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 “El´ements de G´eom´etrie Alg´ebrique”. 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 gen- eral 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 lim- its, 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.