aa r X i v : . [ m a t h . AG ] N ov NON-ADIC FORMAL SCHEMES
TAKEHIKO YASUDA
Abstract.
Our purpose is to make a contribution to the foundationof the theory of formal scheme. We are interested particularly in non-Noetherian or non-adic formal schemes, which have been little studied.We redefine the formal scheme as a proringed space and study its basicproperties. We also find several examples of non-adic formal schemes.
Contents
Introduction 2Acknowledgment 41. Prorings and promodules 41.1. Procategories 41.2. Prorings and promodules 61.3. Mittag-Leffler and epi pro-objects 72. Formal schemes as proringed spaces 92.1. Sheaves on a qsqc basis 102.2. Locally admissibly proringed spaces 112.3. Formal spectra 122.4. Morphisms of formal spectra 122.5. Formal schemes 132.6. Fiber products 143. Semicoherent promodules and formal subschemes 143.1. Semicoherent promodules 143.2. Pullback and pushforward 163.3. Mittag-Leffler O X -promodules 173.4. Stalks and exactness 183.5. Formal subschemes 193.6. Extension of a promodule on an open subset 203.7. Semicoherent promodules on a qsqc formal scheme 213.8. Locally ind-Noetherian formal schemes 224. Formal schemes as functors and formal algebraic spaces 244.1. Formal schemes as functors 244.2. Formal algebraic spaces 255. Mild and gentle formal schemes 285.1. Complete rings and complete modules 28 Mathematics Subject Classification.
Introduction
The formal scheme is an important tool for the infinitesimal analysis in thealgebraic geometry. In the original reference [EGA], Grothendieck definedthe formal scheme, which is not a priori locally Noetherian or adic. But hemade some arguments only under the locally Noetherian assumption. Alsoin most literature, one considers only locally Noetherian or at least adicformal schemes. The aim of this paper is to complement the theory of notnecessarily adic formal schemes. As far as I know, the first attempt in thisdirection after [EGA] is McQuillan’s one [McQ].Our first task is to redefine the formal scheme as a proringed space ( § § C consists of projective systems in C with appropriate Hom-sets. Now, by definition, a proring is an object of theprocategory of the category of rings and a proringed space is a topologicalspace with a sheaf of prorings. For a proring A satisfying some condition,which we call an admissible proring, we define the formal spectrum, Spf A ,and gluing formal spectrums, we obtain a formal scheme. In fact, a formalscheme is not only a proringed space, but also a locally admissibly proringedspace. Then a morphism of formal scheme is defined to be a morphism oflocally admissibly proringed spaces. We will see in § → (Sets) . Then we will define a formal algebraic space as a functor.
ON-ADIC FORMAL SCHEMES 3
We will introduce the notion of semicoherent promodules on a formalscheme ( § A , as the sheafification M △ of an A -promodule M , and do not impose the existence of local free presentationlike O JX → O IX → M → . Indeed, if a semicoherent promodule M is not Mittag-Leffler (see § O IX ։ M . The semicoherent pro-module is much like the quasicoherent module on a scheme. For a technicalreason, for a sheaf M on a formal scheme X , we only consider the values M ( U ) for quasi-compact and quasi-separated open U ⊆ X . It enables us toavoid some troublesome infinite projective limits.The local study of formal schemes reduces to the study of admissible pror-ings. However the proring does not seem suitable to discuss specific examplesand the complete (linearly topologized) ring seems better for this purpose.The category of complete rings embeds naturally and fully faithfully intothat of prorings. We define an admissible ring as a complete ring whoseassociated proring is admissible. It is equivalent to McQuillan’s admissiblering and more general than Grothendieck’s one.Similarly, the category of complete modules over a complete ring embedsinto that of promodules over the corresponding proring. The former categoryis additive but not abelian , while the latter is abelian. (It is the greatestadvantage of the use of promodule.) We say that a proring or promoduleis mild if it is isomorphic to the one associated to a complete one. Themildness is a key when moving from the “pro side” to the “complete side”.It is worth noticing that if a proring or promodule satisfies the Mittag-Lefflercondition (for instance, every admissible proring does) and is indexed by acountable set, then it is mild. So in practice, most admissible prorings aremild and considering a mild admissible proring is equivalent to consideringan admissible ring. The central purpose of the use of prorings is to ensureconsistency with promodules, the use of which is more essential. (See § § § Convention 0.1.
We denote by N the set of positive integers, and by N the set of non-negative integers. A ring means a commutative ring with Correspondingly the category of quasicoherent modules in [McQ] is not abelian andClaim 5.3 in op. cit. is not correct. This observation is due to a referee.
TAKEHIKO YASUDA unit. A projective (resp. inductive) system means a projective (resp. induc-tive) system indexed by a preordered set, which is also called a generalizedprojective (resp. inductive) system in literature. A directed projective (resp.inductive) system means a projective (resp. inductive) system indexed bya directed set. We write a projective or inductive system as ( X d ) d ∈ D orsimply ( X d ). The morphisms in a projective or inductive system are called bonding morphisms . We denote the category of sets by (Sets), that of ringsby (Rings) and so on. For categories C and D which admits finite projectiveand inductive limits, a (covariant) functor C → D is said to be left exact(resp. right exact, exact) if it commutes with finite projective limits (resp.finite inductive limits, finite projective and inductive limits). Acknowledgment.
I would like to thank Fumiharu Kato for useful dis-cussions. Also I gratefully acknowledges the many helpful comments of tworeferees. Especially suggestions of one of them helped me a lot to pursuethe generality. 1.
Prorings and promodules
In this section, we review some of the standard facts on procategories,prorings and promodules, which are required in subsequent sections.1.1.
Procategories.Definition 1.1.
Let C be a category. A pro-object of C is a directed pro-jective system in C . The procategory of C , denoted pro - C , is defined asfollows: An object of pro - C is a pro-object of C . For objects X = ( X d ) and Y = ( Y e ) of pro - C , the set of morphisms is defined byHom pro - C ( X, Y ) := lim ←− e lim −→ d Hom C ( X d , Y e ) . Note that the index sets are not supposed to be equal.The original references on the procategory are [SGA4, AM]. In these ref-erences, projective systems are more generally labeled by filtering categories.But it is proved in [MS] that this leads to an equivalent category.An object X ∈ C is considered as a projective system indexed by asingleton, and as an object of pro - C . This makes C a full subcategory of pro - C . It is tautology that for a directed projective system ( X d ) in C , itsprojective limit in pro - C is ( X d ) itself as an object of pro - C . Even if theprojective limit in C of ( X d ) exists, it is not generally isomorphic to the onein pro - C . In general, we denote the projective limit in C by lim ←− , and theone in pro - C by “ lim ←− ”.Let A = ( A d ) d ∈ D ∈ pro - C and φ : E → D an order-preserving mapof directed sets. We obtain a new pro-object A φ := ( A φ ( e ) ) e ∈ E . If φ iscofinal (that is, ∀ d ∈ D , ∃ e ∈ E , φ ( e ) ≥ d ), then A and A φ are canonicallyisomorphic. We say that A φ is the reindexing of A or that A φ is obtained byreindexing A by φ . ON-ADIC FORMAL SCHEMES 5
For A = ( A d ) d ∈ D , B = ( B e ) e ∈ E ∈ pro - C , a morphism f : A → B is bydefinition a collection of morphisms f e : A → B e , e ∈ E . Each f e is anelement of lim −→ d Hom C ( X d , Y e ). If f de : X d → Y e is a representative of theclass f e , then we say that f de represents f . It is equivalent to the diagramin pro - C A f / / (cid:15) (cid:15) B (cid:15) (cid:15) A d f de / / B e is commutative.A directed projective system ( f d : A d → B d ) of morphisms in C is calleda level morphism and induces a morphism f : ( A d ) → ( B d ) in pro - C in anobvious way. If every f d is a monomorphism or an epimorphism, then so is f . If A ∼ = ( A d ) and B ∼ = ( B d ) are isomorphisms in pro - C and if f ′ denotethe composite, A ∼ = ( A d ) f −→ ( B d ) ∼ = B , then we say that ( f d ) represents f ′ .Conversely, any morphism f : A → B in pro - C can be represented by alevel morphism in C . Moreover f can be represented by a level morphism( f d : A d → B d ) where ( A d ) and ( B d ) are reindexings of A and B respectively. Proposition 1.2. (1) If C has finite projective limits, then pro - C hasprojective and inductive limits. (2) Suppose C has finite projective limits. If ( A d ) is a projective systemin pro - C (each A d is a pro-object of C ) and B ∈ C , then Hom pro - C (“ lim ←− ” A d , B ) = lim −→ Hom pro - C ( A d , B ) . (3) If C is additive (resp. abelian), then so is pro - C .Proof. (1) pro - C has inductive limits, finite projective limits and directedprojective limits [AM, Propositions 4.2, 4.3 and 4.4]. In general, if a categoryhas finite projective limits and directed projective limits, then it has arbi-trary projective limits, which proves the assertion. Indeed, let ( X d ) d ∈ D be aprojective system labeled by a preordered set D in such a category. Let E bethe set of the finite subsets of D and for each A ∈ E , put X A := lim ←− d ∈ A X d .We make E a directed set by A ≤ B for A ⊆ B , A, B ∈ E . For A ≤ B , wehave the projection X B → X A and ( X A ) A ∈ E is a directed projective system.By assumption, lim ←− A ∈ E X A exists. It is now straightforward to check thatlim ←− A ∈ E X A = lim ←− d ∈ D X d .(2) For every X = ( X d ) ∈ pro - C , define the functor F X : C → (Sets) Y Hom pro - C ( X, Y ) = lim −→ Hom C ( X d , Y ) . It determines a fully faithful embedding pro - C ֒ → Fun( C , (Sets)) ◦ . For( A d ) in the assertion, define G ∈ Fun( C , (Sets)) by G ( Y ) := lim −→ Hom pro - C ( A d , Y ) . TAKEHIKO YASUDA
Then G is the projective limit of ( F A d ) in Fun( C , (Sets)). Since the directedinductive limit of sets is exact, G is left exact. From [AM, Cor. 2.8], G ispro-representable, that is, for some C ∈ pro - C , G ( Y ) = Hom pro - C ( C, Y ) . Then C is the projective limit of ( A d ), which shows the assertion.(3) [AM, Prop. 4.5]. (cid:3) Proposition 1.3.
The inclusion functor C ֒ → pro - C is exact. Moreover itcommutes also with (not necessarily finite) inductive limits.Proof. Equivalent statements for the ind-category are [SGA4, I, Prop. 8.9.1and 8.9.5]. (cid:3)
In particular, for an exact sequence in C of the form0 → A → B ⇒ C, is exact also in pro - C . If C is abelian, the same holds for a short exactsequence, 0 → A → B → C → . Prorings and promodules.Definition 1.4.
We define the category of abelian progroups to be the pro-category of the category of abelian groups.From Proposition 1.2, the category of abelian progroups is abelian.
Definition 1.5.
We define the category of prorings to be the procategoryof the category of rings.
Definition 1.6.
Let A be a proring. A ring B endowed with a morphism A → B is called an A - algebra . We define the category of A -proalgebras asthe procategory of the category of A -algebras.We see that giving an A -proalgebra is equivalent to giving a proring B endowed with a morphism A → B .We now fix a proring A = ( A d ) d ∈ D . Definition 1.7.
An abelian group M is called an A -module if it is given acompatible system of A d -actions, d ∈ D M , for some residual subset D M of D (that is, ∃ d ′ ∈ D , ∀ d ≥ d ′ , d ∈ D M ). We define the category of A -modules by defining the Hom-set as follows: For A -modules M and N ,Hom A ( M, N ) := [ d ∈ D M ∩ D N Hom A d ( M, N ) ⊆ Hom ab. gp. ( M, N ) . If every bonding map of A is surjective, then for every d ∈ D M ∩ D N , wesimply have Hom A ( M, N ) = Hom A d ( M, N ) . ON-ADIC FORMAL SCHEMES 7
For an A -module M and for d ∈ D , we say that A d acts on M if there existsan A d -action on M which is compatible with the A d ′ -actions for d ′ ∈ D M .Then we can safely add d to D M .It is easy to see that the category of A -modules is abelian and that in thiscategory, the notions of injection and monomorphism (resp. surjection andepimorphism) coincide. Definition 1.8.
We define the category of A -promodules as the procategoryof the category of A -modules.From Proposition 1.2, the category of A -promodules is abelian.An A -proalgebra B is naturally regarded as an A -promodule too. Inparticular, A itself is an A -promodule, but not an A -module unless its indexset is a singleton. If M is a B -promodule, then M is regarded as an A -promodule too, which we denote by M A . If B is isomorphic to A , then thefunctor M M A is an equivalence.Let M = ( M e ) e ∈ E be an A -promodule and set F := { ( d, e ) | A d acts on M e } ⊆ D × E. Then reindexing A and M by F , we may suppose that A and M has thesame index set D = E and for every d ∈ D , A d acts on M d . Definition 1.9.
Let A = ( A d ) be a proring, and M = ( M e ) and N =( N f ) A -promodules. Then we define the tensor product M ⊗ A N as the A -promodule ( M e ⊗ A d N f ). Here ( d, e, f ) runs over the triplets such that A d acts on M e and N f .If the index sets of A and M are equal and if for every d , A d acts on M d ,then we have a natural isomorphism M ⊗ A N ∼ = ( M d ⊗ A d N f ) , because the projective system ( M d ⊗ A d N f ) is a cofinal subsystem of ( M e ⊗ A d N f ). If the index set of N is also equal, then similarly M ⊗ A N = ( M d ⊗ A d N d ) . Now it is obvious that for each A -promodule M , the functor M ⊗ A − isright exact. Definition 1.10. An A -promodule M is said to be flat if − ⊗ A M is exact.If for each d , M d is a flat A d -module and M = ( M d ), then M is flat.If A is a proring and if B and C are A -proalgebras, then B ⊗ A C isalso an A -proalgebra. We see that B ⊗ A C is the sum in the category of A -proalgebras and hence the fiber sum in the category of prorings.1.3. Mittag-Leffler and epi pro-objects.
