿR^m. Any closed subschemeX of Spf(C_n;m) is cut out by a family of formal power series ff_iji2Ig C_n;m. Each formal power series f 2Cn;m induces a function mⁿR^m!R. The above bijection identifies X(R) with the set of points in mⁿR^m on which the fi are zero.

DEFINITION2.17. LetG:ˆAut_R…R†  Aut_K…K†. For X 2AFS_R, we letGact on X(R) from the left by

GX(R) !X(R)

(g;h)7!gh:G(X;O_X)!^h R !^g R:

LEMMA2.18. Let C be a topological R-algebra of formally finite type. Every homomorphism h:C!K of R-algebras factors through R.

PROOF. The homomorphism h induces a homomorphism CRK! R _RK !K with the last homomorphism given by multiplication. Its kernel is a maximal ideal nof C_RK. By Proposition 2.6 the quotient (C_RK)=n is a finite field extension of K. Hence it suffices to consider homomorphism h:C!K⁰ofR-algebras where K⁰ is a finite overfield ofK contained inK. As- sume that there is such anhwhose image is not contained inR⁰. Then the image of h_RR⁰:C_RR⁰!K⁰_RR⁰!K⁰ is also not contained in R⁰. Hence after replacingKbyK⁰we may assumeK⁰ˆK. Since the image ofhcontainsR, one sees that it must be all of K. Hence the kernel of h is maximal. Let p be a uniformizer ofR. Sincepis topologically nilpotent inCand since by Proposition 2.6 the Jacobson radical J of C is an ideal of definition, the element p is con- tained inJ and hence in the kernel ofh. This contradicts h(p)ˆp. p

For a ringC, we denote by Max(C) the set of maximal ideals ofC.

PROPOSITION 2.19. Let X ˆSpf(C)2AFS_R. Let c be the map X(R) ! Max(C_RK)which associates to h2X(R) the kernel of the induced homomorph- ism h_RK:C_RK!R _RK K.

(i) The mapcmakesMax(CRK)into the set-theoretic quotient ofX(R) by the action ofG.

(ii) Let Y ˆSpf(C⁰) be a closed formal subscheme of X. Then there is a commutative diagram

in whichc⁰is the analogue ofcforY andMax(C⁰_RK)!Max(C_RK) is induced by the surjectionC!C⁰.

PROOF. (i): By Lemma 2.18 any homomorphism C_RK!K of K-alge- bras maps C to R. Hence the assigment h7!h_RK:C_RK!K gives a bijection between X(R) and the set of homomorphisms ~h:C_RK!K of K- algebras.

The kernel of any such homomorphismh~is a maximal ideal ofCRK. On the other hand, for any maximal idealnofC_RK, the quotient (C_RK)=nis a finite field extensionK⁰ofKby Proposition 2.6. Thus givingh~as above amounts to giving a maximal idealnofC_RKand an embedding of (C_RK)=nintoK overK. Any two such embeddings are conjugate underG.

By combining the above facts one gets (i).

(ii) follows by a direct verification. p

DEFINITION2.20. LetR⁰be the valuation ring of a completely valued overfield K⁰ofK. LetX ˆSpf(C) be an affine formal scheme over Spf(R⁰).

(i) The formal schemeX isreducedif the ringCis reduced.

(ii) The formal schemeX isflat overR⁰if the ringCis flat overR⁰.

(iii) We denote byAFS^rf_R0the full subcategory ofAFSR⁰whose objects are those formal schemes which are reduced and flat overR.

LEMMA 2.21. Let X ˆSpf(C)2AFS_R. Let X^rf be the closed formal sub- scheme ofX defined by the idealfc2Cj 9n50:(cm)ⁿ ˆ0g. This formal scheme is reduced and flat over R and the natural mapX ^rf(R) !X(R) is a bijection.

PROOF. Direct verification using the fact that anR-module is flat if and only if it

has nom-torsion. p

PROPOSITION2.22. LetY₁;Y₂ 2AFS^rf_R be two closed formal subschemes of X 2AFSR. If Y₁(R) Y₂(R), then Y₁Y₂.

