Formal Formal Schemes

Michael McQuillan

Abstract. We propose some alterations, or better a putting back on course, of chapters 7 and 10 of [EGA], leading to formal schemes and quasi-coherent modules enjoying not only good exactness properties and determination by global sections as per the Noetherian case, but even complete stalks by virtue of patching not at the level of locally ringed spaces, but rather in a ‘new’- strictly speaking there is no need for the inverted commas around the word new, on the other hand the said category is simply functorial with respect to the ideas of A. Grothendieck, so there is- even in the Noetherian case, category of admissibly ringed spaces.

Introduction Exegesis of texts is not in general considered a mathematical discipline. In so much, however, as the text in question is the cornerstone of modern algebraic geometry, the labour in question takes on a certain value, and even more so if we can offer a little improvement on the original. The specific text in question is of course Grothendieck’s “Eléments de Géométrie Algébrique”. Even more specifically we’re only interested in part I, which we’ll rather abusively denote [EGA], and that therein which pertains to formal schemes. Now, while any modifications we might proffer of [EGA] itself are very much in the non-Noetherian realm, and whence very much a matter for the cognoscenti, it is nevertheless psychologically deeply disturbing that the absolute generality of [EGA] and the ideas therein, seem to break down catostrophically as soon as one seeks to extend them to the formal category without the Noetherian hypothesis. While we may certainly note in passing that problems such as Zariski’s conjecture on the Notherianity of the global functions on a variety completed in in a sub-variety, [H], imply that it’s not wholly obvious that rather natural geometric constructions don’t lead outwith the Noetherian world, it appears equally noteworthy that the stalks of the structure sheaf on a formal scheme in the sense of [EGA] are not in general a complete local ring, nor for that matter are the stalks of coherent modules, complete modules. Should one be happy working in the category of schemes, or algebraic spaces, and other categories built on these such as algebraic stacks, then such questions would appear rather irrelevant. Nonetheless there are no shortage of everyday ob- jects such as infinitesimal groupoids, or to give them their more common name, integrable foliations, whose algebro-geometric study, cf. [B-M], takes place prop- erly at the formal level. In addition, there seems to be no shortage of internal 1 2 MICHAEL MCQUILLAN evidence within the latter parts of [EGA] that schemes should only be considered as a stepping stone to formal schemes. In any case, let’s summarise the problems in a little more detail, and our proposed solutions as indicated by our various chapters. I. Admissibility. This is a highly desirable generalisation of adically complete rings, but the category as defined in [EGA] doesn’t have ‘natural’ limits. To elu- cidate, a natural limit should reflect the topology. Quite generally if Mi, i ∈ I, is a directed system of continuous maps of complete abelian groups in some linear topologies, we say that the system maps finely to a module M in the said category if not only do we have commutative diagrams of continuous maps,

Mi → M ↓ % Mj for j > i, but that M enjoys the finest linear topology with respect to which this holds, and it’s easily checked that fine limits lim∧M exist. The technical definition −→ i i of natural limit is of course fine limit. Ergo, stage I is to slightly alter the definition of admissibility so that fine limits of admissible rings are admissible, sheafily the admissible category, that is, introduce admissibly ringed spaces, and of course fine stalks. II. Complete Modules. Ultimately the sheafication of these will be the quasi- coherent modules on formal schemes, but to begin with, this might be thought of as the category of modules over an admissible ring in which the Artin-Rees lemma holds, that is, we more or less just declare the Artin-Rees lemma to be true. This yields sheaves of complete modules on admissibly ringed spaces, for which exactness is determined by the fine stalks. III. Localisation. By using fine limits, we’re in a position where ultimately the ‘local rings’ of formal schemes will themselves be admissible in contradistinction to [EGA]. Better still, in the complete category localisation proves to be an exact functor. IV. Formal Schemes. Modulo the changes necessitated by our change of definition for admissible ring and introduction of locally admissibly ringed spaces, this follows [EGA] verbatim, and is merely a check that nothing has gone wrong. V. Quasi-coherent Modules. Evidently this is where everything gets put together, and should be considered as our ultimate goal. Indeed we arrive at a category of modules on formal schemes, without any hypothesis of Noetherianity, wholly determined by their global sections over formal affines at one extreme, and by their formal stalks at another. In the specific case of adelically complete Noetherian rings, and their finitely generated modules, the Artin-Rees lemma embeds this latter category as a full sub-category of our quasi-coherent category. It is, however, recommended that the reader does not try to extrapolate the meaning of a quasi-coherent sheaf of modules on a formal scheme from the schematic/locally ringed space notion of the same which in the admissible setting would represent a property of such strength that the category is unlikely to be abelian. Indeed the [EGA] notion must, therefore, be considered as either an unworthy subterfuge or a work of genius, and as such the jury may yet prove FORMAL FORMAL SCHEMES 3 the healty scepticism of J.-B. Bost on this subject, for conversation on which warm thanks are extended, correct.