Fix an abelian category A suchthat any collection ( X i ) of subobjects of X ∈ A has the infimum, denoted T X i ⊆ X . TAKEHIKO YASUDA
Definition 1.11.
A pro-object ( X d ) d ∈ D of A is said to be Mittag-Leffler iffor every d ∈ D , there exists d ′ ∈ D such that \ e ≥ d Im( X e → X d ) = Im( X d ′ → X d ) . A pro-object is said to be epi if its every bonding morphism is an epimor-phism. A proring is said to be
Mittag-Leffler (resp. epi ) if it is so as anabelian progroup.Every epi pro-object is obviously Mittag-Leffler.
Remark . We define the Mittag-Leffler and epi properties for a proringby think of it as an abelian progroup.
Definition 1.13.
For a pro-object X = ( X d ), we define a pro-object X epi =( X epi d ) by for each d , X epi d := T e ≥ d Im( X e → X d ).The inclusions X epi d ֒ → X d induces a natural monomorphism X epi → X . Lemma 1.14.
Let X = ( X d ) be a pro-object. An endomorphism α : X → X is the identity if and only if for each d , there exists d ′ ≥ d such that thebonding morphism X d ′ → X d represents α .Proof. It follows from the definition of procategory. (cid:3)
Proposition 1.15.
For a Mittag-Leffler pro-object X = ( X d ) , the naturalmorphism X epi → X is an isomorphism.Proof. We construct the inverse of X epi → X as follows: For the pairs( d, d ′ ) ∈ D × D such that Im( X d ′ → X d ) = X epi d , the projections X d ′ → X epi d represent a morphism X → X epi . From the preceding lemma, this is theinverse of the X epi → X . (cid:3) Proposition 1.16.
A pro-object which is isomorphic to a Mittag-Lefflerpro-object is Mittag-Leffler. Thus being Mittag-Leffler is stable under iso-morphisms.Proof.
Let X = ( X d ) be a pro-object which is isomorphic to a Mittag-Lefflerpro-object. From the preceding proposition, X is also isomorphic to an epipro-object, say Y = ( Y e ). Let φ : X → Y and ψ : Y → X be isomorphismswhich are the inverse to each other. We now fix a member X d of ( X d ). If ψ ed : Y e → X d represents ψ , then put X ′ d := Im( ψ ed ), which is independent ofthe choice of ψ ed , because Y is epi. For every d ′ ≥ d and for any morphism Y e ′ → X d ′ representing ψ , we haveIm( X d ′ → X d ) ⊇ Im( Y e ′ → X d ′ → X d ) = X ′ d . On the other hand, for the above ψ ed : Y e → X d , and for a morphism X d ′′ → Y e representing φ , we have X ′ d ⊆ Im( X d ′′ → X d ) = Im( X d ′′ → Y e → X d ) ⊆ Im( Y e → X d ) = X ′ d . ON-ADIC FORMAL SCHEMES 9
Hence Im( X d ′′ → X d ) = X ′ d ⊆ \ d ′ ≥ d Im( X d ′ → X d ) ⊆ Im( X d ′′ → X d ) . It follows that X is Mittag-Leffler. (cid:3) Thus the essential image of the class of epi pro-objects in pro - A is thatof Mittag-Leffler pro-objects. Lemma 1.17.
Let φ : M = ( M d ) → N = ( N e ) be a morphism in pro - A .Suppose that ( N e ) is epi. Then φ is an epimorphism if and only if everymorphism M d → N e representing φ is an epimorphism.Proof. The “if” is obvious and holds without the assumption that N isepi. Suppose that φ is an epimorphism and some morphism ψ : M d → N e is not an epimorphism. Put N ′ e := Im( ψ ). Then the natural morphism N → N e is an epimorphism and so is the composite M → N → N e . Butthe last morphism factors as M → N ′ e → N e , and is not an epimorphism, acontradiction. (cid:3) Lemma 1.18.
Let M = ( M d ) ∈ pro - A . Then M ∼ = 0 if and only if M isMittag-Leffler and for every d , M epi d = 0 .Proof. If M is Mittag-Leffler and for every d , M epi d = 0, then 0 = M epi ∼ = M .Conversely if M ∼ = 0, then M is Mittag-Leffler and M epi ∼ = M ∼ = 0. Hencethere exists an isomorphism 0 → M epi . From Lemma 1.17, for every d , M epi d = 0. (cid:3) Corollary 1.19.
Let A be a proring. Then a sequence → L → M → N → of A -promodules is exact if and only if it is exact as a sequence of abelianprogroups.Proof. Consider a level morphism ( φ d : X d → Y d ) of A -modules and theinduced morphism φ : ( X d ) → ( Y d ) of A -promodules. Then both in thecategory of A -promodules and that of abelian progroups, we have Ker ( φ ) ∼ =(Ker ( φ d )). Hence φ is a monomorphism in the category of A -promodules ⇔ Ker ( φ ) ∼ = 0 as an A -promodule ⇔ Ker ( φ ) ∼ = 0 as an abelian progroup (the preceding lemma) ⇔ φ is a monomorphism in the category of abelian progroups . A similar statement holds for the epimorphism. Now the corollary is obvious. (cid:3) Formal schemes as proringed spaces
In this section, we define formal schemes as proringed spaces.
Sheaves on a qsqc basis.Definition 2.1.
Let X be a topological space. We say that X is quasi-separated if for any two quasi-compact open U, V ⊆ X , U ∩ V is quasi-compact. We say that X is qsqc if it is quasi-separated and quasi-compact.A qsqc basis is a basis of open subsets consisting of qsqc open subsets.For instance, qsqc are the underlying topological space of an affine schemeand a Noetherian topological space.Throughout the paper, we assume that every topological space has a qsqcbasis. For instance, the underlying topological space of any scheme satisfiesthis. Definition 2.2.
Let X be a topological space and B a basis of open subsetsof X . For open U ⊆ X , a (necessarily open) covering U = S U i is called a B -covering if for every i , U i ∈ B . A bicovering of an open subset U ⊆ X consists of a covering U = S U i and coverings U i ∩ U j = S U ijk for each twodistinct indices i, j , which we denote by { U i , U ijk } . A bicovering { U i , U ijk } is said to be finite if it consists of finite coverings, that is, { U i , U ijk } is afinite set. A bicovering { U i , U ijk } is called a B -bicovering if U i , U ijk ∈ B forevery i and ( i, j, k ).Let X be a topological space and B a qsqc basis. Then every qsqc open U ⊆ X has a finite B -bicovering. Let K be a category which admits finiteprojective limits. We say that a presheaf in K on B is a contravariant functor B → K . Here we think of B as a category so that the only morphisms arethe inclusion maps. A presheaf F is called a sheaf if for every U ∈ B andits every finite B -bicovering { U i , U ijk } , the sequence0 → F ( U ) → Y F ( U i ) ⇒ Y F ( U ijk )is exact. When B consists of all qsqc open subsets, we just say that F isa presheaf or sheaf on X respectively. For any qsqc basis B and a sheaf F on B , we can extend F to all qsqc open subsets so that it becomes a sheafon X : For any qsqc open U ⊆ X , take a finite B -bicovering { U i , U ijk } of U and put F ( U ) := lim ←− V ∈{ U i ,U ijk } F ( V ) . It is easy to show that the F ( U ) defined in this way is independent of thechoice of bicovering and the extended F is a sheaf on X . Remark . An advantage of considering only qsqc open subsets is that weneed to consider only finite bicoverings, which corresponds to considering finite projective limits. For instance, the following proposition is not true ifwe consider values at all open subsets.
Proposition 2.4.
Let X be a topological space and C a category havingfinite projective limits. Then a sheaf in C on X is a sheaf in pro - C as well. ON-ADIC FORMAL SCHEMES 11
Proof.
It is because the inclusion C ֒ → pro - C is exact (Proposition 1.3). (cid:3) Proposition 2.5.
Let X = S i ∈ I U i be an open covering of a topologicalspace, and F i , i ∈ I , sheaves in K on U i respectively with a gluing data,Then we can glue them to obtain a sheaf F on X with F | U i = F i .Proof. We first define F as a sheaf on the qsqc open subsets V ⊆ X suchthat V ⊆ U i for some i . Then we uniquely extend it to all qsqc open subsetsas above. (cid:3) Locally admissibly proringed spaces.Definition 2.6. (1) An admissible system of (affine) schemes is a di-rected inductive system ( X d ) of (affine) schemes such that everybonding morphism X d → X d ′ is a bijective closed immersion.(2) A proring ( A d ) is said to be admissible if every bonding morphism A d ′ → A d is surjective and induces an isomorphism ( A d ′ ) red → ( A d ) red of the associated reduced rings, or equivalently if (Spec A d )is an admissible system of affine schemes. For an admissible proring A = ( A d ), we define the associated reduced ring A red by A red := A d, red for any d .A morphism α : A → B of admissible rings induces a morphism α red : A red → B red in an obvious way. If α is an isomorphism, so is α red . Definition 2.7. A proringed space is a topological space X endowed witha sheaf O X of prorings, which is called the structure sheaf . A proringedspace X is called an admissibly proringed space if there exists a qsqc basis B such that for every U ∈ B , O X ( U ) is an admissible proring (moduloisomorphisms).For an admissibly proringed space X , if B is as above, we have the sheafof rings on B , B ∋ U ( O X ( U )) red . Extending it to all qsqc open subsets, we obtain a sheaf of rings on X anddenote it by O X red . Then we denote by X red the ringed space obtained byreplacing O X with O X red . Definition 2.8. A locally admissibly proringed space is an admissibly proringedspace X such that for every x ∈ X , O X red ,x is a local ring. A morphism φ : Y → X of locally admissibly proringed spaces consists of a contin-uous map φ : Y → X denoted by the same symbol and a φ -morphism φ ∗ : O X → O Y such that for every y , the induced map O X red ,φ ( y ) → O Y red ,y is a local homomorphism.In the definition, a φ -morphism O X → O Y means a compatible collectionof morphisms O X ( U ) → O Y ( V ), where U and V run over the qsqc opensubsets U ⊆ X and V ⊆ Y with φ ( V ) ⊆ U . Formal spectra.Definition 2.9.
For an admissible proring A = ( A d ), we define the formalspectrum Spf A as a locally admissibly proringed space as follows: As atopological space, defineSpf A := Spec A red = Spec A d , ∀ d. We have a directed projective system ( O Spec A d ) of sheaves of rings. Now wedefine O Spf A := “ lim ←− ” O Spec A d (defined on qsqc open subsets).As easily checked, it is actually a locally admissibly proringed space. Wehave (Spf A ) red = Spec A red . If A is a ring, then it is also a admissibleproring and Spf A = Spec A .Let A = ( A d ) be an admissible proring. For f ∈ A red , write D ( f ) :=Spec A red ,f ⊆ Spec A red , which is called an distinguished open subset . Here A red ,f is the localization of A red by f . If f ′ ∈ A d is a lift of f , then O Spec A d ( D ( f )) = A d,f ′ . Since A d,f ′ is independent of the lift f ′ , we write A d,f := A d,f ′ and A f := ( A d,f )Then ( D ( f ) , O Spf A | D ( f ) ) = Spf A f . Definition 2.10.
Let ( A d ) be an admissible proring. Then for each point x ∈ Spf A , we define the stalk O Spf
A,x of O Spf A at x as the proring ( O Spec A d ,x ).If p ⊆ A red is a prime ideal and p d ⊆ A d is its inverse image, then O Spec A d , p = A d, p d =: A d, p , where A d, p d is the localization of A d with respect to p , and O Spf A, p = ( A d, p ) =: A p . Every stalk is local in the following sense:
Definition 2.11.
An admissible proring A = ( A d ) is said to be local ifsome (and every) A d is a local ring, or equivalently if A red is a local ring. Amorphism α : ( A d ) → ( B e ) of local admissible rings is said to be local if some(and every) morphism A d → B e representing α is a local homomorphism,or equivalently if α red is a local homomorphism.2.4. Morphisms of formal spectra. If α : A = ( A d ) → B = ( B e ) isa morphism of admissible prorings, then for every morphism A d → B e representing it, we have the induced morphism of schemes, Spec B e → Spec A d . Then the morphisms Spec B e → Spec A d determines a morphism φ α : Spf B → Spf A of locally admissibly proringed spaces, which is calledthe morphism induced from α . For each y ∈ Spf B , we have the naturallocal morphism O Spf
B,y → O
Spf
A,φ α ( x ) . Proposition 2.12.
Let A and B be admissible prorings and φ : Spf B → Spf A a morphism of locally admissibly proringed spaces. Then φ is themorphism induced from the morphism φ ∗ : A = O Spf A (Spf A ) → B = O Spf B (Spf B ) . ON-ADIC FORMAL SCHEMES 13
Proof. φ induces a morphism φ red : Spec B red → Spec A red of locally ringedspaces, which is induced from φ ∗ red : A red → B red . In particular, if f ∈ A red and g := φ ∗ red ( f ) ∈ B red , then φ − ( D ( f )) = D ( g ). Hence we have thecommutative diagram A φ ∗ / / (cid:15) (cid:15) B (cid:15) (cid:15) A f φ ∗ / / B g . It easily follows from the universality of localization that the bottom arrowis uniquely determined by the top one. This shows the assertion. (cid:3)
Formal schemes.Definition 2.13. An affine formal scheme is a locally admissibly proringedspace which is isomorphic to the formal spectrum of some admissible proring. Corollary 2.14.
The category of affine formal schemes is equivalent to thedual category of the category of admissible prorings.Proof.
It follows from Proposition 2.12. (cid:3)
Definition 2.15.
A locally admissibly proringed space ( X, O X ) is called a formal scheme if there exists an open covering X = S U i such that for every i , ( U i , O X | U i ) is an affine formal scheme.If X is a scheme, then the structure sheaf O X , which is a sheaf of rings,is also a sheaf of prorings (Proposition 2.4), and X is regarded as a formalscheme as well. Thus the category of schemes is a full subcategory ofthat of formal schemes. By abuse of terminology, a formal scheme which isisomorphic to a scheme is also called a scheme . Convention 2.16.
