Version current as of Sat. June 12. Recent additions are marked with "+++". These notes are a companion to EGA Chapter 0, which may be useful. At this point, I've only gone in detail to near the end of 6 (flatness), and I've only had a chance to type in comments up to the end of 5. I'll do 6 and 7 (Adic rings) in a week and a half, when I get back from St. John's. First of all, I have to appoligize for the quality of the photocopying of the first part of EGA0 (although I didn't do it myself). Some parts were illegible to me. In what follows, I'll try to remember to fill in illegible bits. (This is a fifth generation photocopy: original --> Tom Hagedorn --> Henry Shahrouz --> me --> Johan --> you.) We'll start with a glossary of french terms, and then some preliminary reminders on inductive (=direct) and projective (=indirect) limits, and presheaves and sheaves. Throughout, we'll try to remember which functors are (left/right)-exact, etc. It should go without saying that many of the insights here were explained by others (most often Brian Conrad, Johan de Jong, Benji, Kalle, or Andreas). Some references are to (T), Grothendieck's article in the Tohoku journal of math, called "Sur quelques points d'algebre homologique". I've copied it in case it's useful. *** (Mostly french) Glossary: "unitary" module - means 1 acts as the identity (1.0.1). partie - subset (1.2.1) +++ probleme d'application universelle - universal mapping problem (1.2.4) application ... dans - maps ... to (1.4.1) en effet - +++ Benji offers what seems to me to be the best translation: "Indeed". Brian translates this as "in fact", but I think it might be better to loosely think of it as "Proof:" or "Here's why." monogenic - generated by 1 thing (1.7.4) rare - nowhere dense (2.1.1) adherence - closure (2.1.8) faisceau - sheaf (3.1.2) sousjacent - underlying (3.1.4) parcourt - runs through (3.4.2) fibre - stalk (3.4.4 and earlier), not fiber! -- CAUTION! s'annuler - vanishes (5.5.1) unique a un isomorphisme pres - unique up to isomorphism (3.1.1), (3.5.3), (4.3.4) (_not_ necessarily up to _unique_ isomorphism) +++ revenir au meme - It comes to the same thing as (Matthew) (e.g. 3.1.2) +++ se borner au case ou - reduce to the case where (I.1.1.13) +++ integre - domain (I.1.1.13) +++ or - now +++ complements - further information *** Preliminary reminders: limits +++* Definitions in Set Theory (from Bourbaki's "Theorie des Ensembles", E.R.25.) Side fact: For the French, 0 is a positive number -- but it isn't "strictly positive". Transitivity: x <= y, y<= z implies x <= z. Reflexivity: x <= x. - An _order relation_ on a set is a relation that is i) transitive, and ii) satisfies x<=y, y<= x iff x=y. ii) implies reflexivity. (If furthermore for each (x,y) either x<=y or y<=x, the set is _totally ordered_.) An ordered set is _inductive_ if every totally ordered subset is bounded above by some element of the set. Zorn's lemma (requiring AC of course) can be stated as: every inductive (ordered) set has at least one maximal element. - A set is pre-ordered if it has a relation that has transitivity and reflexivity. You then have an equivalence relation: y ~ x if y is both <= and => x. If you quotient by this equivalence relation then you get an ordered set. If E is a pre-ordered set, then a subset X of E is _cofinal_ with (to?) E if each element of E is dominated by some element in X. Example: a base for a topology is cofinal for the open sets.) _Coinitial_ is the dual condition. A preordered set E is _filtered_ (filtrant) on the left if every non-empty finite subset of E is bounded above by an element of E. Similarly for filtered on the right. I find it tricky to remember what left and right means... A map from a preordered set A to a preordered set F is increasing (croissante) if x <= y imples f(x) <= f(y). Decreasing (decroissante) is defined analogously; also strictly increasing and strictly decreasing. - Less important definitions: open and closed intervals in an ordered set. An ordered set is well-ordered if each nonempty subset has a smallest element; every set can be well-ordered (Zermelo's theorem, a consequence of the axiom of choice). +++- Is there a word for when a pre-ordered satisfies: for any pair (x,y), there is a z that is >= both? * Inductive limit (EGA), also known as direct(ed) limit (e.g. Atiyah-Macdonald) Suppose you have a partially ordered index set I (which means you have some relations x x K, say, with the composition M(x) --> M(y) --> K the same as M(x)--> K where x M commuting with the M(x)--> M(y), and if N is an object of K, and M(x) --> K (x runs through I) are morphisms commuting with M(x) --> M(y), then there is a unique morphism M --> K that all of the M(x) --> K factor through. That's abstract nonsense, so here are concrete applications. i) If I is any index set, with no relations (i.e. "x M(x) factor through N --> M --> M(x). i) If I is any index set with no relations, you get the direct product. ii) Aside: Depending on your choice of universal properties, there are two a canonical map from the direct sum to the direct product; they are the same map. iii) The canonical example of the p-adics: the system is given by Z/p <-- Z/p^2 <-- ... Note in this case that a general p-adic isn't "induced" from any "finite level". We need information at "all levels at once" that is "coherent". Again, Atiyah-Macondald works through this in a special case in Ch. 2. iv) Projective limits are left-exact. There's an example showing that they aren't in general exact, from AM p. 113; can anyone think of an easy one? v) If however the system if a surjective system (I is totally ordered and where all the morphisms are surjective, e.g. Z_p) then the limit is indeed exact. +++ Benji expands on this: If I remember correctly, a weaker