PROOF. LetCbe the ring of global sections ofX andI₁;I₂the ideals defining Y₁;Y₂. We want to proveI2I1. Letpbe a uniformizer ofR. The fact that theY_i are flat over Rmeans that pis not a zero-divisor inC=I_i. This implies that it is enough to proveI2_RKI1_RKinsideC_RKˆC[1=p]. Since by assumption the idealsI_iare radical, so are the idealsI_iRK. Since by Proposition 2.6 the ring C_RK is Jacobson it suffices to prove that each maximal ideal ofC_RK which containsI_1RK also containsI_2RK. This follows from the fact thatY₁(R)

Y₂(R) and Proposition 2.19. p

For i50, we endow the ring R=(m R) ⁱ with the quotient topology induced from the valuation topology onR, with respect to which it is adic. Hence there is a formal scheme Spf(R=(m R) ⁱ) and forX 2AFSRwe denote byX(R=(m R) ⁱ) the set of morphisms Spf(R=(m R) ⁱ)!X over Spf(R). There is a natural map X(R) ! X(R=(m R) ⁱ) for alli50.

COROLLARY2.23. LetX 2AFS_R. The setX(R) is nonempty if and only if for all i50 the set X(R=m ⁱR) is nonempty.

PROOF. The ``only if'' direction is clear. Conversely, assume thatX(R) is empty.

Let X^rf2AFS^rf_R be the closed formal subscheme given by Lemma 2.21. Since X^rf(R) ˆX(R) is empty, it follows from Proposition 2.22 that X^rf is the empty formal scheme. Hence, if we letC:ˆG…X;O_X†andpis a uniformizer ofR, it follows from the definition ofX^rfthat for eachc2Cthere existsn50 such that (cp)ⁿˆ0.

Forcˆ1 we get that there is ann50 such that the image ofpⁿinCis zero. Hence for alli>nthere is no homomorphismC!R=(m R) ⁱofR-algebras since for suchi the image ofpⁿinR=(m R) ⁱis not zero. This proves the claim. p

LEMMA 2.24. LetX ˆSpf(C)2AFSR be be the union of closed subschemes X₁;. . .;X_m. ThenX(R) ˆ [_iX_i(R).

PROOF. Let I₁;. . .;I_mC be the ideals defining theX_i. Leth:C!R. We want to show that h(Ii)ˆ0 for some i. If this is not the case we pick 06ˆri2 h(Ii). Then the productrof theriis a non-zero element which lies inh(Ii) for all i. For eachipickc_i2I_isuch thath(c_i)ˆr. Then the product of thec_ilies in the intersection of the I_i which by assumption is zero. Thus by applying h to this product we get r^mˆ0 and hence rˆ0, which is a contradiction. p

2.2 ±Irreducibility

DEFINITION2.25. LetR⁰be the valuation ring of a finite field extensionK⁰ofK.

LetX ˆSpf(C)2AFS^rf_R0be non-empty.

The formal schemeX isirreducibleif and only ifSpec(C) is irreducible.

An irreducible component ofX is a maximal irreducible closed formal sub- scheme.

The formal scheme X is geometrically irreducible if and only if X_R⁰⁰ is irreducible for all valuation ringsR⁰⁰ of finite field extensionsK⁰⁰ of K⁰. Note that the irreducible components of X correspond to the irreducible components of Spec(C), that is to the minimal prime ideals ofC. In particular there are finitely many such components. Also, sinceCis reduced, the intersection of all its minimal prime ideals is the zero ideal. Thus X is the union of its irreducible components in the language of Definition 2.12.

PROPOSITION2.26. LetX 2AFS^rf_R be non-empty. Each irreducible component ofX is reduced and flat over R.

PROOF. LetC:ˆG…X;OX†. It suffices to show that each irreducible component of Spec(C) is reduced and flat overR. Reducedness is clear. AsRis a discrete val- uation ring, a schemeXover Spec(R) is flat if and only if its generic fiber is sche- matically dense inX. By a direct verification, if Spec(C) satisfies this condition, then

so does any irreducible component of Spec(C). p

LEMMA 2.27. Let X ˆSpf(C)2AFS^rf_R be irreducible. If X1;. . .;Xm are closed formal subschemes ofX such thatX is the union of theX_i, thenX ˆX_i for some i.