For an open subset U of a formal scheme X , ( U, O X | U )is again a formal scheme, which is called an open formal subscheme . Unlessotherwise noted, we identify an open formal subscheme with its underlyingopen subset. We say that an open subset U ⊆ X is affine if it is an affineformal scheme. If U ∼ = Spf A , then we write Spf A ⊆ X . Proposition 2.17.
We can glue formal schemes along open formal sub-schemes.Proof.
We can first glue them as a topological space and then glue thestructure sheaves as in Proposition 2.5. (cid:3)
Proposition 2.18.
Every formal scheme admits a basis B of open subsetsconsisting of affine ones, which is a qsqc basis. It holds because we consider only values of sheaves at qsqc open subsets. See Remark2.3.
Proof.
Every formal scheme X has an affine covering X = S U i and each U i has a basis of open subsets consisting of distinguished open subsets. Thisproves the proposition. (cid:3) Proposition 2.19.
Let A be an admissible proring. Then Spf A is a schemeif and only if A is isomorphic to a ring.Proof. The “if” is trivial. Suppose that X := Spf A is a scheme. Thenwe have a finite open covering Spf A = S U i by affine schemes. Put A i := O X ( U i ), which are rings. For each i, j , A ij := O X ( U i ∩ U j ) is also a ring.Then A is isomorphic to the difference kernel of natural maps Q A i ⇒ Q A ij ,which is a ring. It proves the “only if.” (cid:3) Definition 2.20.
For a formal scheme X and x ∈ X , if Spf A ⊆ X is anaffine neighborhood of x , we define the stalk O X,x of O X at x to be O Spf
A,x .(It is clearly independent of the affine neighborhood.)2.6.
Fiber products.
The category of formal schemes has fiber products.For morphisms Y → X and Z → X of formal schemes, we denote the fiberproduct by Y × X Z . To show the existence of fiber products, we only haveto consider the case where X , Y and Z are affine. If we write X = Spf A , Y = Spf B and Z = Spf C , then since B ⊗ A C is the fiber sum, dually thefiber product Y × X Z exists and is isomorphic to Spf B ⊗ A C .3. Semicoherent promodules and formal subschemes
In this section, we introduce the notions of semicoherent promodule andformal subschemes, and study their basis properties.3.1.
Semicoherent promodules.Definition 3.1.
Let X be a formal scheme. An O X - (pro)module is a sheafof abelian (pro)groups, M , such that for each qsqc open U ⊆ X , M ( U ) isan O X -(pro)module in a compatible way.A morphism α : M → N of O X -(pro)modules is a compatible system ofmorphisms, α ( U ) : M ( U ) → N ( U ) , qsqc U ⊆ X, of O X ( U )-(pro)modules.Let A = ( A d ) be an admissible proring, X := Spf A and M an A -module.Reindexing A , we may suppose that every A d acts on M . Put X d := Spec A d .For each d , we have the O X d -module ˜ M associated to M . As a sheaf ofabelian groups, ˜ M is independent of d . All the O X d -module structuresmake ˜ M an O X -module. Next, for an A -promodule M = ( M e ), we definethe associated O X -promodule M △ by M △ := “ lim ←− ” ˜ M e . For f ∈ A red anda lift f ′ ∈ A d , ˜ M ( D ( f )) = M ,f ′ = M ⊗ A d A d,f =: M ,f , ON-ADIC FORMAL SCHEMES 15 and hence M △ ( D ( f )) = ( M e,f ) = M ⊗ A A f =: M f . For x ∈ X , if p ⊆ A red is the corresponding prime ideal, the stalk ˜ M ,x is an O X,x -module and ˜ M ,x = M ⊗ A A p =: M , p . Then we define the stalk M △ x by M △ x = M p := M ⊗ A A p = ( M e, p ) , which is an O X,x -promodule.Let N = ( N c ) be another A -promodule and α : M → N a morphism of A -promodules. The morphisms α ec : M e → N c representing α determines˜ α ec ( U ) : ˜ M e ( U ) → ˜ N c ( U ) , qsqc U ⊆ X. These morphisms then determine M △ ( U ) → N △ ( U ), U ⊆ X , and a mor-phism α △ : M △ → N △ of O X -promodules. Conversely if β : M △ → N △ is a morphism of O X -promodules, then β = ( β ( X )) △ . Indeed, for each f ∈ A red , we have the commutative diagram M (cid:15) (cid:15) β ( X ) / / N (cid:15) (cid:15) M f β ( D ( f )) / / N f . From the universality of localization, β ( D ( f )) should be the one inducedfrom β ( X ), which shows β = ( β ( X )) △ . Definition 3.2.
Let X be a formal scheme. An O X -promodule M is said tobe semicoherent if every point of X has an affine neighborhood Spf A ⊆ X such that M| Spf A ∼ = M △ for some A -promodule M . For a semicoherent O X -promodule M and for x ∈ X , we define the stalk M x by M △ x for Spf A and M as above.For a scheme X , every quasi-coherent O X -module is a semicoherent O X -module and vice versa:(Quasi-coherent O X -modules) = (Semicoherent O X -modules) . However in general there are much more semicoherent O X - pro modules thanquasi-coherent O X -modules (see Example 3.24). Lemma 3.3.
Let A be an admissible proring, M an A -promodule, and Spf B ⊆ Spf A an affine open subset. Then M △ | Spf B = ( M ⊗ A B ) △ .Proof. Obvious. (cid:3)
Proposition 3.4.
Let M be a semicoherent O X -promodule on an affineformal scheme Spf A . Then M = ( M (Spf A )) △ . Proof.
There exists a Zariski covering φ : Spf B → Spf A such that φ ∗ M =( M ′ ) △ for the B -promodule M ′ := ( φ ∗ M )(Spf B ). Put C := B ⊗ A B andlet ψ : Spf C → Spf A be the natural morphism, which is again a Zariskicovering. Then ψ ∗ M ∼ = ( M ′′ ) △ , where M ′′ := ( ψ ∗ M )(Spf C ).Then we have the exact sequence of A -promodules0 → M → M ′ A ⇒ M ′′ A . For each affine open Spf D ⊆ Spf A , since − ⊗ A D is exact, the sequence0 → M ⊗ A D → ( M ′ A ) ⊗ A D ⇒ ( M ′′ A ) ⊗ A D is exact. It follows that M (Spf D ) = M ⊗ A D , which implies the assertion. (cid:3) The following is a direct consequence:
Corollary 3.5.
For an affine formal scheme X = Spf A , we have the equiv-alence ( A -promodules) ∼ = (Semicoherent O X -promodules) M M △ M ( X ) ← [ M . Corollary 3.6.
For a formal scheme X , the category of semicoherent O X -promodules is abelian.Proof. From the preceding corollary, for a morphism
M → N of semico-herent O X -promodules, its kernel and image is defined on each affine opensubset. Gluing the local ones, we obtain the globally defined kernel andimage. The rest is easy to check. (cid:3) Pullback and pushforward.Definition 3.7.
Let φ : Y → X be a morphism of formal schemes and M a semicoherent O X -promodule. We define the pullback φ ∗ M , whichis a semicoherent O Y -promodule, as follows: If V ⊆ Y is an affine opensubset such that φ ( V ) is contained in an affine open U ⊆ X , then put( φ ∗ M )( V ) := M ( U ) ⊗ O X ( U ) O Y ( V ), which is independent of U from Lemma3.3. Such V ’s form a qsqc basis of Y and φ ∗ M is a sheaf on this basis. Wecan now uniquely extend it to all qsqc open subsets.With the above notation, If Y = Spf B , X = Spf A and M = M △ , then φ ∗ ( M △ ) = ( M ⊗ A B ) △ . Definition 3.8.
A continuous map φ : Y → X is said to be qsqc if for everyqsqc open U ⊆ X , φ − ( U ) is qsqc. Definition 3.9.
Let φ : Y → X be a qsqc morphism of formal schemesand N a semicoherent O Y -promodule. We define the pushforward φ ∗ N ,which is an O X -promodule, as follows: For qsqc open U ⊆ X , ( φ ∗ N )( U ) := N ( φ − ( U )) O X ( U ) . ON-ADIC FORMAL SCHEMES 17
Lemma 3.10.
Let φ : Spf B → Spf A be a morphism of affine formalschemes, which is always qsqc. For a B -promodule N , φ ∗ ( N △ ) = ( N A ) △ .Proof. Obvious from the definition. (cid:3)
Proposition 3.11.
Let the notation be as in Definition 3.9. Then φ ∗ N is semicoherent. In addition, if N is an O Y -module, then φ ∗ N is an O X -module.Proof. We may suppose that X is affine, say X = Spf A . Then Y is qsqc,so there exists an affine bicovering { U i , U ijk } of Y . Put N i := N ( U i ) and N ijk := N ( U ijk ). If we denote the restrictions of φ to U i and U ijk alsoby φ , then φ ∗ ( N | U i ) = (( N i ) A ) △ and φ ∗ ( N | U ijk ) = (( N ijk ) A ) △ , which aresemicoherent. Since φ ∗ N is the kernel of a morphism of semicoherent O X -promodules M φ ∗ ( N | U i ) → M φ ∗ ( N | U ijk ) ,φ ∗ N is semicoherent too, which proves the first assertion.If N is an O Y -module, then L φ ∗ ( N | U i ) and L φ ∗ ( N | U ijk ) are O X -modules, and hence so is φ ∗ N . (cid:3) Proposition 3.12.
Let φ : Y → X be a qsqc morphism of formal schemes, N a semicoherent O Y -promodule and M a semicoherent O X -promodule.Then we have a natural equation Hom( M , φ ∗ N ) = Hom( φ ∗ M , N ) . Namely φ ∗ is the right adjoint of φ ∗ . Hence φ ∗ is left exact and φ ∗ is rightexact.Proof. In general, let A be a proring, B an A -proalgebra, M an A -promodule,and N a B -promodule. ThenHom A ( M, N A ) = Hom B ( M ⊗ A B, N ) . Giving a morphism
M → φ ∗ N is equivalent to giving a compatible systemof O X ( U )-morphisms M ( U ) → N ( V ) O X ( U ) for affine open subsets U ⊆ X and V ⊆ Y with φ ( V ) ⊆ U . In turn, it is equivalent to giving a compatiblesystem of O Y ( V )-morphisms M ( U ) ⊗ O X ( U ) O Y ( V ) → N ( V ) for such pairs( U, V ). Finally it is equivalent to giving a morphism φ ∗ M → N , whichcompletes the proof. (cid:3)
Mittag-Leffler O X -promodules.Definition 3.13. Let X be a formal scheme and M a semicoherent O X -promodule. We say that M is Mittag-Leffler if every x ∈ X has an affineneighborhood U ⊆ X such that M ( U ) is Mittag-Leffler. Proposition 3.14.
Suppose that X = Spf A is an affine formal schemeand that M is a Mittag-Leffler semicoherent O X -promodule. Then M ( X ) is Mittag-Leffler Proof.
Write M = ( M e ) := M ( X ). There exists a finite open covering X = S D ( f i ) such that the M f i are Mittag-Leffler. Then for each i , theprojective system ( ˜ M e | D ( f i ) ) of sheaves satisfies the Mittag-Leffler conditionand so does ( ˜ M e ). Therefore M is Mittag-Leffler. (cid:3) Proposition 3.15.
Let φ : F → G be an epimorphism of semicoherent O X -promodules. If F is Mittag-Leffler, then so is G .Proof. Obvious. (cid:3)
Proposition 3.16.
Let φ : Y → X be a morphism of formal schemes and M a Mittag-Leffler semicoherent O X -promodule. Then φ ∗ M is also Mittag-Leffler.Proof. It follows from the construction of pullback. (cid:3)
Stalks and exactness.
Let X be a formal scheme. Given a morphism α : F → G of semicoherent O X -promodules, for each x ∈ X , we have theinduced morphism α x : F x → G x of stalks. ThenKer ( α x ) = Ker ( α ) x , Im( α x ) = Im( α ) x . As a consequence, we obtain the following:
Corollary 3.17.
Let → F → G → H → be an exact sequence of semicoherent O X -promodules. Then for every x ∈ X ,the induced sequence of stalks → F x → G x → H x → , is exact. The converse of the corollary holds only under some condition:
Proposition 3.18.
Let D be an abelian subcategory of the category of semi-coherent O X -promodules. Suppose that if M ∈ D has Mittag-Leffler stalks,then M is Mittag-Leffler. Then a sequence → F → G → H → in D is exact if for every x ∈ X , the induced sequence of stalks → F x → G x → H x → , is exact.Proof. Put K := Ker ( F → G ). By assumption the stalks of K are zero,in particular, Mittag-Leffler. So K is Mittag-Leffler. From the followinglemma, K = 0. Hence the F → G is a monomorphism. The rest can beproved similarly. (cid:3)
Lemma 3.19.
Let X be a formal scheme and F a semicoherent O X -promodule.Then F = 0 if and only if F is Mittag-Leffler and for every x ∈ X , F x = 0 . ON-ADIC FORMAL SCHEMES 19
Proof.
The “only if” is trivial. We prove the “if.” We may suppose that X = Spf A . Then F = ( F d ) d ∈ D := F ( X ) is a Mittag-Leffler A -promoduleand so we may suppose also that F is epi. Then for each prime ideal p ⊆ A red , F x = ( F d, p ) d ∈ D = 0. From Lemma 1.18, for each d ∈ D , F d, p = 0. Hence F d = 0 and F = 0. (cid:3) Proposition 3.20.
Let φ : F → G be a morphism of semicoherent O X -promodules. Suppose that G is Mittag-Leffler and that for every x ∈ X , φ x : F x → G x is an epimorphism. Then φ is an epimorphism.Proof. Put H := Coker ( φ ), which is Mittag-Leffler from Proposition 3.15.Moreover from the assumption, its stalks are zero. So, from the precedinglemma, H = 0 and φ is an epimorphism. (cid:3) Formal subschemes.Definition 3.21.