1. Admissible Rings In [EGA] §7 one finds a definition of admissible rings. Unfortunately this definition is too restrictive, and doesn’t lead to a category with direct limits. Con- sequently while freely using the terminology of op. cit. we will employ technically distinct definitions, beginning with: Definition 1.1. Let A be a linearly topologised ring, that is, 0 has a basis of open neighbourhoods consisting of ideals, then for I an ideal of A we say In tends weakly to zero if f n → 0, n ∈ N, for all f ∈ I. An open ideal tending weakly to zero is called an ideal of definition. The difference with [EGA] of the corresponding notion is that in op. cit., In → 0, not just weakly. This is in itself the only difference in regard to: Definition 1.2. A complete separated linear topologised ring with an ideal of definition is said to be admissible. If it is neither separated nor complete then it is pre-admissible. Observe the key point that if I is an ideal of definition, then in contrast to the adelic case In, n ∈ N, need not be. However if J is open and I an ideal of definition then I ∩ J is also an ideal of definition, so we have:

Fact/Notation 1.3. The totality of ideals of definition {Iλ : λ ∈ Λ} is a basis of open neighbourhoods of zero. The advantage of our definition over [EGA] is that trivially: Observation/Definition 1.4. Let A be a linearly topologised ring and N the topological nil-radical, that is, N = {f ∈ A : f n → 0}. Then A is admissible iff N is open. An immediate consequence of which is: Sub-observation 1.4 bis. The category of admissible rings has fine direct limits, limc . −→ Indeed let A , i ∈ I be a directed system, and let B = limA be the Proof. i −→ i i usual direct limit, which we topologise by way of the finest linear topology such −1 that the maps, ϕi : Ai → B are continuous, that is, J/B is open iff, ϕi (J) is open −1 for all i. Denoting by NB the nil-radical of B, then certainly ϕi (NB) ⊃ NAi , so B is pre-admissible, and its completion in the said topology is admissible. Evidently we’ve employed in passing the basic fact:

Fact 1.5. If A is pre-admissible and {Iλ : λ ∈ Λ} the ideals of definition then lim A/I is admissible with A being the same iff the natural map, A → limA/I is ←− λ ←− λ λ λ an isomorphism. A combination of these remarks shows that these definitions sheafify well, that is: 4 MICHAEL MCQUILLAN

Definition 1.6. A ringed space (X, OX) is said to be an admissibly ringed space, if there is a collection {Iλ : λ ∈ Λ} of sheaves of ideals such that: 0 0 (i): ∀λ, λ ∈ Λ there is µ ∈ Λ with Iµ = Iλ ∩ Iλ, in particular the Iλ(U) give a basis of a linear topology on OX(U) for all open sets U. (ii): For all opens sets U, OX(U) is complete and separated for the induced topology. n (iii): For all open sets U, and λ ∈ Λ, Iλ(U) →0. w Necessarily the ﬁne stalks O := limO (U), x ∈ X, and U open are to be bX,x −→c X U3x understood in the category of admissible rings, with of course:

Definition 1.7. An admissible ringed space (X, OX) is said to be an admissible locally ringed space if the ﬁne stalks ObX,x taken in the category of admissible rings are local rings.