PROOF. LetI₁;. . .;I_mCbe the ideals defining theX_i. By assumption their intersection is zero and C is integral. If all Ii were non-zero, we could pick ele-

ments 06ˆx_i2I_iwhose product would be zero. Thus one of the I_iis zero, which

is what we wanted. p

In [9, Section 7], de Jong gives a construction, due to Berthelot, of a ``generic fiber'' functor from AFS_Rto the category of quasi-separated rigid analytic spaces over K. We denote this functor by X 7!X^rig. It can be described on objects as follows: The formal scheme Spf(Cn;m) is sent to the product of the openn-dimen- sional unit discDⁿ_K overK and the closedm-dimensional unit discB^m_K over K. A closed formal subscheme X as above is cut out by a family of power series ff_iji2Ig C_n;m. These f_i induce global sections of Dⁿ_KB^m_K, and X^rig is the closed rigid analytic subspace ofD^m_KBⁿ_K cut out by these global sections.

In [7], Conrad introduces the notion of irreducibility of a quasi-separated rigid analytic space and that of an irreducible component of such a space. He also shows that this notion is well-behaved under the functorX 7!X^rig. The following is a slight reformulation of [7, Theorem 2.3.1]:

THEOREM 2.28. Let X 2AFS^rf_R be non-empty and X1;. . .;Xm be the irre- ducible components ofX. The closed rigid analytic subvarietiesX_i^rigofX ^rigare the irreducible components ofX^rig.

PROOF. In fact, [7, Theorem 2.3.1] says the following: LetC:ˆG…X;O_X†and let C~ be the normalization of C, that is the integral closure of Cin its total ring of fractions. LetI1;. . .;Inbe the preimages inCof the minimal prime ideals ofC. Let~ Y₁;. . .;Y_n be the closed formal subschemes ofX defined by the idealsI_i. Then Y^rig₁ ;. . .;Y_n^rigare the irreducible components ofX^rig.

Since C~ is the normalization of C, the preimages of the minimal prime ideals of C~ are exactly the minimal prime ideals of C. Thus Theorem 2.28 is a re-

formulation of [7, Theorem 2.3.1]. p

COROLLARY2.29. A non-empty formal schemeX 2AFS^rf_R is irreducible if and only if the rigid analytic space X^rig is irreducible.

PROPOSITION 2.30. LetX 2AFS^rf_R be non-empty. There exists a finite field extension K⁰K of K with valuation ring R ⁰such that the irreducible components ofX_Spf(R⁰₎ are geometrically irreducible.

PROOF OFPROPOSITION2.30. In [7, Section 3.4], Conrad calls a quasi-separated rigid analytic spaceXoverKgeometrically irreducible if for all completely valued overfieldsK⁰ofKthe rigid analytic spaceX_K⁰is irreducible. By [7, Theorem 3.4.3], for any quasi-separated rigid analytic space Xover K having finitely many irre- ducible components, there exists a finite field extensionK⁰K ofKwith valuation ring R⁰ such thatX^rig_K0 has finitely many irreducible components which are geo- metrically irreducible. Using the compatibility of the functorX 7!X^rigwith base

change to finite extensions ofK(c.f. [9, 7.2.6]) the claim follows from this result and

Theorem 2.28. p

2.2.1 - Formal Schematic Image

DEFINITION2.31. Letf:X !X⁰be a morphism of affine formal schemes. We define the formal schematic image f(X) of f to be the intersection of all closed formal subschemes of X⁰ through which f factors.

Thus f(X) is the smallest closed formal subscheme of X⁰ through which f factors. IfX ˆSpf(C) andX⁰ˆSpf(C⁰) then the ideal corresponding tof(X) is the kernel of the homomorphismC⁰!Ccorresponding toX !X⁰.

LEMMA2.32. LetG;G⁰be connected p-divisible groups overSpf(R)considered as formal schemes. Let f:G !G⁰be an isogeny andY G⁰be a closed formal subscheme. Let X:ˆf ¹…Y†:ˆY _G⁰G be its preimage in G. Then Y is the formal schematic image ofX inG⁰.