Let X be a formal scheme. A semicoherent proideal sheaf on X is a semicoherent O X -subpromodule of O X .For a semicoherent proideal sheaf I on X , the quotient O X -promodule O X / I is naturally regarded as a sheaf of prorings. Definition 3.22.
Let
I ⊆ O X be a semicoherent proideal sheaf and Y := { x ∈ X | ( O X / I ) x = 0 } its support. We say that the subspace Y ⊆ X endowed with the sheaf O X / I is a closed formal subscheme of X if it is aformal scheme. (Unlike the scheme case, ( Y, O X / I ) is not a priori a formalscheme.) Then we say that I is the defining ideal sheaf of Y . If Y is even ascheme, then we call it a closed subscheme .For a formal scheme X , the X red defined in § X . Lemma 3.23. If Y is a closed formal subscheme of a formal scheme X ,then Y red is a closed subscheme of X red . In particular, Y is set-theoreticallya closed subset of X .Proof. Since the problem is local, we may suppose that X = Spf A for someadmissible proring. Then put I := I ( X ), a proideal of A . The quotient ring B := A/I is an admissible proring up to isomorphisms. From Lemma 1.17,the epimorphism A → B (of A -promodules) induces a surjection A red → B red , and a closed immersion Y red → X red , which proves the assertion. (cid:3) Example 3.24.
For a scheme X and a closed subscheme Y , we define the completion X /Y of X along Y , which is a closed formal subscheme of X , asfollows: Let I ⊆ O X be the defining ideal sheaf of Y . For each n ∈ N , wehave an epimorphism O X → O X / I n . Put O := “ lim ←− ” O X / I n . Then thenatural morphism O X → O is an epimorphism and we have O = O X / J for some semicoherent proideal sheaf J ⊆ O X . Let Y n ⊆ X be the closedsubscheme with the defining ideal I n and Y the underlying topological spaceof Y . Then the support of O is Y and the proringed space ( Y , O ) is a formal scheme, which is isomorphic to lim −→ n Y n . We write X /Y := ( Y , O ). The O X -promodule O is not generally an O X -module. Definition 3.25.
A closed formal subscheme of an open formal subschemeof X is called a formal subscheme of X . We identify formal subschemes Y and Z of X if the natural morphisms Y → X and Z → X are isomorphic.A subscheme of X is a formal subscheme of X which is a scheme.A morphism W → X of formal schemes is called a (closed, open) im-mersion if it is an isomorphism onto a (closed, open) formal subscheme of X . Proposition 3.26.
Let φ : Y → X be a morphism of formal schemes. Then φ is an immersion if and only if φ is, as a continuous map, a homeomor-phism onto a locally closed subset of X , and for every y ∈ Y , the naturalmorphism O X,φ ( y ) → O Y,y is an epimorphism (of O X,φ ( y ) -promodules).Proof. The “only if” part is obvious and we prove the “if” part. Take anopen formal subscheme U ⊆ X such that φ ( Y ) is a closed subset of U .Then the map φ : Y → U is qsqc and φ ∗ O Y is well-defined as a sheaf on U ,which is clearly Mittag-Leffler. From the assumption and Proposition 3.20,the natural morphism O U → φ ∗ O Y is an epimorphism of semicoherent O U -promodules. Its kernel is a proideal sheaf and defines a closed subscheme Y ′ ⊆ U . Then φ is equal to the composite Y ∼ = Y ′ ֒ → U ֒ → X and hence animmersion. (cid:3) Proposition 3.27.
Let Y → X and W → X be morphisms of formalschemes. If the Y → X is an immersion (resp. closed immersion, openimmersion), then so is the natural morphism W × X Y → W .Proof. The assertion follows form the preceding proposition and the factthat the tensor product is right exact. (cid:3)
Extension of a promodule on an open subset.
The following twopropositions generalizes [EGA, Prop. 9.4.2 and 9.5.10] to formal schemes.
Proposition 3.28.
Let X be a formal scheme and U ⊆ X an open sub-set such that the inclusion map ι : U ֒ → X is quasi-compact, hence qsqc.Let N be a semicoherent O X -promodule, M ⊆ N | U a semicoherent O U -subpromodule, and Q := N | U / M the quotient O U -promodule. Then thereexists the largest semicoherent O X -subpromodule ¯ M ⊆ N such that ¯ M| U = M . Correspondingly there exists the smallest semicoherent quotient O X -promodule N ։ ¯ Q such that ¯ Q| U = Q .Proof. We have a natural morphism φ : N → ι ∗ Q . Then ¯ M := Ker ( φ ) and¯ Q := Im( φ ) have the desired properties. (cid:3) Proposition 3.29.
Let X be a formal scheme and Y ⊆ X a formal sub-scheme. Suppose that the inclusion map Y ֒ → X is quasi-compact, henceqsqc. Then there exists a smallest closed formal subscheme ¯ Y ⊆ X which ON-ADIC FORMAL SCHEMES 21 contains Y as an open formal subscheme. If in addition Y is a scheme, thenso is ¯ Y .Proof. We can construct ¯ Y as follows: The underlying topological space of¯ Y is the set-theoretic closure of Y . If ι : Y ֒ → X denotes the inclusion map,then the structure sheaf of ¯ Y is the image of O X → ι ∗ O Y .We now check that the above construction gives a closed formal subschemeof X . To do this, we may suppose that X is affine, and hence Y is qsqc.Let ( X d ) be an admissible system of affine scheme with X = lim −→ X d and put Y d := Y ∩ X d = Y × X X d . Then ( Y d ) is an admissible system of schemeswith Y = lim −→ Y d . Now we can define the closure ¯ Y d of Y d in X d so that thestructure sheaf of ¯ Y d is the image of O X d → ( ι d ) ∗ O Y d , where ι d denotes theinclusion morphism Y d ֒ → X d . It is now easy to see that ¯ Y = lim −→ ¯ Y d andthat ¯ Y is a closed formal subscheme of X .The second assertion of the proposition follows from the construction. (cid:3) Definition 3.30.
With the notation in the preceding proposition, we call¯ Y the closure of Y in X .It also follows from the construction that if Y admits an admissible system( Y d ) of schemes with Y = lim −→ Y d , and if ¯ Y d denotes the closure of Y d in X ,then ¯ Y = lim −→ ¯ Y d .3.7. Semicoherent promodules on a qsqc formal scheme.Proposition 3.31.
For a qsqc formal scheme X , we have a natural equiv-alence (Semicoherent O X -promodule) ∼ = pro - (Semicoherent O X -module) . Proof.
If ( M d ) and ( N e ) are directed projective systems of semicoherent O X -modules, then it is easy to seeHom(“ lim ←− ” M d , “ lim ←− ” N e ) = lim ←− e lim −→ d Hom( M d , N e ) . So it suffices to show that for every semicoherent O X -promodule M , thereexists a directed projective system ( M d ) of semicoherent O X -modules with M = “ lim ←− ” M d . Let { U i , U ijk } be a finite affine bicovering of X . If ι i : U i ֒ → X denotes the inclusion, we put M U i := ( ι i ) ∗ ( M| U i ) and sim-ilarly define M U ijk . We set M ′ := Q M U i and M ′′ := Q M U ijk . Thenthere exist directed projective systems ( M ′ d ) and ( M ′′ e ) of semicoherent O X -modules such that M ′ = “ lim ←− ” M ′ d and M ′′ = “ lim ←− ” M ′′ e respectively. Thenreindexing ( M ′ d ) and ( M ′′ e ), we may suppose that the natural morphisms M ′ ⇒ M ′′ are represented by level morphisms ( M ′ d ⇒ M ′′ d ). For each d , ifwe put M d to be the difference kernel of M ′ d ⇒ M ′′ d , then M = “ lim ←− ” M d ,which completes the proof. (cid:3) Proposition 3.32.
Every qsqc formal scheme is the inductive limit of someadmissible system of schemes.Proof.
Let X be an arbitrary qsqc formal scheme. Take a finite affine cover-ing X = S i ∈ I U i . For each i , U i is by definition the inductive limit of someadmissible system of affine schemes, say ( U i,d ). Without loss of generality,we may suppose that the index sets of ( U i,d ), i ∈ I , are equal. We denote¯ U i,d the closure of U i,d , which is a closed subscheme of X . For each d , define X d to be the scheme-theoretic union of ¯ U i,d , i ∈ I , whose defining proidealis the intersection of those of ¯ U i,d . Then X d is a subscheme of X and ( X d )is an admissible system such that X = lim −→ X d . (cid:3) Locally ind-Noetherian formal schemes.Definition 3.33.
An admissible proring ( A d ) is said to be pro-Noetherian if every A d is Noetherian.From Lemma 1.17, the pro-Noetherian property for an admissible proringdepends only on its isomorphism class in the category of admissible prorings. Definition 3.34.
A formal scheme X is said to be locally ind-Noetherian if every x ∈ X admits an affine neighborhood x ∈ Spf A ⊆ X with A pro-Noetherian. Proposition 3.35.
Let A be an admissible proring such that Spf A is locallyind-Noetherian. Then A is pro-Noetherian.Proof. If we write A = ( A d ), then Spec A d is a locally Noetherian scheme,and hence A d is Noetherian, and A is pro-Noetherian. (cid:3) Definition 3.36.
Let A = ( A d ) be a pro-Noetherian admissible ring. An A -module N is said to be Noetherian if the set of A -submodules of N satisfiesthe maximal condition, or equivalently if for some (and every) d such that A d acts on N , N is a Noetherian A d -module. An A -promodule M = ( M e )is said to be pro-Noetherian if for every e , M e is a Noetherian A -module.Note that the pro-Noetherian property for promodules is not invariantunder isomorphisms. However if M and N are epi A -promodules isomorphicto each other and if M is pro-Noetherian, then so is N .It is clear that the category of pro-Noetherian A -promodules is abelian.In particular, every A -subpromodule and quotient A -promodule of a pro-Noetherian A -promodule are again pro-Noetherian up to isomorphisms. Definition 3.37.
Let X be a locally ind-Noetherian formal scheme. Asemicoherent O X -(pro)module M is said to be locally (pro-)Noetherian ifevery point x ∈ X admits an affine neighborhood U = Spf A such that M| U ∼ = M △ for some (pro-)Noetherian A -(pro)module M .Again it is clear that the category of locally Noetherian O X -modules andthat of locally pro-Noetherian O X -promodules are abelian. ON-ADIC FORMAL SCHEMES 23
Proposition 3.38.
Every formal subscheme of a locally ind-Noetherian for-mal scheme is again locally ind-Noetherian.Proof.
Obvious. (cid:3)
Lemma 3.39.
Let A be a pro-Noetherian admissible proring, X := Spf A and M a locally Noetherian semicoherent O X -module. Then M ( X ) is aNoetherian A -module.Proof. Write A = ( A d ). Then M is a coherent sheaf on some Spec A d . So M ( X ) is Noetherian. (cid:3) Proposition 3.40.
Let A be a pro-Noetherian admissible proring, X :=Spf A and M a locally pro-Noetherian semicoherent and Mittag-Leffler O X -promodule. Then M ( X ) is a pro-Noetherian A -promodule up to isomor-phisms.Proof. Let M = ( M e ) be an epi A -promodule with M ( X ) ∼ = M and X = S U i , U i = Spf A i , an affine covering such that for each i , M ( U i ) is a pro-Noetherian A i -promodule up to isomorphisms. Then for each i , being epi,( ˜ M e ( U i )) is actually a pro-Noetherian A i -promodule. Hence every ˜ M e islocally Noetherian, and M e is Noetherian. We conclude that ( M e ) is pro-Noetherian, which completes the proof. (cid:3) Corollary 3.41.
Let A be a pro-Noetherian admissible ring, X := Spf A and D the smallest abelian full subcategory of the category of locally pro-Noetherian O X -promodules which contains all Mittag-Leffler and locally pro-Noetherian O X -promodules. Then for every M ∈ D , M ( X ) is a pro-Noetherian A -promodule up to isomorphisms. In particular, for every semi-coherent proideal sheaf I ⊆ O X , I ( X ) is pro-Noetherian up to isomor-phisms.Proof. The property that M ( X ) is pro-Noetherian up to isomorphisms isstable under taking direct sums, subobjects and quotient objects. Thisproves the corollary. (cid:3) Proposition 3.42.
Let X be a locally ind-Noetherian formal scheme and M a locally pro-Noetherian O X -promodule. Suppose that for every x ∈ X ,the stalk M x is Mittag-Leffler. Then M is Mittag-Leffler.Proof. Without loss of generality, we may suppose that X = Spf A for anadmissible proring A and that M ∼ = M △ for a pro-Noetherian A -promodule M = ( M e ). We have to show that M is Mittag-Leffler. Put M e := ˜ M e , M e ′ e := Im( M e ′ → M e ), e ′ ≥ e , and M ∞ e := T e ′ ≥ e M e ′ e . We now fix anindex e . For a point x ∈ X , since M x is Mittag-Leffler, there exists e ≥ e such that ( M e e / M ∞ e ) x = 0, that is, x / ∈ Supp ( M e e / M ∞ e ). Here Supp ( − )denotes the support of a sheaf. If Supp ( M e e / M ∞ e ) = ∅ , then we choose x ∈ Supp ( M e e / M ∞ e ) and take e ≥ e such that x / ∈ Supp ( M e e / M ∞ e ).Then Supp ( M e e / M ∞ e ) ( Supp ( M e e / M ∞ e ) . We can continue this procedure until we get empty Supp ( M e i e / M ∞ e ). Sincethe underlying topological space of X is Noetherian and the Supp ( M e i e / M ∞ e )are closed subsets, for some e i , Supp ( M e i e / M ∞ e ) = ∅ . It proves the propo-sition. (cid:3) Corollary 3.43.