2. Complete Modules

To reiterate our notations, {Iλ : λ ∈ Λ} will be the ideals of deﬁnition of an admissible ring A. For any A-module M we have the A-linear topology, that is, that with basis of open neighbourhoods of zero, {IλM : λ ∈ Λ}. The appropri- ate category of A-modules comprises those that can be recovered from their data modulo Iλ, that is: Definition 2.1. An A-module M is said to be complete if it is complete and separated in any topology coarser than the A-linear topology. ¿From which follows: Fact 2.2. If M is a complete A-module, and Φ: A × M → M the A-action, then Φ is continuous. Proof. By deﬁnition if M 0 ⊂ M is an open sub-module, then being open in the −1 0 A-linear topology it contains IλM for some λ ∈ Λ, whence Φ (M ) ⊃ Iλ × M. Naturally between complete A-modules we consider only continuous homomor- phisms, and so view complete A-modules as a sub-category of complete separated linearly topologised abelian groups. The latter is an abelian category, but also the former, by way of: Fact 2.3. Let M 0 be a closed sub-module of a complete A-module M. Then M 0 is also complete.

Proof. Again if V ⊂ M is open, then for some λ, IλM ⊂ V , so we have,

0 0 0 M ∩ V ⊃ M ∩ IλM ⊃ IλM . Whence by just taking closures we have that images, quotients, kernels, etc., exist. In particular a short exact sequence

0 → M 0 → M → M 00 → 0 means that M 0 is a closed sub-module of M with the induced linear topology, that is, we just declare the Artin-Rees lemma to be true. Evidently we can sheaﬁfy this, that is: FORMAL FORMAL SCHEMES 5

Definition 2.4. A sheaf F of complete OX-modules of an admissibly ringed space comprises an OX-module F in a linear topology given by a set of sub-modules {Fσ : σ ∈ Σ} closed under ﬁnite intersections such that for any σ ∈ Σ there is a λ ∈ Λ with Fσ ⊃ IλF and,

F = lim F/F . ←− σ σ Observe the unambiguity of the term on the right. Indeed for U open we have the pre-sheaf,

F(U) U 7−→ Fσ(U) together with a commutative diagram,

F(U) ↓ & F(U) lim ,→ lim(F/Fσ)(U). ←− Fσ (U) ←− σ σ Consequently the bottom arrow is an isomorphism, and F(U) is a complete OX(U)-module. Manifestly we should compute stalks within the category of complete separable linearly topologised abelian groups, and to this end we note: Observation/Definition 2.5. ( ) F := lim lim(F/F )(U) = lim F(U) bx ←− −→ σ −→c σ x∈U x∈U where the latter limit over U 3 x open, is the ﬁne directed limit of the F(U) in the category of complete separable linearly topologised abelian groups. Better still, Fbx is a complete ObX,x module.

Proof. Certainly we have continuous compatible maps F(U) → Fbx as U varies, and of course Fbx is in the said category. Now if it’s not the limit then there is a group F to which all the F(U) map in a compatible way, but Fbx does not. In particular we have a map of the usual stalk, Fx → F , with each F(U) → F continuous in the ﬁnest possible topology so the images of the Fσ,x are open in F , and Fbx maps to F. The next observation shows that ﬁne directed limits encapsulate exactness in the category of complete separated linearly topologised abelian groups, that is: Claim 2.6. Let 0 → F0 → F → F00 → 0 be a sequence of sheaves of complete OX-modules in an admissibly ringed space then,

0 00 0 → Fbx → Fbx → Fbx → 0 is exact ∀x ∈ X, iﬀ

0 00 0 → F → F → F → 0 0 00 is exact where F is the categorical image of F0, that is, its closure, and F the categorical quotient of the same. 6 MICHAEL MCQUILLAN