PROOF. Let C:ˆG…G;OG†andC⁰:ˆG…G⁰;O⁰_G†. By [14, Proposition 4.4], the ringsCandC⁰ are isomorphic toR[[x1;. . .;xn]] for somen50. First we want to show thatC⁰!Cis flat. SinceR[[x₁;. . .;x_n]] is a regular local ring, by [12, Theorem 18.16] for this it suffices to show that dim (C⁰)ˆdim (C=n⁰C)‡dim (C⁰) wheren⁰is the maximal ideal ofC⁰. This follows from the fact thatC=n⁰Cis finite overk. Thus as a finite flat module over the local ringC⁰, the ringCis finite free overC⁰.

Let I⁰C⁰ be the ideal defining Y. Then X is the formal spectrum of C⁰=I⁰^_C⁰C. SinceC⁰!Cis finite free we haveC⁰=I⁰^_C⁰CˆC⁰=I⁰_C⁰CC=I⁰C.

Now letY⁰be the formal schematic image ofX inG⁰. It is contained inY. Thus it is defined by an ideal ~I⁰containing I⁰. Its preimage inG must coincide with X. Thus the induced homomorphism C⁰=I⁰C⁰C!C⁰=~I⁰C⁰C is an iso- morphism. Since C⁰!C is finite free this implies thatC⁰=I⁰!C⁰=~I⁰ is an iso-

morphism. This means Y⁰ˆY. p

2.3 ±Base Change and Formal Schematic Closure

PROPOSITION2.33. Let TmⁿR^mfor some n;m50. Let R⁰be the valua- tion ring of a complete overfield K⁰K of K. Let I^ C_n;mand I⁰C_n;m⁰ be the ideals consisting of those power series which vanish at all elements of T. Then IC⁰_n;mˆI⁰.

PROOF. The idealIis characterized by the left exact sequence 0!I!Cn;m^(ev!^t)t2T Y

t2T

R

where fort2Twe let evt:Cn;m!Rbe the function given by evaluation att.

Let DCn;m=I be the image of Cn;m in Q

t2TR endowed with the quotient topology from C_n;m. This is a topological R-algebra of formally finite type by Proposition 2.6. Lemma 2.11 yields an exact sequence

0!IC_n;m⁰ !C_n;m⁰ !D^RR⁰!0:

…2:34†

LetD_ibe the kernel ofD!Q

t2TR!Q

t2TR=mⁱ. We haveDˆlim_iD=J_n;mⁱ D and the ringsD=J_n;mⁱ D, being quotients ofCn;m=Jⁱ_n;m, are Artinian. Furthermore

\_i50D_iˆ0. By [6, Prop. III.2.7.8] these facts imply that there exists a sequence (j(i))_i50 of positive integers going to infinity such that D_iJ_n;m^j(i) D. Since Jⁱ_n;mDDithis shows that the topology onDdefined by theDi is theJn;m-adic topology. ThusD^RR⁰can be written as lim_iD=D_iRR⁰.

The inclusion D,! Q

t2TR induces injections D=D_i,! Q

t2TR=mⁱ and hence injectionsD=D_iRR⁰,!(Q

t2TR=mⁱ)_RR⁰. Hence we get a homomorphism …2:35† D^RR⁰ˆlim

i

…D=DiRR⁰† !lim

i

……Y

t2T

R=mⁱ† RR⁰†

!lim

i

…Y

t2T

R⁰=…mR⁰†ⁱ† Y

t2T

R⁰ and by a direct verification the composition of this homomorphism with the homomorphism C_n;m⁰ !D^_RR⁰ from (2.34) is the homomorphism which evalu- ates a power series at the elements of T.

We want to show that the homomorphism (2.35) is injective. That the first arrow in (2.35) is injective follows from the choice of theD_i, the flatness ofR!R⁰and the left exactness of the inverse limit functor. By the left exactness of inverse limits, in order to prove injectivity of the second arrow in (2.35), it is enough to show that

(Y

t2T

R=mⁱ)RR⁰!Y

t2T

R⁰=(mR⁰)ⁱ …2:36†

is injective for all i. Since any element of (Q

t2TR=mⁱ)RR⁰ is contained in the image of (Q

t2TR=mⁱ)RR⁰⁰for the valuation ringR⁰⁰of a finite field extension K⁰⁰K⁰ofKfor this step we may assume thatK⁰is finite overK. But thenR⁰is finite free overRand hence (2.36) is even an isomorphism in this case.