Let X be a locally ind-Noetherian formal scheme. Then asequence of locally pro-Noetherian O X -promodules → L → M → N → is exact if and only if for every x ∈ X , the induced sequence of stalks → L x → M x → N x → is exact.Proof. We can prove it like Proposition 3.18. (cid:3) Formal schemes as functors and formal algebraic spaces
In this section, we see that a formal scheme can be considered as a sheafon the category of schemes. Along this line, we also define a formal algebraicspace.4.1.
Formal schemes as functors.
Let F denotes the category of con-travariant functors (Schemes) → (Sets) , and F Zar , F ´et ⊆ F the full subcategories of Zariski and ´etale sheaves respec-tively. For a formal scheme X , we define F X ∈ F by F X ( Y ) := Hom form. sch. ( Y, X ) , which is clearly a sheaf for both the Zariski and ´etale topologies. As iswell-known, the functor F • : (Schemes) → F , X F X is fully faithful. So, by abuse of terminology, we say that a functor F ∈ F is a scheme if F ∼ = F X for some scheme X . Definition 4.1.
A morphism F → G in F is said to be schematic if forevery scheme X and for every morphism X → G , the fiber product F × G X ,which exists in F , is a scheme.We can generalize various properties of morphisms of schemes to schematicmorphisms: Definition 4.2.
Let P be a property of morphisms of schemes which isstable under base changes. We say that a schematic morphism F → G of F has a property P if for every scheme X and for every morphism X → G ,the projection F × G X → X has the property P . ON-ADIC FORMAL SCHEMES 25
Theorem 4.3.
The functor F • : (Formal schemes) → F , X F X is fully faithful.Proof. Let X and Y be formal schemes. First consider the case where X =lim −→ X d for some admissible system ( X d ) of schemes. Given a morphism F X → F Y , we have canonical elements ( X d → X ) ∈ F X ( X d ) and theirimages ( X d → Y ) ∈ F Y ( X d ), which uniquely determine a morphism X → Y .It proves that the natural mapHom( X, Y ) → Hom( F X , F Y )is bijective.Next consider the general case. Take an affine Zariski covering X = S U i and put V := ` U i and W := V × X V . Then there exist admissible systems( V d ) and ( W d ) of schemes such that V = lim −→ V d and W = lim −→ W d . Amorphism F X → F Y induces F V → F Y and F W → F Y , and hence V → Y and W → Y . The last two morphisms is actually a gluing data of morphismswith respect to the Zariski topology, so we obtain a morphism X → Y . (cid:3) Remark . Note that if X is a formal scheme and ( X d ) is an admissiblesystem of schemes with X = lim −→ X d , then F X is not isomorphic to theinductive limit G of F X d ’s in F , but isomorphic to its sheafification, whichis the inductive limit of F X d in F Zar and F ´et . However G is a sheaf on qsqcformal schemes.4.2. Formal algebraic spaces.
Again by abuse of terminology, we say that F ∈ F is a formal scheme if it is isomorphic to F X with X a formal scheme.For a schematic morphism of formal schemes, the immersions of Definitions3.25 and 4.2 coincide thanks to the following: Lemma 4.5.
Let φ : Y → X be a schematic morphism of formal schemes.Then φ is an immersion in the sense of Definitions 3.25 if and only if forevery morphism W → X with W a scheme, so is the natural morphism W × X Y → Y .Proof. The “only if” follows from Proposition 3.27. We now prove the “if”.Considering the case W = X red , we easily see that φ is a homeomorphismonto a locally closed subset. For y ∈ Y , if we write O X,φ ( y ) = ( A d ) wherethe A d are rings, then by assumption, B d := A d ⊗ O X,φ ( y ) O Y,y is a ring up toisomorphisms, and the natural morphism f d : A d → B d is surjective. Thenthe natural morphism O X,φ ( y ) → O Y,y is equal to “ lim ←− ” f d and hence anepimorphism, which proves the lemma. (cid:3) Lemma 4.6. (1)
For a formal scheme X , the diagonal morphism X → X × X is schematic and an immersion. (2) Let Y → X be a morphism of F which is schematic and an immer-sion. If X is a formal scheme, then so is Y . (3) Let F ∈ F such that the diagonal morphism F → F × F is schematicand an immersion, and let U → F and V → F be morphisms of F with U and V formal schemes. Then U × F V is a formal scheme.Proof. (1). We may suppose that X is affine and there exists an admissiblesystem of affine schemes ( X d ) with X = lim −→ X d . Let V be a quasicompactscheme and V → X × X an arbitrary morphism. Then X × X × X V ∼ = lim −→ X d × X d × X d V, where d runs over those indices such that V → X × X factors through X d × X d . Since for d ′ ≥ d , the natural diagram X d / / (cid:15) (cid:15) X d × X d (cid:15) (cid:15) X d ′ / / X d ′ × X d ′ is cartesian, the inductive system ( X d × X d × X d V ) is constant. It proves theproposition.(2). First suppose that there exists an admissible system of schemes ( X d )with X = lim −→ X d . Then for each d , Y d := Y × X X d is a subscheme of X d ,and ( Y d ) is an admissible system of schemes. Then Y is the inductive limitof ( Y d ) say in F Zar , which is a formal subscheme of X and hence a formalscheme. (I do not know if for any admissible system ( X d ) of schemes, itslimit in F Zar is a formal scheme. So I have to add the condition of being animmersion. The problem is that I do not know if for a bonding morphism X d → X d ′ , the image of an affine open is affine, and if the limit of ( X d ) asa proringed space is covered by affine formal schemes.)In the general case, we take a Zariski covering U → X such that U =lim −→ U d for some admissible system ( U d ). If we put V := U × X U ,then thereexists an admissible system ( V d ) with V = lim −→ V d . Then Y U := Y × X U and Y V := Y × X V are formal schemes. From the gluing data ( Y U , Y V ) of formalschemes, we obtain a formal scheme, which is nothing but Y .(3). The natural morphism U × F V → U × V is a base change of F → F × F , so schematic and an immersion. From (2), U × F V is a formalscheme. (cid:3) As the definition of formal algebraic space, we adopt the following one:
Definition 4.7.
An ´etale sheaf X ∈ F ´et is called a formal algebraic space if the diagonal morphism X → X × X is schematic and an immersion, andthere exists a schematic ´etale morphism U → X with U a formal scheme.From Lemma 4.6 (1), a formal scheme is a formal algebraic space. For aformal algebraic space X and U → X as in the definition, from Lemma 4.6(3), R := U × X U is a formal scheme. The natural morphism R → U × U is schematic and an immersion, because it is a base change of X → X × X .The two projections R ⇒ U are schematic and ´etale. Thus R is an ´etale ON-ADIC FORMAL SCHEMES 27 equivalence relation on U , and X is the quotient R/U in F ´et . We have alsothe ´etale equivalence relation R red on the reduced scheme U red and obtain areduced algebraic space X red := U red /R red , which is a generalization of theone defined for a formal scheme.Conversely given an equivalence relation R → U × U in F ´et such that R and U are formal schemes, R → U × U is schematic and an immersion, and R ⇒ U are schematic and ´etale, then the quotient R/U is a formal algebraicspace.
Definition 4.8.
For a formal algebraic space X , we define the ´etale site , X ´et , as the category of formal algebraic spaces Y which are schematic and´etale over X with the obvious notion of covering.We define the structure sheaf O ´et X on X ´et as follows, which makes X ´et a “proringed site”: For ( U → X ) ∈ X ´et with U a qsqc formal scheme , O ´et X ( U ) := O U ( U ). From the following lemma, this defines a sheaf on thebasis consisting of all such U ’s. Lemma 4.9.
Let V → U be an ´etale covering of formal schemes and W := V × U V . Then the sequence → O U ( U ) → O V ( V ) ⇒ O W ( W ) is exact.Proof. Take an admissible system ( U d ) of schemes with U = lim −→ U d , and put V d := U d × U V and W d := U d × U W . Then ( V d ) and ( W d ) are admissiblesystems of schemes such that V = lim −→ V d and W = lim −→ W d . For each d , wehave the exact sequence of rings0 → O U d ( U d ) → O V d ( V d ) ⇒ O W d ( W d ) . Taking the projective limit, we obtain the exact sequence in the lemma. (cid:3)
Then we can uniquely extend O ´et X to X ´et as a sheaf on qsqc objects. Herethe qsqc object is defined just like the qsqc topological space. Then weeasily see that for a formal scheme ∈ X ´et , being qsqc as an object of X ´et isequivalent to being qsqc as a topological space. Definition 4.10. An O ´et X -promodule M is said to be semicoherent if forsome ´etale covering V → X with V a formal scheme, the restriction of M to the Zariski site of V is a semicoherent O V -promodule.If X is a formal scheme and M is a semicoherent O X -promodule, thenwe define a semicoherent O ´et X -promodule M ´et so that for an ´etale morphism φ : U → X with U a formal scheme, M ´et ( U ) = ( φ ∗ M )( U ). Restricting M ´et to the Zariski site of X , we can recover M . Lemma 4.11. If X = Spf A and if M is a semicoherent O ´et X -promodule,then M ∼ = ( M ( X ) △ ) ´et .Proof. It is proved in the same manner as Proposition 3.4. (cid:3)
As a consequence, for a formal scheme X , we obtain the equivalence(Semicoherent O X -promodules) ∼ = (Semicoherent O ´et X -promodules) . Definition 4.12.
A morphism φ : Y → X of formal algebraic spaces is formally schematic if for every morphism W → X with W a formal scheme, Y × X W is also a formal scheme. A formally schematic morphism φ : Y → X of formal algebraic spaces is a (resp. closed, open) immersion if for everymorphism W → X with W a formal scheme, the projection Y × X W → W is a (resp. closed, open) immersion. For a formal algebraic space X , a (resp.closed, open) formal algebraic subspace of X is an isomorphism classes of(resp. closed, open) immersions Y → X .We can now translate all the results in § U ⊆ X not withan ´etale morphism but with an open immersion U → X of formal algebraicspaces. 5. Mild and gentle formal schemes
This section establishes the relation between prorings (resp. promodules)and complete rings (resp. complete modules). Also we introduce two classesof formal schemes which are well-behaved when completing the structuresheaf.5.1.
Complete rings and complete modules. A linearly topologized ring is a topological ring which admits a basis of open neighborhoods of 0 con-sisting of ideals. Such a basis is called a basis of open ideals . Conversely ifa ring A is given a collection ( I d ) of ideals which is directed with respectto the preorder I d ≤ I d ′ ⇔ I d ⊇ I d ′ , then there exists a unique topologyon A for which A is linearly topologized and ( I d ) is a basis of open ideals.We call this topology the ( I d )- topology . A linearly topologized ring is saidto be gentle if it has a countable basis of open ideals. If A is a linearlytopologized ring and ( I d ) is a basis of open ideals, then the completion ˆ A of A is defined to be the projective limit lim ←− A/I d which is linearly topologizedso that the kernels of ˆ A → A/I d form a basis of open ideals. A completering is a linearly topologized ring A such that the natural map A → ˆ A isbijective. As a special case, every ring with the discrete topology is a com-plete linearly topologized ring. We define a morphism of complete rings asa homomorphism of rings which is continuous.Let A be a linearly topologized ring and M an abelian topological groupendowed with an A -module structure. We call M a linearly topologized A -module if it has a basis ( M e ) of open neighborhoods of 0 consisting of A -submodules M e ⊆ M and if for every open neighborhood 0 ∈ V ⊆ M ,there exists an open ideal I ⊆ A with IM ⊆ V . Such a basis is calleda basis of open submodules . A linearly topologized A -module is said tobe gentle if it admits a countable basis of open submodules. If M is a ON-ADIC FORMAL SCHEMES 29 linearly topologized A -module, with the above notation, the completion ˆ M is defined to be lim ←− M/M e , which is a linearly topologized ˆ A -module aswell as a linearly topologized A -module. A complete A -module is a linearlytopologized A -module M with M = ˆ M . We define a morphism of complete A -modules as a homomorphism of A -modules which is continuous.5.2. Pro vs. complete.
Let A = ( A d ) be a proring. We give to the pro-jective limit, ˆ A := lim ←− A d , in the category of rings the topology such thatˆ A is linearly topologized and the kernels Ker ( ˆ A → A d ) form a basis ofneighborhoods of 0. This defines a functor ∧ : (Prorings) → (Complete rings) , A ˆ A. Conversely if B is a complete ring and ( I d ) is a basis of open ideals, then( B/I d ) is an epi proring. The isomorphism class of ( B/I d ) is independentof the choice of ( I d ). To kill the ambiguity, taking the set ( I d ) of all openideals, we put ˇ B := ( B/I d ), which defines a functor ∨ : (Complete rings) → (Epi prorings) , B ˇ B. This is a fully faithful embedding and the composite functor ∧ ◦ ∨ is isomor-phic to the identity.
Definition 5.1.
A proring A is said to be mild if A ∼ = ˇ B for some completering B .A mild proring is by definition isomorphic to an epi proring, and henceMittag-Leffler. But there exists an epi proring which is not mild (see Ex-ample 5.5). Similarly, for a proring A , we have ∧ : ( A -promodules) → (Complete ˆ A -modules) , M ˆ M and for a complete ring B , ∨ : (Complete B -modules) → (Epi ˇ B -promodules) , N ˇ N . which is fully faithful.
Definition 5.2.
Let A be a mild proring. An A -promodule M is said to be mild if M ∼ = ˇ N for some complete ˆ A -module N .For a mild proring A , we have(Complete ˆ A -modules) ∼ = (Mild A -promodules) . Proposition 5.3. (1)
An epi proring A = ( A d ) is mild if and only ifthe natural maps ˆ A → A d are surjective. (2) Let A be a mild proring. Then an epi A -promodule M = ( M e ) ismild if and only if the natural maps ˆ M → M e are surjective. It was a referee who let me know the existence of non-mild and epi proring.
Proof.