Proof. Let’s start with the if direction, with {Fλ : λ ∈ Λ} the topology on F. 0 For the if direction we may simply suppose F0 = F , and we have an exact square,

0 0 0 ↓ ↓ ↓ 0 00 0 → F ∩ Fλ → Fλ → Fλ → 0 ↓ ↓ ↓ 0 → F0 → F → F00 → 0 ↓ ↓ ↓ 0 0 00 00 0 → F /F ∩ Fλ → F/Fλ → F /Fλ → 0 ↓ ↓ ↓ 0 0 0 00 where Fλ is the image of Fλ. Consequently we have an exact row,

0 0 00 00 0 → Fx/Fx ∩ Fλ,x → Fx/Fλ,x → Fx/Fλ,x → 0 for any x ∈ X, and since Mittag-Leﬄer holds we get the if direction in the topologised category. Conversely there are a few more things to check. Observe in the ﬁrst place that if F is complete then we always have an injection,

F(U) ,→ Π Fbx x∈U for any open set U. Indeed if f ∈ F(U) goes to zero then ∀x ∈ U, and λ ∈ Λ, f 7→ 0 ∈ Fx/Fλ,x, so ∀x ∈ U, f ∈ Fλ,x for any given λ, from which f ∈ Fλ(U), and whence by completeness f = 0. It thus remains to check the surjectivity aspects of the assertion, where arguing as above we use the exact row

0 0 00 00 0 → Fbx/Fbx ∩ Fbλ,x → Fbx/Fbλ,x → Fbx/Fbλ,x → 0 00 and let f ∈ Fx be such that f 7→ 0 in Fx. From what we have already from the if direction,

Fbx/Fbλ,x = Fx/Fλ,x and similarly for the other terms, so ∀x ∈ U, where U is an open on which f is 0 0 defined and goes to zero, f ∈ Fλ + F . Since this holds for all λ, f ∈ F , while by 00 construction F → F is surjective. Obviously, taking the closure of F0 is a little regrettable, but without some sort of finite generation it seems difficult to relate the topology on F0 to that on F purely from the stalks.

3. Localisation Recall, [EGA] 0.7.7, that given complete A-modules M,N over an admissible ring we have a complete tensor product M⊗b AN, that is, consider all triples, I, U, V of admissible ideals and open sub-modules with IM ⊂ U, and IN ⊂ V , then M⊗b AN is the inverse limit over such triples of

M/U ⊗A/I N/V. FORMAL FORMAL SCHEMES 7

This is the complete A-module corresponding to the completion of the usual tensor product M ⊗A N by the sub-modules generated by U ⊗A N ⊕ M ⊗A V . In the particular case that B,C are continuous A-algebras, B⊗b AC satisﬁes the obvious universal property in the category of admissible rings. Furthermore for S ⊂ A multiplicatively closed we have the complete ring of −1 fractions A{S} (A{S } by the way in the notation of [EGA] 0.7.6) deﬁned as:

A = lim(A/I ) {S} ←− λ (S) λ where (S) denotes the usual localisation. This is the smallest (continuous) admissible A-algebra in which S is invertible. It’s critical to note:

Fact 3.1. A{S} = 0 iff 0 is in the closure of S. That is, being not just nilpotent, but topologically nilpotent is a problem. In any case with these two definitions in mind, we introduce a definition which is not in [EGA], that is: Definition 3.2. For M a complete A-module define its complete localisation at S by:

M{S} := M⊗b AA{S}. Observe that by construction:

Fact 3.3. M 7→ M{S} is an exact functor on complete A-modules. Proof. Let

0 → M 0 → M → M 00 → 0 be a short exact sequence of complete A-modules, with {Mγ : γ ∈ Γ} a basis of 00 open sub-modules of M and Mγ their images in M, then we have an exact square:

0 0 0 ↓ ↓ ↓ 0 00 0 → Mγ ∩ M → Mγ → Mγ → 0 ↓ ↓ ↓ 0 → M 0 → M → M 00 → 0 ↓ ↓ ↓ 0 0 00 00 0 → M /Mγ ∩ M → M/Mγ → M /Mγ → 0 ↓ ↓ ↓ 0 0 0 which for any λ, with IλM ⊂ Mγ has the bottom row as an exact sequence of A/Iλ modules. Applying the usual localisation functor to this row gives an exact sequence

0 0 00 00 0 → M /Mγ ∩ M ⊗A/Iλ A(S) → M/Mγ ⊗A/Iλ A(S) → M /Mγ ⊗A/Iλ A(S) → 0 0 Observing once more that by deﬁnition {Mγ ∩M : γ ∈ Γ} is a basis of open sub- modules and that the Mittag-Leﬄer condition holds yields the result after taking limits in λ and γ. 8 MICHAEL MCQUILLAN

As such we have a good localisation functor, so let’s pause for some examples, which put together some of our previous deﬁnitions. Examples 3.4. (1) Restricted Formal Power Series. Considering Z and Z[T1,...,Tn] as admissible rings in the discrete topology, then

A{T ,...,T } = A⊗ [T ,...,T ] = limA/I [T ,...,T ]. 1 n bZZ 1 n ←− λ 1 n λ (2) Localisation again. For f ∈ A, put A = A ; then A = lim∧A , where {f} {f n:n∈N} {S} −→ {f} f∈S the directed limit is understood ﬁnely. More generally, for M a complete A-module, M{S} is the ﬁne direct limit of the M{f}, f ∈ S, in the category of complete separated linearly topologised abelian groups.

4. Formal Schemes Here we do nothing but recapitulate [EGA] I.10.1 for the convenience of the reader. Bear in mind that we have changed the definition of admissible ring, and introduced a definition only implicit in op. cit., that is, that of admissably ringed space, so some minor changes result. As ever, A is an admissible ring, and {Iλ : λ ∈ Λ} the ideals of definition; we begin with: Observation 4.1. There is a one-to-one correspondence between open prime ideals of A and prime ideals of A/Iλ for any λ ∈ Λ.

Proof. A prime of A/Iλ for some λ manifestly corresponds to an open prime. 0 Conversely if p / A of A is open, then p ⊃ Iλ for some λ. However if f ∈ A is n n topologically nilpotent, f → 0, so f ∈ p for n 0, n ∈ N. Deﬁne therefore: Definition 4.2. Spf (A), the formal spectrum of A, is the admissably locally ringed space (X, OX) corresponding to the data:

(i): X is the topological space Spec (A/Iλ) for any λ. (ii): O is the sheaf of rings lim(A/I] ), where ∼ is the usual sheafication X ←− λ λ functor on affines. (iii): OX is topologised by Iλ, λ ∈ Λ, where, I = lim(I^/I ) λ ←− λ µ µ≥λ with the partial ordering on Λ being that induced by set inclusion in A. For any f ∈ A, we have the open set D(f) of open primes not containing f, and despite the minor changes in definition the obvious affinisation lemma still holds, that is:

Fact 4.3. The admissibly locally ringed space (D(f), OX |D(f)) is isomorphic to Spf A{f}. Proof. By op. cit. 10.1.4, we only need to check our condition (iii), which is clear by the usual ring ←→ aﬃne scheme correspondence. FORMAL FORMAL SCHEMES 9

We have the obvious notion of morphisms between admissible locally ringed spaces, that is, ϕ :(X, OX) → (Y, OY) is a continuous map ϕ : X → Y together # ˆ ˆ with a continuous map, ϕ : OY → ϕ∗OX such thatϕ ˆ : OY,ϕ(x) → OX,x is a map of local rings on ﬁne stalks. Since EGA works with morphisms of locally ringed spaces it’s worth checking: Fact 4.4. A 7→ Spf (A) is a fully faithful functor from admissable rings to admissable locally ringed spaces. Proof. Obviously the root of the proposition is to check that ϕ : Spf (A) → Spf (B) comes from a map of rings. Certainly we have a continuous map