Thus by combining (2.34) and (2.35) we get a left exact sequence 0!IC⁰_n;m!C⁰_n;m^(ev!^t)t2T Y

t2T

R⁰

which showsIC⁰_n;mˆI⁰. p

DEFINITION2.37. LetR⁰be the valuation ring of a complete overfieldK⁰K^ of K. LetX be an affine formal scheme over Spf(R⁰) andTX(R⁰). We define the formal schematic closureofTinX to be the intersection of all closed subschemes Y ofX for whichTY(R⁰).

IfX⁰is the formal schematic closure ofT, then we say thatTisformal-sche- matically denseinX⁰.

Thus the formal schematic closure ofTis the smallest closed subscheme ofX which containsT.

COROLLARY 2.38. Let X 2AFS_R and let TX(R). Let Y be the formal schematic closure of T inX. Let R⁰be the valuation ring of a complete overfield K⁰K of K. Then^ Y_Spf(R0) is the formal schematic closure of TX(R)X(R⁰) insideX_Spf(R⁰₎.

PROOF. It suffices to prove this for X ˆSpf(C_n;m) in which case it is a re-

formulation of Proposition 2.33. p

2.4 ±Transporters

LetG be an abelian group object in the category AFS_R. We will need the ex- istence of (strict) transporters inG:

CONSTRUCTION 2.39. Let X;Y G be two closed formal subschemes. For i50 letG_ibe thei-th infinitesimal neighborhood of the zero section inG and let X_i;Y_i:ˆG_i\X;G_i\Y. These are finite schemes over R=mⁱ. The group structure on G makes G_i into a group scheme over R=mⁱ andX_i and Y_i are closed subschemes ofGi. Fori50 let TransGi(Xi;Y_i) be the strict transporter in Gi, that is the closed subscheme ofGiwhose points are those pointsgofGifor whichX_i‡gˆY_i. It exists by [3, Exemple VI.6.4.2 e)]. Then for eachione has a decreasing sequence (Trans_Gi‡j(X_i‡j;Y_i‡j)\G_i)_j50 of subschemes of G_i. By noetherianity this sequence stabilizes; let Trans_G(X;Y)_i be its eventual value.

Then TransG(X;Y)_i‡1\GiˆTransG(X;Y)_ifor alli. Hence the inductive limit of these schemes is a closed formal subscheme TransG(X;Y) ofG.

For a formal schemeZover Spf(R) and an affine formal scheme Spf(C) over Spf(R) we denote byZ(C) the set of morphisms Spf(C)!Zover Spf(R).

PROPOSITION2.40. LetX;Y be closed formal subschemes ofG and letSpf(C) be an affine formal scheme overSpf(R). Then

TransG(X;Y)(C)ˆ fg2G(C)jg‡XSpf(C)ˆY_Spf(C)g:

PROOF. LetJbe an ideal of definition ofC. Letg:R[[x₁;. . .;x_n]]!C2G(C).

Since g is continuous, there exists N50 such that g(J^N_n;0)J. Then g induces homomorphisms g_j:R[[x₁;. . .;x_n]]=J^jN_n;0!C=J^j for each j51. Denote as in

Construction 2.39 byGjˆSpec(R[[x1;. . .;xn]]=J^j_n;0) thej-th infinitesimal neigh- bourhood of the origin inG. Theng_j2G_jN(C=J^j)G(C=J^j) is the image ofgin G(C=J^j). The construction of Trans_G(X;Y) implies:

g2TransG(X;Y)(C)()8j50:gj 2TransG(X;Y)_Nj(C=J^j)

()8j509N_j5Nj8i5N_j:g_j‡(X_i)_C=Jj ˆ(Y_i)_C=Jj

()g‡X_Spf(C)ˆY_Spf(C):

p 2.5 ±Descent

LetR⁰be the valuation ring of a complete overfieldK⁰K^ ofK. LetX 2AFS_R and X⁰X_Spf(R⁰₎ a closed formal subscheme. If there exists a closed formal subscheme X⁰⁰ of X such thatX⁰⁰_Spf(R0) ˆX⁰ then it follows from Lemmas 2.9, 2.10 and 2.11 that such anX is unique. In this case we will say thatX⁰is defined over R.