Since the proofs of (1) and (2) are parallel, we only prove (1). Wefirst prove the “if” part. Put I d := Ker ( ˆ A → A d ). Then by definition, ( I d )is a basis of open ideals and A = ( A d ) = ( ˆ A/I d ) ∼ = ˇˆ A. So A is mild.Next we prove the “only if” part. Let B be a complete ring such that A ∼ =ˇ B . We write ˇ B = ( B e ). By construction, we have the natural surjections B → B e . From Lemma 1.17, for each d , there exists a surjection B e → A d which represents an isomorphism ˇ B → A . So the natural morphismˆ A ∼ = B → B e → A d is surjective. (cid:3) Definition 5.4.
A pro-object of any category is said to be gentle if it is iso-morphic in the procategory to a pro-object indexed by a countable directedset, or equivalently to one indexed by N .From Proposition 5.3 and the construction of the projective limit, everygentle and Mittag-Leffler proring is mild, and every gentle and Mittag-Leffler A -promodule for a mild proring A is mild. Hence(Gentle complete rings) ∼ = (Gentle and Mittag-Leffler prorings) , and if B is a complete ring and A is the associated mild proring,(Gentle complete B -modules) ∼ = (Gentle and Mittag-Leffler A -promodules) . Example 5.5.
There exists a directed projective system ( S d ) of sets withsurjective bonding maps and lim ←− S d = ∅ (see [Hen, HS, Wat]). From such asystem, imitating a construction of Higman and Stone [HS], we can constructan admissible proring which is not mild: For d ′ ≥ d , we have a natural sur-jective homomorphism of polynomial rings (possibly with infinite variables)(5.1) k [ x s ; s ∈ S d ′ ] → k [ x s ; s ∈ S d ] . Here k is a filed. Consider the quotient ring R d := k [ x s ; s ∈ S d ′ ] / ( x s ; s ∈ S d ) , which is, as a vector space, isomorphic to k · ⊕ M s ∈ S d k · x s . The homomorphism (5.1) induces a surjective homomorphism R d ′ → R d and yields an epi admissible proring R := ( R d ). We easily see that ˆ R = k .So R is not mild. Corollary 5.6. (1)
Let A → B be a morphism of prorings which is anepimorphism of A -promodules. If A is mild, then so is B . (2) Let A be a mild proring and M → N an epimorphism of A -promodules.If M is mild, then so is N . ON-ADIC FORMAL SCHEMES 31
Proof.
Again we prove only (1). Being mild, A is Mittag-Leffler and so is B .Hence we may suppose that A = ( A d ) and B = ( B e ) are epi. For each e , takea homomorphism A d → B e representing the given A → B . From Lemma1.17, it is surjective. Since A is mild, from Proposition 5.3, the natural mapˆ A → A d → B e is surjective. Since ˆ A → B e factors as ˆ A → ˆ B → B e , thenatural map ˆ B → B e is surjective, which proves the assertion. (cid:3) Corollary 5.7. (1)
Let A → B and A → C be morphisms of prorings.If B and C are mild, then so is B ⊗ A C . If A , B and C are gentle,then so is B ⊗ A C . (2) Let A be a mild (resp. gentle) proring, and M and N mild (resp.gentle) A -promodules. Then M ⊗ A N is (resp. gentle) mild.Proof. Again we prove only (1). We first consider the case where B and C are mild. Then we may suppose that B and C are epi and that A , B and C has a same index set and the given A → B and A → C are representedby level morphisms ( A d → B d ) and ( A d → C d ) respectively. Choose anindex d and arbitrary elements b ∈ B d and c ∈ C d . From Proposition 5.3,there exists ( b d ) ∈ ˆ B = lim ←− B d and ( c d ) ∈ ˆ C = lim ←− C d such that b d = b and c d = c . Then ( b d ⊗ c d ) ∈ \ B ⊗ A C = lim ←− B d ⊗ A d C d . Hence b ⊗ c ∈ Im( \ B ⊗ A C → B d ⊗ A d C d ). It follows that \ B ⊗ A C → B d ⊗ A d C d is surjective. From Proposition 5.3, this means that B ⊗ A C is mild.Next we consider the case where A , B and C are gentle. Then we maysuppose that A = ( A d ), B = ( B e ) and C = ( C f ) are indexed by countablesets. Then B ⊗ A C = ( B e ⊗ A d C f ) is also indexed by a countable set, sogentle. (cid:3) Lemma 5.8.
Let A be a mild proring, φ : M → N a morphism of mild A -promodules and ˆ φ : ˆ M → ˆ N the corresponding morphism of complete ˆ A -modules. Then the following are equivalent: (1) φ is an epimorphism in the category of A -promodules. (2) φ is an epimorphism in the category of mild A -promodules. (3) ˆ φ is an epimorphism in the category of complete ˆ A -modules.Proof. We obviously have (1) ⇒ (2) ⇔ (3). It remains to show (2) ⇒ (1).Suppose (2) and write M = ( M d ) and N = ( N e ), which may be supposedto be epi. For every e , the composite morphism φ e : M → N → N e isan epimorphism in the category of mild A -promodules. Therefore everymorphism φ de : M d → N e which represents φ e must be surjective. Hence φ is an epimorphism also in the category of A -promodules. (cid:3) Admissible rings.
An open ideal I of a topological ring is called an ideal of definition if every element a ∈ I is topologically nilpotent (that is, a n →
0, as n → ∞ ). A linearly topologized ring is called an admissible ring if it is complete and admits an ideal of definition. Especially every discretering is admissible. Every admissible ring has the largest ideal of definition,which is the ideal of all the topologically nilpotent elements. A collection of ideals of definition in a topological ring A is called a basisof ideals of definition if it is a basis of open ideals. If A is an admissiblering, J ⊆ A is an ideal of definition and ( I d ) is a basis of open ideals, then( I d ∩ J ) is a basis of ideals of definition. Thus every admissible ring admitsa basis of ideals of definition. In particular, the collection of all ideals ofdefinition is a basis of ideals of definition.If A = ( A d ) is an admissible proring, then ˆ A is an admissible ring. Con-versely if B is an admissible ring and ( I d ) is the set of all ideals of definition,then ( B/I d ) is a mild admissible proring. Hence(Mild admissible prorings) ∼ = (Admissible rings) . This induces the equivalence of subcategories,(Gentle admissible prorings) ∼ = (Gentle admissible rings) . Mild and gentle formal schemes.Definition 5.9.
A formal scheme is said to be mildly (resp. gently) affine if it is isomorphic to Spf A with A mild (resp. gentle). A formal scheme X is said to be mild (resp. gentle) if every point x ∈ X admits a mildly (resp.gently) affine neighborhood. For a mild (resp. gentle) formal scheme X , asemicoherent O X -promodule M is said to be mild (resp. gentle) if every point x ∈ X admits a mildly (resp. gently) affine neighborhood U = Spf A ⊆ X such that M ( U ) is a mild (resp. gentle) A -promodule.By definition, every gentle formal scheme is mild. Proposition 5.10.
Suppose that an affine formal scheme X = Spf A isgentle. Then A is gentle.Proof. Take a finite gently affine covering X = S Spf A i . Put A ij := Spf A ij , B := Q A i and C := Q C ij . We have an exact sequence0 → A → B ⇒ C. Since B and C are gentle, the morphisms B ⇒ C are represented by levelmorphisms ( B i ⇒ C i ) i ∈ N . Then A ∼ = (Ker ( B i ⇒ C i )) i ∈ N , and hence A isgentle. (cid:3) Proposition 5.11. (1)
Every distinguished open subscheme of a mildly(resp. gently) affine formal scheme is mildly (resp. gently) affine. This definition is due to McQuillan [McQ]. The one in [EGA] is more restrictive: In op. cit. , an ideal I is an ideal of definition if for every open neighborhood V of 0, thereexists n ∈ N with I n ⊆ V . This fails if we adopt the definition in [EGA].
ON-ADIC FORMAL SCHEMES 33 (2)
Let X be a mild (resp. gentle) formal scheme. Then for every x ∈ X , O X,x is mild (resp. gentle).Proof. (1) Let A := ( A d ) be a mild admissible proring. Set X := Spf A and X d := Spec A d . The natural map ˆ A → A red is surjective. Take anarbitrary f ∈ A red . Then D ( f ) = Spf A f , where A f = ( A d,f ). For anylift f ′ ∈ ˆ A of f , the natural map ( ˆ A ) f ′ → A d,f is surjective and factors as( ˆ A ) f ′ → c A f → A d,f . So c A f → A d,f is surjective and A f is mild. It shows theassertion for the mildly affine formal scheme. The assertion for the gentlyaffine formal scheme is trivial.(2) The proof is parallel to the one of (1). (cid:3) Proposition 5.12.
Let X := Spf A be a gentle affine formal scheme and M a gentle semicoherent O X -promodules. Then M ( X ) is a gentle A -promodule. Hence we have the equivalence of abelian categories(Gentle A -promodules) ∼ = (Gentle semicoherent O X -promodules)Proof. It can be proved in the same way as Proposition 5.10. (cid:3)
Proposition 5.13.
Let X be a mild formal scheme, M a mild semicoherent O X -promodule and N a quotient semicoherent O X -promodule of M . Then N is mild.Proof. It is a direct consequence of Corollary 5.6. (cid:3)
Corollary 5.14.
Every formal subscheme of a mild formal scheme is mild.Proof.
It is clear that every open formal subscheme of a mild formal schemeis mild. So it is enough to show that every closed formal subscheme of a mildformal scheme is mild, which follows from the preceding proposition. (cid:3)
Proposition 5.15. (1)
Let Y → X and Z → X be morphisms of formalschemes. If Y and Z are mild, then so is Y × X Z . If X , Y and Z are gentle, then so is Y × X Z . In particular, the category of mild(resp. gentle) formal schemes is closed under fiber products. (2) Let φ : Y → X be a morphism of mild (resp. gentle) formal schemesand M a mild (resp. gentle) semicoherent O X -promodule. Then φ ∗ M is mild (resp. gentle).Proof. The assertions follow from Corollary 5.7. (cid:3)
Complete sheaves.Definition 5.16.
Let X be a formal scheme. We define the complete struc-ture sheaf ˆ O X , which is a sheaf of complete rings, by ˆ O X ( U ) := \ O X ( U ) forqsqc U ⊆ X , and the complete stalk ˆ O X,x to be the complete ring corre-sponding to the proring O X,x .Since the projective limit is left exact, the complete structure sheaf isindeed a sheaf of complete rings. If X is mild, then every stalk O X,x is mildand ˆ O X,x is a local ring.
For mild formal schemes X and Y , a morphism φ : Y → X gives thedata of a continuous map φ : Y → X denoted by the same symbol and a φ -morphism ˆ O X → ˆ O X which induces local homomorphisms ˆ O X,φ ( y ) → ˆ O Y,y ,and vice versa.
Convention 5.17.
Let A be a mild admissible proring and B the corre-sponding admissible ring. Then by abuse of notation, we also write Spf B for Spf A .The underlying topological space of Spf B is identified with the set ofopen prime ideals of B . Definition 5.18.
Let X be a formal scheme. A complete ˆ O X -module is asheaf M of complete abelian groups such that for each qsqc U ⊆ X , M ( U )is given a complete ˆ O X ( U )-module structure in a compatible way.If X is a formal scheme and M is an O X -promodule, then putting ˆ M ( U ) := \ M ( U ), we obtain a complete ˆ O X -module ˆ M . Definition 5.19.
Let X be a mild formal scheme. A complete ˆ O X -module N is said to be semicoherent if N ∼ = ˆ M for some mild semicoherent O X -promodule M . If in addition X is gentle and M is gentle and Mittag-Leffler,then we say that N is gentle .From the definition, we obtain: Proposition 5.20. (1)
For a mild formal scheme X , (Mild semicoherent O X -promodules) ∼ = (Semicoherent complete ˆ O X -modules) . (2) For a gentle formal scheme X , (Gentle Mittag-Leffler semicoherent O X -promodules) ∼ = (Gentle semicoherent complete ˆ O X -modules) . (3) For a gentle admissible ring A , if we put X := Spf A , (Gentle semicoherent complete ˆ O X -modules) ∼ = (Gentle complete A -modules) . Local properties of mild formal schemes
We will study local properties of mild formal schemes and their formalsubschemes in terms of complete rings.6.1.
Adic and Noetherian admissible rings.Definition 6.1.
A complete ring A is said to be adic if there exists an ideal I ⊆ A such that I n , n ∈ N , form a basis of open ideals.By definition, an adic complete ring is admissible and gentle. ON-ADIC FORMAL SCHEMES 35
Definition 6.2.
An admissible ring is said to be
Noetherian if it is Noe-therian as a ring. An admissible ring A is said to be pro-Noetherian if itscorresponding admissible proring is pro-Noetherian, or equivalently if forevery open ideal I ⊆ A , A/I is Noetherian.It is obvious that every Noetherian admissible ring is pro-Noetherian.
Example 6.3.
Let A be the ring k [[ x, y ]] of formal power series in variables x and y over a field k with the (( xy n )) n ∈ N -topology. Then A is a Noetherianadmissible ring which is not adic. Definition 6.4.
A mild formal scheme X is said to be locally Noetherian (resp. adic ) if every x ∈ X has an affine neighborhood x ∈ Spf A ⊆ X withˆ A Noetherian and adic (resp. adic).The definition is due to [EGA]. We note that if A is Noetherian but notadic, then Spf A is not locally Noetherian. Proposition 6.5.
Let A be an admissible ring such that Spf A is locallyNoetherian. Then A is Noetherian and adic.Proof. [EGA, Chap. I, Cor. 10.6.5]. (cid:3) Proposition 6.6.
Let A be a pro-Noetherian admissible ring and I ⊆ A anideal of definition. Then for any neighborhood V of , there exists n ∈ N with I n ⊆ V . (Namely, in the sense of [EGA] , I is an ideal of definitionand A is admissible. See Footnote 5.3, page 32)Proof. Since A is linearly topologized, we may suppose that V is an ideal.Then A/V is Noetherian. Therefore I ( A/V ) is finitely generated. Sinceevery element of I ( A/V ) is nilpotent, so is I ( A/V ). This means that forsome n , I n ⊆ V . (cid:3) From this proposition, our locally ind-Noetherian mild formal schemedetermines a formal scheme in the sense of [EGA].