# Φ = Γ(ϕ ) : Γ(Y, OY) = B → Γ(X, OX) = A where (X,OX) = Spf (A), (Y, OY) = Spf (B), cf. op. cit. 10.1.2 & 10.2.3. Indeed the only thing that the proof in op. cit. doesn’t yield is that Spf (Φ), with the obvious deﬁnition of that notion, is the map ϕ : X → Y on spaces, since our notion of local is weaker. To remedy this requires: Sub-fact 4.4. bis. Let p be an open prime ideal in an admissible ring A, and A the formal localisation at p. Denote by m the maximal ideal lim(p/I ) . {p} {p} ←− λ (p) λ Then

p = A ∩ m{p}.

Sub-proof 4.5. Let f ∈ A ∩ m{p}, then for every λ, f ∈ p + Iλ, that is, f ∈ p, the closure of p. However, p is open, so it’s also closed. Using the symbol ∩ to denote equally inverse image by Φ when oppropriate we therefore have,

ϕ(p) = m{ϕ(p)} ∩ B = m{p} ∩ B{ϕ(p)} ∩ B = p ∩ B that is, Spf (ϕ) = Φ on spaces, and the rest of the proof proceeds as in [EGA].

5. Quasi-coherent modules Let M be a complete A-module over an admissible ring. By deﬁnition M is complete in a linear topology {Mγ : γ ∈ Γ}, such that for every γ there is a λ with Mγ ⊃ IλM. We can therefore consider the system

Σ = {(λ, γ) ∈ Λ × Γ: Mγ ⊃ IλM} together with the corresponding A/Iλ-modules M/Mγ , and sheaves of the same

M/M^γ over X the topological space for Spf (A), then take limits over (λ, γ) in the 4 said set to define a sheaf of OX-modules, M . A situation of such generality is not discussed in [EGA], so let’s check what’s involved step by step. Checklist 5.1. 4 (a) M 7→ M is a function from complete A-modules to complete OX-modules. As per the definition of the latter notion we have a family of sheaves of sub- 4 modules Mγ closed under finite intersections, given the same for the family of 4 sub-modules Mγ . Over an affine D(f) the sections of M are 10 MICHAEL MCQUILLAN

lim (M/M ) ⊗ (A/I ) ←− γ A λ (f) (λ,γ)∈P which is M{f} by deﬁnition. Equally, for any Mγ the sections in the induced topology are Mγ,{f}, and whether M{f} or Mγ,{f} is just the completion of the usual localisation ?(f) in the M-linear topology, so by the exactness of localisation we have a short exact sequence,

0 → Mγ,{f} → M{f} → (M/Mγ )(f) → 0 from which

M M lim {f} = lim f = M ←− ←− {f} γ Mγ,{f} γ Mγ,f M 4 4 4 so the naive pre-sheaf, lim 4 is a sheaf, and equal to M , that is, M is complete, ←− Mγ γ and functoriality equally follows from the above discussion. (b) Better still, M 7→ M 4 is fully faithful. ◦ ◦ Let M, with topology {M ω : ω ∈ Ω} be another complete A-module and consider the map

◦ ◦ Hom(M, M) → Hom(M 4, M4) In the ﬁrst place this is injective. Indeed by the previous discussion, Γ(X,M 4) = M, and the induced map arises via commutative diagrams with exact columns:

0 0 ↓ ↓ ◦ Mγ → M ω ↓ ↓ ◦ M → M ↓ ↓ ◦ ◦ M/Mγ → M/M ω ↓ ↓ 0 0 ◦ as ω varies over all of Ω, so if the induced map from M 4 to M4 is zero, it’s zero on global sections, and since this map is the map we started with, we must have injectivity. ◦ ◦ For surjectivity, given α : M 4 → M4 we need only check that Γ(α): M → M induces α. Now Γ(α)4 is a map, and α − Γ(α)4 is zero on global sections, so we’re reduced to checking Γ(α) = 0 =⇒ α = 0, which is clear. At this point one realises that quasi-coherence in the sense of [EGA] 0.5.1.2 is either a regrettable subterfuge or work of genius. Speciﬁcally if a complete A- module M were to be quasi-coherent, then it should have a presentation in the category of complete A-modules as

AJ → AI → M → O FORMAL FORMAL SCHEMES 11 for some index sets I and J. However this is a very strong condition since M would have to enjoy the A-linear topology, and the same for the kernel of AI → M. Whence we propose:

Intermission/Definition 5.2. A sheaf F of complete OX modules is quasi- coherent iﬀ ∀x ∈ X there is an aﬃne neighbourhood D(f) 3 x and an A{f}-module M such that,

e 4 F |D(f) →M . Note that the quasi-coherence condition in the case of affine schemes is purely employed via the full fidelity of the usual sheafication functor to guarantee this. If moreover every point has a neighbourhood U such that F enjoys a presentation

⊕I ⊕J OX → OX → F → 0 for some index sets I,J in the category of linearly topologised abelian groups, then we say F is almost-coherent. As we have said, one should note that almost-coherence is very strong. Apart from the discrete topology or the adelic-Noetherian-ﬁnitely generated case, it’s diﬃcult to imagine a substantial class of examples. It also seems highly improbable that almost coherent sheaves form an abelian category. On the other hand it would be great if they did, since they’re manifestly much closer to the adelic-Noetherian situation than anything else that could be imagined. Intermission over, it’s back to the last point on the checklist, viz: (c) M 7→ M 4 is an equivalence of categories between complete A-modules and quasi-coherent complete OX-modules.

Proof. Let F be a quasi-coherent complete OX-module with {Fγ : γ ∈ Γ} the open sub-modules. Now X is quasi-compact so we can cover it by ﬁnitely many aﬃnes D(fi), 1 ≤ i ≤ n, such that on each we have an isomorphism

4 e Mi →F |D(fi) for Mi a complete A{fi} module. Necessarily this also encodes the data of open sub-modules M with, M 4 being the restriction of F , for the same index set Γ. γi γi γ

Equally for any γ we have a λ such that IλF ⊂ Fγ , so F/Fγ is an A/I]λ-module on Spec (A/Iλ). Better still, it’s quasi-coherent in the usual sense, so there is a unique A/Iλ-module Qγ such that

e Mi/Mγi →Qγ,fi as (A/Iλ)fi -modules. Appealing again to the usual aﬃne case, we have for µ ≥ γ maps Q → Q which we can put together to form a complete A-module Q = limQ µ γ ←− γ γ with open sub-modules the kernels Kγ of the projections Q → Qγ . In addition we e 4 have local isomorphisms, Mi→Q{fi}, so Q is indeed F. It necessarily follows from the checklist that we have: 12 MICHAEL MCQUILLAN

Claim 5.3. Quasi-coherent sheaves of complete OX-modules are an abelian category, indeed a sub-category of sheaves of complete separated linearly topologised abelian groups. For the said category we have co-homology functors Hi extending the global sections functor Γ = H0, and

Hi(X, F) = 0, i ≥ 1 for all quasi-coherent sheaves of complete OX-modules. References [BM] Bogomolov, F.A., McQuillan, M.L., “Rational curves on foliated varieties” IHES preprint Feb. 2001. [EGA] Grothendieck, A., with the collaboration of DieudonnéJ.-A., “Eléments de Géométrie Algébrique”, Publ. Math. I.H.E.S. 4. [H] Hartshorne, R. “Ample Subvarieties of Algebraic Varieties”, Springer-Verlag.