In the following we will always endow k[[x1;. . .;xn]] with the (x1;. . .;xn)-adic topology. LetX ˆSpf(C) be a formal scheme over Spec(k) which is isomorphic to Spf(k[[x₁;. . .;x_n]]).

LEMMA2.41. LetX⁰be a closed formal subscheme ofX defined by an ideal I of C. Then X⁰_Spf(R) is the closed formal subscheme of X_Spf(R) defined by the ideal I(C^_kR)of C^_kR. This ideal is equal to I^_kR.

PROOF. Since the homomorphismsk[x1;. . .;xn]=(x1;. . .;xn)ⁱ!R[x1;. . .;xn]=

(x1;. . .;xn)ⁱ are faithfully flat, this follows from Lemma 2.9 in the same way as

Lemma 2.11. p

Let X⁰X_Spf(R) a closed formal subscheme. If there exists a closed formal subschemeX⁰⁰ofX such thatX⁰⁰_Spf(R)ˆX⁰then it follows from Lemmas 2.9 and 2.41 that such anX is unique. In this case we will say thatX⁰is defined over k.

LetF:X !X be the Frobenius endomorphism ofX with respect tok. IfX⁰ is a closed formal subscheme of X_Spf(R) defined by formal power series ff_iji2Ig R[[x₁;. . .;x_n]], then the formal schematic imageF(X) ofX underF is defined by the power series obtained by applyingFto the coefficients of thefi. LEMMA2.42. LetX⁰be a closed formal subscheme ofX_Spf(R). IfX⁰is defined over R^pifor all i50, thenX⁰is defined over k.

PROOF. We may assume thatX ˆSpf(k[[x1;. . .;xn]]). Leti50. As any element ofR^piis congruent to an element ofkmodulom^pi, the fact thatX⁰can be defined over R^pi implies that the intersection of X⁰ with Spec((R=m^pi)[x1;. . .;xn]=

(x1;. . .;xn)^pi)X can be defined overk. AsX⁰is the direct limit of these inter-

sections, the claim follows by varyingi. p

LEMMA2.43. A closed formal subschemeX⁰ofX_Spf(R)is defined over k if and only if F(X)ˆX.

PROOF. The ``only if'' direction is straightforward. For the other direction, the fact thatFⁱ(X)ˆX implies thatX is defined overR^pi for alli50. Thus we can

conclude using Lemma 2.42. p

PROPOSITION2.44. LetG be a formal abelian group scheme over k which as a formal scheme is isomorphic toSpf(k[[x1;. . .;xn]]). LetX G_Spf(R) be a closed formal subscheme. If for each i50there exists a finite field extension K⁰K of K and g2G(R⁰)such thatX_Spf(R⁰₎‡g is defined over R^pi, then there exists a finite field extension K⁰K of K with valuation ring R ⁰ and g2G(R⁰) such that XSpf(R⁰)‡g is defined over k.

PROOF. We consider the closed formal subscheme Trans_GSpf(R)(X;F(X)) G_Spf(R)given by Construction 2.39. Leti50 and pickR⁰as in the claim together with g2G(R⁰) such thatX_Spf(R⁰₎‡gis defined overR^pi. Letp2Rbe a uniformizer. By identifying Rwith k[[p]] and considering defining equations forXSpf(R⁰)‡g with coefficients from k[[p^pi]] one sees that XR⁰‡g F(XR⁰‡g)( modp^pi). Thus g F(g)2Trans_G(X;F(X))(R=(m R) ⁱ) by Proposition 2.40. Thus Proposition 2.23 implies that there existsg⁰2Trans_G(X;F(X))(R).