Proposition 6.7.
Every pro-Noetherian adic ring A is Noetherian. Fur-thermore for every ideal I of definition in A , the topology on A is identicalto the I -adic topology.Proof. Let I ⊆ A be an ideal such that ( I n ) n ∈ N is a basis of ideals of defi-nition. By definition, A/I and
A/I are Noetherian. Consequently I/I isfinitely generated, and from [EGA, 0, Cor. 7.2.6], A is Noetherian.Let J be an arbitrary ideal of definition. Then for some m ∈ N , I m ⊆ J .Hence for every n ∈ N , I mn ⊆ J n , and so J n is open. Conversely, since J isfinitely generated, for every n ∈ N , there exists m ∈ N with J m ⊆ I n . Thisproves the lemma. (cid:3) Corollary 6.8.
Every mild, locally ind-Noetherian and adic formal schemeis locally Noetherian.Proof.
Obvious. (cid:3)
Strict formal subschemes.
Let A be an admissible ring. The cate-gory of complete A -modules is an additive category with kernels and cok-ernels. For a morphism φ : N → M of complete A -modules, we have thekernel K ⊆ N in the category of A -modules. Since K ⊆ N is closed, it isa complete A -module with respect to the subspace topology, and the kernelof φ also in the category of complete A -modules.For every closed A -submodule P of a complete A -module M , we canconstruct the complete quotient , denoted M (cid:12) P , as the completion of theusual quotient M/P which is endowed with the quotient topology.If M is gentle, then M/P is complete, so M (cid:12) P = M/P (for example, see[Mat, Th. 8.1]). Now the cokernel of a morphism φ : N → M of complete A -modules is M (cid:12) Im φ . Here Im φ is the closure of Im φ ⊆ M . Definition 6.9.
Let M → N be an epimorphism of complete A -modules.We say that N is a normal quotient of M if N ∼ = M (cid:12) Ker ( φ ).Let M be a complete A -module, ( M d ) a basis of open A -modules and P ⊆ M a closed A -submodule. Then we have the exact sequences of discrete A -modules 0 → P/ ( P ∩ M d ) → M/M d → M/ ( P + M d ) → A -promodules0 → ( P/ ( P ∩ M d )) → ( M/M d ) → ( M/ ( P + M d )) → . Applying the completion functor to the last exact sequence, we obtain theexact sequence of complete A -modules0 → P → M → M (cid:12) P → . Lemma 6.10.
Let A be an admissible ring and M → N an epimorphism ofcomplete A -modules. Then N is a normal quotient if and only if the kernelof the natural epimorphism ˇ M → ˇ N of the corresponding mild ˇ A -promodulesis mild.Proof. First suppose that N is a normal quotient and write N = M (cid:12) P .Take a basis ( M d ) of open A -submodules of M . Then the kernel of ˇ M → ˇ N is ( P/ ( P ∩ M d )), which is clearly mild.Conversely suppose that the kernel K of ˇ M → ˇ N is mild. Then we maysuppose that ˇ M = ( M d ) d ∈ D , K = ( K d ) d ∈ D and for each d ∈ D , K d ⊆ M d .Put L d := T d ′ ≥ d Im( K d ′ → K d ) and L := ( L d ). Since K is Mittag-Leffler,the natural morphism L → K is an isomorphism, and L is equal to K as asubobject of ˇ M . So ˇ N = ˇ M /K = ˇ
M /L = ( M d /L d ) . Now we easily see that N = M (cid:12) ˆ L and so N is a normal quotient. (cid:3) If A is an admissible ring, ( I d ) is a basis of ideals of definition and J ⊆ A is a closed ideal, then B := ( A/ ( J + I d )) is an admissible proring. So thenormal quotient A (cid:12) J = ˇ B is an admissible ring. ON-ADIC FORMAL SCHEMES 37
Definition 6.11.
Let X be a formal scheme. A formal subscheme Y ⊆ X is said to be strict if for every y ∈ Y , there exists an affine neighborhood U ⊆ X such that Y ⊆ U is closed and ˆ O Y ( U ) is a normal quotient of ˆ O X ( U ).If A is a Noetherian adic ring, then every ideal I of A is closed (see [ZS,page 264] or [Mat, Th. 8.2 and 8.14]). Then A/I = A (cid:12) I and Spf A/I is a strict closed formal subscheme of Spf A . Thus for a locally Noetherianformal scheme X , the formal closed subscheme of X is the same as the closedsubscheme in [EGA]. Proposition 6.12.
Let X be a mild formal scheme and Y a closed formalsubscheme. Then Y is strict if and only if the defining proideal sheaf of Y is mild.Proof. The proposition follows from Lemma 6.10. (cid:3)
Proposition 6.13.
Every strict formal subscheme of a gentle formal schemeis gentle.Proof.
Obvious. (cid:3)
Corollary 6.14.
Every strict closed formal subscheme of a gentle affineformal scheme
Spf A is Spf
A/I for some closed ideal I ⊆ A .Proof. Let Y ⊆ X := Spf A be a closed formal strict subscheme and I ⊆ O X the defining proideal sheaf, which is gentle and Mittag-Leffler. Now thecompletion I of I ( X ) is a closed ideal of A and Y = Spf A (cid:12) I . (cid:3) Proposition 6.15.
Let A be an admissible ring, J ⊆ A a closed idealand B := A (cid:12) J . If A is adic (resp. Noetherian and adic), then so is B . Correspondingly every closed formal strict subscheme of an adic (resp.locally Noetherian) formal scheme is adic (resp. locally Noetherian).Proof. In both cases, since A is gentle, B = A/J . So if A is Noetherian, so is B . It remains to show that if A is adic, so is B . If A has the I -adic topologyfor some ideal I ⊆ A , then B = lim ←− A/ ( J + I n ) = lim ←− B/ ¯ I n , where ¯ I := IB .It shows that B has the ¯ I -adic topology. We have proved the assertion. (cid:3) Proposition 6.16.
Every (non-formal) subscheme of a mild formal schemeis strict.Proof.
It is enough to consider a closed subscheme Y of an affine formalscheme X := Spf A . Then Y is also affine, say Y = Spec B . If we write A = ( A d ), then every map A d → B representing A → B is surjective. Soˆ A → B is surjective. Its kernel I ⊆ ˆ A is an open ideal. So the quotienttopology on ˆ A/I is discrete, so B = ˆ A/I = ˆ A (cid:12) I . Thus Y is strict. (cid:3) Proposition 6.17.
Let X be a mild formal scheme, Y ⊆ X a strict formalsubscheme and φ : W → X a morphism of mild formal schemes. Then theformal subscheme Y × X W ⊆ W is strict. Proof.
We may suppose that Y ⊆ X is closed. If I Y ⊆ O X denotes thedefining proideal sheaf, then we have the exact sequence0 → I Y → O X → O W → . Since φ ∗ is right exact, the sequence φ ∗ I Y → O W → O Y × X W → Y × X W ⊆ W is the image of φ ∗ I Y →O W . It is mild because φ ∗ I Y is so, which proves the proposition. (cid:3) Proposition 6.18.
Let A be a Noetherian adic ring and ( I d ) d ∈ D a directedset of ideals of definition in A (not necessarily a basis of ideals of definition),and let B := lim ←− A/I d . Suppose that B is adic. Then the natural map A → B is surjective. Moreover for every d , the topology on B is the ( I nd B ) n ∈ N -topology.Proof. Being pro-Noetherian and adic, from Proposition 6.7, B is Noether-ian. For each d , put ˆ I d := lim ←− d ′ ≥ d I d /I d ′ . Fix e ∈ D , and set J := ˆ I e . Since J is an ideal of definition, again fromProposition 6.7, the topology on B is identical to the J -adic topology. Since( ˆ I d ) is also a basis of ideals of definition of B , for every n , there exists d ∈ D such that ˆ I d ⊂ J n . Then we have B/J n = ( B/ ˆ I d ) / ( J n / ˆ I d ) = ( A/I d ) / ( I e /I d ) n . Hence there exists d ∈ D such that the kernel of B/J m → B/J n is( I e /I d ) n / ( I e /I d ) m = I ne ( B/J m ) . Besides
B/J = A/I e is clearly a finitely generated A -module. As a result,the projective systems ( A/I ie ) and ( B/J n ) satisfy the conditions of [EGA,0, Prop. 7.2.9], and hence J n = Ker ( B → B/J n ) = I ne B. This shows the second assertion.For each d , the map A/I d → B/I d B is surjective and B is separated forthe ( I nd B )-topology. From [Mat, Th. 8.4], A → B is surjective. (cid:3) Corollary 6.19.
Let X be a locally Noetherian formal scheme and Y ⊆ X a closed formal subscheme with Y red = X red . Then Y is strict if and only if Y is adic.Proof. It is a consequence of Propositions 6.15 and 6.18. (cid:3)
Example 6.20.
Let A = k [[ x, y ]] be the admissible ring as in Example 6.3and A adic be the same ring k [[ x, y ]] endowed with the ( xy )-adic topology.The identity map A adic → A is a morphism of admissible rings. Then A isnot a normal quotient. So Spf A is a closed formal subscheme of Spf A adic which is not strict. ON-ADIC FORMAL SCHEMES 39
Example 6.21.
Suppose that the ring C [ x ][[ t ]] is endowed with the ( t )-adictopology. Put X := Spf C [ x ][[ t ]]. The underlying topological space of X isidentified with that of A = Spec C [ x ]. For each a ∈ C , we define Y a := Spec C [ x ][[ t ]] / ( t , ( x − a ) t ) , which is a closed subscheme of X and has an embedded point at a ∈ C = A ( C ). For a finite subset S ⊆ C , we define Y S to be the subscheme of X that is isomorphic to Y a around each a ∈ S and to A outside S .Let T be a subset of C . Then the Y S with finite S ⊆ T form an inductivesystem. Define a closed formal subscheme Y of X by Y := lim −→ finite S ⊆ T Y S . Then ˆ O Y,p ∼ = ( k [ x, y ] / ( y , xy )) ( x,y ) ( p ∈ T ) k [ x ] ( x ) ( p ∈ C \ T ) k ( x ) ( p the generic point) . Thus all complete stalks of ˆ O Y are discrete. If Y is locally Noetherian, thenit is impossible that infinitely many complete stalks of O Y have an embeddedprime. Therefore if T is an infinite set, then Y is not Noetherian nor a strictformal subscheme. Moreover for every open subscheme U ⊆ X , Y ∩ U is nota strict closed formal subscheme of U either. If T is uncountable, then Y isnot gentle (but mild). Theorem 6.22.
Let X be a locally Noetherian formal scheme. Every closedformal subscheme of X is strict if and only if the underlying topological spaceof X is discrete.Proof. The “if” direction is essentially due to Chevalley [Che, Lem. 7] (seealso [ZS, Ch. VIII, §
5, Th. 13]). To show this, we may suppose that theunderlying topological space of X consists of a single point. Then for someNoetherian complete local ring ( A, m ) with the m -adic topology, we have X ∼ = Spf A . There exists a directed set ( I d ) of open ideals of A such that Y = lim −→ d Spec
A/I d . Replacing A with A/ T I d , we may suppose that T I d = 0. Then for each n ∈ N , since A/ m n is Artinian, there exists d such that I d ( A/ m n ) = 0,equivalently I d ⊆ m n . Conversely for every d , there exists n ∈ N with m n ⊆ I d . Thus the ( I d )-topology coincides with the m -adic topology, and so Y = X .We now prove the “only if” direction. Suppose that the underlying topo-logical space of X is not discrete. Then there exists a closed but not openpoint x of X . Let Spf A ⊆ X be an affine neighborhood of x . Then Spf A has at least two points. Let A red be the reduced ring associated to A , that is,the ring A modulo the ideal of nilpotent elements. Then Spf A and Spf A red0 TAKEHIKO YASUDA have the same underlying topological space. If ˆ A red is the m -adic comple-tion of A red with m the maximal ideal of x , then Spf ˆ A red is a closed formalsubscheme of Spf A red consisting of a single point, hence not isomorphic toSpf A red . Being injective, the natural map A red → ˆ A red does not factors as A red → A red /J ∼ = ˆ A red for any nonzero ideal J . Hence Spf ˆ A red is not aclosed formal strict subscheme of either Spf A red or of Spf A . (cid:3) As a consequence of a theorem in [HR], Bill Heinzer showed the following(see the first page of [AJL]):
Theorem 6.23.
Let k be a field. There exists a nonzero ideal I ⊆ k [ x ± , y, z ][[ t ]] with I ∩ k [ x, y, z ][[ t ]] = (0) . Suppose that k [ x, y, z ][[ t ]] and k [ x ± , y, z ][[ t ]] are given the ( t )-adic topolo-gies. Then X := Spf k [ x ± , y, z ][[ t ]] is an open formal subscheme of ˜ X := k [ x, y, z ][[ t ]]. Let I ⊆ k [ x ± , y, z ][[ t ]] be an ideal as in the theorem and Y := Spf k [ x ± , y, z ][[ t ]] /I . The theorem says that there is no closed strictformal subscheme ˜ Y ⊆ ˜ X with Y = ˜ Y ∩ X . So, The closure ¯ Y of Y in ˜ X isnot strict.There exists also a simpler example: Theorem 6.24.
Consider an element of C [ x ± , y ][[ t ]] f := y + a x − t + a x − t + a x − t + · · · , a i ∈ C \ { } . Suppose that the function i
7→ | a i | is strictly increasing and lim i →∞ | a i +1 || a i | = ∞ . Then ( f ) ∩ C [ x, y ][[ t ]] = (0) . Proof.