The morphismG !G;g7!g F(g) is the identity on the tangent space at the identity ofG and thus is an isomorphism by [17, A.4.5]. Henceg⁰can be written as g⁰⁰ F(g⁰⁰) for some g⁰⁰2G(R). By Lemma 2.15 there exists R⁰ as in the claim such thatg⁰⁰2TransGSpf(R)(X;F(X))(R⁰). The fact thatg⁰2TransG(X;F(X))(R⁰) translates to X_Spf(R⁰₎‡g⁰⁰ˆF(X_Spf(R⁰₎‡g⁰⁰). By Lemma 2.43 this shows that

X_Spf(R⁰₎‡g⁰⁰is defined overk. p

2.6 ±Formal schemes arising from schemes

LetX be a scheme locally of finite type overRtogether with ak-valued point s:Spec(k)! Xof the special fiber ofX. We letX^be the completion ofXalong the closed subschemes. We denote byOX;s the stalk ofOX at the closed point in the image ofs. ThenX^is the formal spectrum of the completionO^X;s of the local ring O_X;swith respect to its maximal ideal.

PROPOSITION2.45. (i)IfX is smooth over R at s of relative dimension n the formal schemeX^is isomorphic toSpf(R[[x1;. . .;xn]]).

(ii) The formal schemeX^is inAFSR.

(iii) The setX(^R) can be naturally identified with the set of elements ofX(R) which map the closed point ofSpec(R) to the closed point in the image of s.

PROOF. (i) The fact that X is smooth at s implies (in fact is equivalent to) that there exist an affine neighbourhood U of s and an eÂtale morphism U!Spec(R[x1;. . .;xn]) over Spec(R) which maps the zero section of the special fiber of Spec(R[x1;. . .;xn]) to s (c.f. [26, Tag 054L]). Such an eÂtale morphism induces a finite eÂtale morphism O^_X;s!R[[x₁;. . .;x_n]] of R-alge- bras. Since O^_X;s, being a complete local ring, is Henselian, the fact that both O^X;s and R[[x1;. . .;xn]] have residue field k implies that this is an iso- morphism.

(ii) From a closed embedding of an affine neighborhood of s into Spec(R[x₁;. . .;x_n]) for some n50 one gets a closed embedding of X^ into Spf(R[[x₁;. . .;x_n]]).

(iii) This follows from the fact thatX ˆ^ Spf(O^_X;s). p PROPOSITION2.46. LetX be a reduced scheme which is flat and of finite type over R and let s:Spec(k)! X a k-valued point ofX.

(i) The formal schemeX^is reduced and flat overR.

(ii) Assume thatXis integral and letY be an irreducible component ofX. The^ setY(R) X(R) is schematically dense inXR.

PROOF. (i) SinceXis reduced, so is the local ringOX;s0. The ringOX;s0 is also excellent. Since the completion of any excellent reduced local ring is reduced (c.f.

[16, 7.8.3]) the formal scheme X^ is reduced. Flatness follows from the flatness of R! OX;s0and the flatness ofOX;s!O^X;s.

(ii) LetY Xbe the schematic closure ofY(R) X(R) and let I OX be the sheaf of ideals definingY.

As O_X;s is a Noetherian local ring, the homomorphism O_X;s!O^_X;s is faith- fully flat and for any finitely generated O_X;s-module M, its completion with respect to the topology induced by the maximal ideal of OX;s is isomorphic to O^X;sOX;sM. By applying this toMˆ OY;s one sees thatY^is the formal closed subscheme of X^ corresponding to the ideal IsO^X;s. Since by construction Y(R) Y^(R) Proposition 2.22 implies Y Y^. Thus I_sO^_X;s is contained in a minimal prime ideal of O^_X;s and hence the rings O^_X;s and O^_Y;s have the same dimension.

Using the flatness ofOX;s!O^X;sand the fact that the maximal ideal ofO^X;sis generated by the image of the maximal ideal ofOX;s, Theorem 10.10 of [12] implies that dim (O_X;s)ˆdim (O^_X;s). Analogously we get dim (O_Y;s)ˆdim (O^_Y;s). Thus dim (Y)5dim (O_Y;s)ˆdim (O^_Y;s)ˆdim (O^_X;s)ˆdim (O_X;s). SinceX is irreducible dim (O_X;s)ˆdim (X) and thus we get dim (Y)ˆdim (X) which using the irredu-

cibility ofX impliesY ˆ X. p