We prove the assertion by contradiction. So we suppose that thereexists 0 = g = P i ∈ N g i t i ∈ C [ x ± , y ][[ t ]] with g i ∈ C [ x ± , y ] such that h := f g ∈ C [ x, y ][[ t ]]. If we write h = P i ∈ N h i t i with h i ∈ C [ x, y ], then for every i ∈ N , we have h i = yg i + i X j =1 a j x − j g i − j = yg i + h ′ i , ( h ′ i := i X j =1 a j x − j g i − j ) . In what follows, we will show that for sufficiently large i , the bottom term of h ′ i +1 (that is, the lowest term in the lexicographic order) is lower in y -orderthan that of h ′ i , which leads to a contradiction.For each i ∈ N , write g i = X m ∈ Z ,n ∈ N g imn x m y n , g imn ∈ C . ON-ADIC FORMAL SCHEMES 41
We set d i := inf { m ∈ Z |∃ n, g imn = 0 } (the order of g i in x ) ,e i := inf { n ∈ N | g id i n = 0 } (the order of X n g id i n in y ) ,D i := inf { d i − j − j | ≤ j ≤ i } = inf { d j ′ − i + j ′ | ≤ j ′ ≤ i − } (the infimum of the orders of x − j g i − j , 1 ≤ j ≤ i ) ,E i := inf { e i − j | ≤ j ≤ i, d i − j − j = D i } = inf { e j ′ | ≤ j ′ ≤ i − , d j ′ − i + j ′ = D i } (the infimum of the y -orders of those termsin the x − j g i − j , 1 ≤ j ≤ i which are of x -order D i ) . Here by convention, inf ∅ = + ∞ . We easily see that for every i ′ > i , D i ′ < D i and E i ′ ≤ E i . If for i ∈ N , D i < x D i y E i in h ′ i is nonzero,then the coefficient of x D i y E i in yg i is also nonzero. Moreover if either“ m < D i ” or “ m = D i and n < E i ”, then the coefficient of x m y n in yg i vanishes. It follows that d i = D i and e i = E i , and that D i +1 = D i − E i +1 = E i − , and that the coefficient of x D i y E i in h ′ i +1 is again nonzero. As a result, E i +1 = E i − i ≥ i . Since E i ∈ N for every i , it is impossible.Now it remains to show that for some i ∈ N , D i < x D i y E i in h ′ i is nonzero. Suppose by contrary that for every i ∈ N with D i <
0, the coefficient of x D i y E i in h ′ i is zero. Since i D i is strictlydecreasing, there exists i ∈ N such that for every i ≥ i , D i <
0. Thenfor every i ≥ i , the coefficient of x D i y E i in yg i must be zero. Therefore wehave D i = D i − ( i − i ) and E i = E i . LetΛ := { j | the coefficient of x D i y E i in x − j g i − j is nonzero } ⊆ { , , . . . , i } and let 0 = c j ∈ C be the coefficient of x D i y E i in x − j g i − j , j ∈ Λ. Forevery i ≥ i , the coefficient of x D i y E i in h ′ i is X j ∈ Λ a j + i − i c j = 0 . Let j ∈ Λ be the largest element and j ∈ Λ the second largest one. (Notethat ♯ Λ ≥ a i , for i ≫ i , we have | a j + i − i | − ( ♯ Λ − | a j + i − i | ( max j ∈ Λ \{ j } | c j /c j | ) > . Therefore, for i ≫ | X j ∈ Λ a j + i − i c j |≥ | c j | | a j + i − i | − X j ∈ Λ \{ j } | a j + i − i c j /c j | ≥ | c j | (cid:18) | a j + i − i | − ( ♯ Λ − | a j + i − i | ( max j ∈ Λ \{ j } | c j /c j | ) (cid:19) > (cid:3) If we remove one more variable, then there is no ideal as in Theorems6.23 and 6.24:
Proposition 6.25.
Let k be a field. Then for any nonzero ideal I of k [ x ± ][[ t ]] , I ∩ k [ x ][[ t ]] = (0) .Proof. It suffices to prove the assertion in the case where I is principal, say I = ( f ), f ∈ k [ x ± ][[ t ]]. Write f = X i ≥ n f i t i ∈ k [ x ± ][[ t ]] , f i ∈ k [ x ± ] , f n = 0 . Define g i ∈ k [ x ± , f − n ] inductively as follows; g := f − n , g i +1 := − ( X ≤ j ≤ i g j f n + i +1 − j ) /f n . Then f ( X i ≥ g i t i ) = X m ≥ n (( f n g m − n + X i + j = mj Locally pre-Noetherian formal schemes and plain formal sub-schemes.Definition 6.26. A mild formal scheme X is said to be locally pre-Noetherian if for every x ∈ X , ˆ O X,x is Noetherian (not necessarily adic). Definition 6.27. A formal subscheme Y of a mild formal scheme X is saidto be plain if for every y ∈ Y , the map of complete stalks ˆ O X,y → ˆ O Y,y issurjective. Proposition 6.28. A plain formal subscheme of a locally pre-Noetherianformal scheme is again locally pre-Noetherian.Proof. Obvious. (cid:3) Proposition 6.29. A strict formal subscheme of a gentle formal scheme isplain.Proof. Let X be a gentle formal scheme and Y be a gentle and strict formalsubscheme. If necessary, replacing X with an open formal subscheme, wemay suppose that Y is a closed formal subscheme, say with the definingproideal I ⊆ O X . Then for each y ∈ Y , we have the short exact sequenceof mild O X,y -modules 0 → I y → O X,y → O Y,y → . Hence we have ˆ O Y,y = ˆ O X,y (cid:12) ˆ I y . Since ˆ O X,y is gentle, ˆ O X,y (cid:12) ˆ I y = ˆ O X,x / ˆ I y .As a consequence, the natural map ˆ O X,y → ˆ O Y,y is surjective and Y is aplain formal subscheme. (cid:3) Proposition 6.30. Every (non-formal) subscheme of a mild formal schemeis plain.Proof. Let Y be a subscheme of a mild formal scheme X . For each y ∈ Y , O Y,y is a ring and O X,y is an admissible ring. If we write O X,y = ( A d ), thenevery morphism A d → O Y,y representing O X,y → O Y,y is surjective. Since O X,y is mild, ˆ O X,y → A d → O Y,y = ˆ O Y,y is surjective, which proves theproposition. (cid:3) Example 6.31. Let A and A adic be as in Example 6.3 and 6.20. Theformal schemes X := Spf A and X adic := Spf A adic have the same underlyingtopological space, which consists of three open prime ideals, ( x, y ), ( x ) and( y ). The complete stalks of O X and O X adic at ( x, y ) and ( y ) are identical asrings, but not at ( x ). We haveˆ O X, ( x ) = k (( y ))[[ x ]] / ( x ) = k (( y )) and ˆ O X adic , ( x ) = k (( y ))[[ x ]] . It follows that via the morphism X → X adic induced by the identity map A adic → A , X is a plain formal subscheme of X adic but not strict. Example 6.32. With the notation as in Example 6.21, if T ⊆ C is infinite,then Y ⊆ X is plain but not strict. Formal separatrices of singular foliations In this section, we construct non-adic formal schemes from singularitiesof foliations.7.1. Formal separatrices. Let X be a smooth algebraic variety over C ,and Ω X = Ω X/ C the sheaf of (algebraic) K¨ahler differential forms. A (one-codimensional) foliation on X is an invertible saturated subsheaf F of Ω X satisfying the integrability condition: F ∧ d F = 0. We say that a foliation F is smooth at x ∈ X if the quotient sheaf Ω X / F is locally free around x , andthat F is singular at x otherwise. We say that F is smooth if F is smoothat every point. The pair ( X, F ) of a smooth variety X and a foliation on X is called a foliated variety . Definition 7.1. Let ( X, F ) be a foliated variety, x ∈ X ( C ), X /x := Spf ˆ O X,x , Y ⊆ X /x a strict closed formal subscheme of codimension one defined by0 = f ∈ ˆ O X,x , and ω ∈ Ω X,x a generator of F x . We say that Y is a formalseparatrix (of F ) at x if f divides ω ∧ df .Frobenius theorem says that if F is smooth at x , there exists a uniquesmooth formal separatrix of F at x . Miyaoka [Miy] proved that the familyof smooth formal separatrices at smooth points of a foliation is a formalscheme: Theorem 7.2. [Miy, Cor. 6.4] Let ( X, F ) be a foliated variety. Supposethat F is smooth. Then there exists a strict closed formal subscheme L of ( X × C X ) / ∆ X such that for every point x ∈ X , p ( p − ( x )) is the smoothformal separatrix of F at x . Here ∆ X ⊆ X × C X is the diagonal, ( X × C X ) / ∆ X is the completion of X × C X along ∆ X and p , p : L → X are thefirst and second projections. Let ( X, F ) be a foliated variety and C ⊆ X a closed smooth subvariety ofdimension 1. Suppose that C meets only at a single point o with the singularlocus of F . Let U ⊆ X be the smooth locus of F and L ⊆ ( U × C U ) / ∆ U thefamily of formal separatrices as in the theorem. Then C \ { o } is a closedsubvariety of U . The fiber product L C \{ o } := ( C \ { o } ) × U,p L is the family of the smooth formal separatrices over C \ { o } , and a strictformal subscheme of Z C := ( C × C X ) / ∆ C . Let L C := L C \{ o } be the closureof L C \{ o } in Z C . Proposition 7.3. The following are equivalent: (1) L C is locally Noetherian. (2) L C is adic. (3) L C is locally pre-Noetherian. (4) L C ⊆ Z C is strict. (5) L C ⊆ Z C is plain. ON-ADIC FORMAL SCHEMES 45 Proof. ⇒ ⇒ 3: Trivial.2 ⇒ 1: Corollary 6.8.3 ⇒ 2: Let p ∈ L C be the point over o ∈ C , A := ˆ O L C ,p and I ⊆ A thelargest ideal of definition, which is prime. From the construction of L C , thesymbolic powers I ( n ) form a basis of ideals of definition in A . It is easy tosee that A is a domain. If m ⊆ A is the maximal ideal of p , then the m -adiccompletion of A is also a domain. From [Zar, page 33, Lem. 3] (see also [ZS,Ch. VIII, § 5, Cor. 5]), the topology on A is equal to the I -adic topology.Now let Spf B ⊆ L C be an affine open with B an admissible ring and J ⊆ B the largest ideal of definition. Then we see that the topology on B is the J -adic topology, which prove the assertion.2 ⇒ 4: Corollary 6.19.4 ⇒ 5: Proposition 6.29.5 ⇒ 3: Proposition 6.28. (cid:3) Theorem 7.4. Suppose that one of the conditions in Proposition 7.3 holds.Then the fiber of L C → C over o is a formal separatrix at o .Proof. We need to use complete modules of differentials of locally Noetherianformal schemes. For a morphism f : W → V of locally Noetherian formalschemes, we have a complete module of differentials, ˆΩ W/V , which is a semi-coherent complete ˆ O W -module, and a derivation ˆ d W/V : ˆ O W → ˆΩ W/V . Werefer to [AJP] for details.If necessary, shrinking X , we can take a nowhere vanishing ω ∈ F ( X ).Let ψ : Z C → X. be the projection. Pulling back ω , we obtain a global section ψ ∗ ω of ˆΩ Z C /C .Since L C is a hypersurface in Z C , it is defined by a section f of ˆ O Z C . Sincethe restriction of L C to C \ { o } is the family of formal separatrices along C \ { o } , f divides ψ ∗ ω ∧ ˆ d Z C /C f .Let Y be the fiber of L C → C over o , which is a hypersurface of X /o defined by the image ¯ f ∈ ˆ O X,o of f . Then ¯ f divides ω ∧ ˆ d X /o / C ¯ f . Hence Y is a formal separatrix. (cid:3) Jouanolou’s theorem. We recall Jouanolou’s result on Pfaff forms.We refer to [Jou] for details.An algebraic Pfaff form of degree m on P C is a one-form ω = ω dx + ω dy + ω dz such that ω i are homogeneous polynomials of degree m and the equation xω + yω + zω = 0holds. A Pfaff equation of degree m on P C is a class of algebraic Pfaff formsmodulo nonzero scalar multiplications.Let ω be an algebraic Pfaff form on P C and [ ω ] its Pfaff equation class.An algebraic solution of ω or [ ω ] is a class of homogeneous polynomials f ∈ C [ x, y, z ] modulo nonzero scalar multiplications such that f divides ω ∧ df .Let V m be the vector space of the algebraic Pfaff forms of degree m on P C . Then the set of the Pfaff equations of degree m on P C is identified withthe projective space P ( V m ) = ( V m \ { } ) / C ∗ . Define Z m ⊆ P ( V m )to be the set of the Pfaff equations that have no algebraic solution. Theorem 7.5. [Jou, § Suppose m ≥ . Then Z m is the intersection ofcountably many nonempty Zariski open subsets of P ( V m ) and contains theclass of the algebraic Pfaff form ( x m − z − y m ) dx + ( y m − x − z m ) dy + ( z m − y − x m ) dz. From [Jou, page 4, Prop. 1.4], every algebraic Pfaff form ω on P C isintegrable: dω ∧ ω = 0. So ω defines also a foliation F ω on C . From [Jou,page 85, Prop. 2.1], the only singular point of F ω is the origin. Accordinglywe can define the family L ω,C \{ o } of formal separatrices along C \ { o } andits closure L ω,C for any line C ⊂ C through the origin.Let f = P i ≥ n f i ∈ C [[ x, y, z ]]. Here f i is a homogeneous polynomialof degree i and f n = 0. Suppose that f defines a formal separatrix at theorigin, equivalently that f divides ω ∧ df . Then the class of f n is an algebraicsolution of the Pfaff equation [ ω ]. Hence if [ ω ] ∈ Z m , then F ω has no formalseparatrix at the origin. Corollary 7.6. For [ ω ] ∈ Z m and a line C ⊆ C through the origin, theformal subscheme L ω,C of Z C ∼ = Spf C [ w ][[ x, y, z ]] is neither strict, plain,locally pre-Noetherian nor adic.Proof. If L ω,C is either strict, plain, locally pre-Noetherian or adic, thenfrom Theorem 7.4, the foliation F ω has a formal separatrix at the origin.Hence [ ω ] / ∈ Z m , a contradiction. 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