condition suffices: the system should satisfy the Mittag-Leffler condition. I think this means that for fixed m and n getting larger, the image of M_n in M_m is eventually stable. You can make this definition for a directed system (in AM terminology). Also, if 0 --> M' --> M --> M" --> 0 is the system (subscripts supressed) then only M' need satisfy any condition. In other words, Mittag-Leffler implies that the first derived functor of (proj lim) is zero on M' . * Note that tensor products are right-exact. When you tensor two sheaves together, you tensor on open sets to get a presheaf, and then sheafify. Example showing you need to sheafify: On P^1, O(-1) has no global section, but O(2) \otimes O(-1) has global sections. Hom is left-exact on _both_ sides. I always get confused what left-exact means for contravariant functors. It means that right-exact sequences turn into left-exact sequences. * Presheaves and sheaves. Fix a topological space X. A presheaf on X is just a contravariant functor (from the category on open subsets of X, where morphisms are inclusions, to some other category). For convenience, I'll assume that we are dealing with sheaves of sets (or sets with some additional structure, e.g. rings etc.). A sheaf has two extra properties, which I'll call "identity" and "gluability" to set notation. Identity: If some section over some open subset U is 0 when restricted to each element of an open cover of U, then it is 0 on U. Gluability: If there are sections on each element an open cover of U, who agree on overlaps, then they glue together. To each presheaf P you can associate a sheaf S with a universal property: each presheaf morphism P--> F (F a _sheaf_) factors through P-->S-->F uniquely. The stalks of P are the same as the stalks of S. The presheaf morphism P-->S has a kernel and a cokernel. In some sense, the kernel corresponds to failure of "identity" of P. If P injects into S, one says P is a _separated_ presheaf. (Perhaps I should replace "identity" by "separatedness".) The cokernel corresponds to the failure of "gluability" of P. There are various ways of constructing S from P. One way involves a descent construction, and you do it twice. It turns a presheaf into a separated presheaf, and a separated presheaf into a fully fledged sheaf. I should find a reference for that and put details in. I think I like this better: it gives a bit more insight, or at least different insight. You can also do it all at once (see or Harshorne Prop-Def II.1.2) by gluing together the stalks. This basically corresponds to the second iteration of the above method: the stalks don't see failure of identity, and the construction takes care of gluability. (That was admittedly vague.) A third way, essentially the same, is the "espace etale" method mentioned in EGA (and Hartshorne Ex. II.1.13). You can check exactness of morphisms of sheaves at the level of stalks. *** Examples to keep on hand There are likely some canonical counterexamples to keep on hand. Here are a few possible starters. - non-finitely presented modules. The module could be non-finitely generated (e.g. A^{\infty}, where A is some nice ring), or non-finitely presented (e.g. M=A/I where A not noetherian, I a non-finitely generated ideal). +++- Example of a "big" non-reduced scheme, where in order to "reduce" it, sheafification is required. To be precise: the presheaf of nilpotent elements of the structure sheaf isn't always a sheaf. For example, consider the union of Spec k[t]/t^n as n varies through the positive integers - a non-quasi-coherent sheaf of ideals on A^2 (pointed out by Kamal Khuri-Makdisi). Let I be the ideal sheaf defined as follows: if U is any open set not containing the origin, Gamma(U,I) consists of regular functions divisible by y (i.e. vanishing on the x-axis). If U contains the origin, I is the ideal 0. Clearly this is a sheaf of ideals. Just as clearly, it isn't quasicoherent (e.g. it doesn't define a closed subscheme). *** Chapter 0: Preliminaries 1. Rings of fractions. 1.0 Rings and algebras Basics here. (1.0.5) Module of finite type (f.t.) = finitely generated. Finite presentation (hereafter f.pr.; warning f.p. often means fidelement plat = faithfully flat) means M is cokernel of some A^q --> A^p where q and p are both finite. Note that if A is Noetherian, every A-module of f.t. adminits a finite presentation. (1.0.6) Brian reminds us not to concern ourselves with Zorn: "Since Zorn's Lemma is needed to ensure the viability of the theory of injective resolutions, without which the general theory of sheaf cohomology would not exist, it seems pointless to worry about using Zorn's Lemma." +++ Benji: "My reason for always assuming Zorn's Lemma is that I like to believe that every non-zero ring has a maximal ideal." 1.1 Radical of an ideal. Nilradical and radical of a ring. (1.1.1) The radical of an ideal is the intersection of the minimal primes containing it. If A is noetherian, there are a finite number of minimal primes containing it. Brian's counterexample if A is not Noetherian: product of infinitely many copies of a field. 1.2 Modules and ring of fractions. 1.3 Functorial properties. (1.3.2) localization is exact. (1.3.3) localization commutes with inductive limits (1.3.5) Say A is a ring, with multiplicative subset S, and M and N and modules over A. Then there's a homomorphism, functorial in M and N, S^{-1} Hom_A(M,N) --> Hom_{S^{-1} A} ( S^{-1} M , S^{-1} N). If M is finitely presented (i.e. there are integers m and n such that there is an exact sequence A^m --> A^n --> M --> 0), then this is an isomorphism. In short, if we want Hom to behave well with respect to localization, we may need f.pr. of the "source". Proof: If M = A^n, this is true: Hom_A(M,N) is canonically N^n, etc. etc. Then apply Hom(.,N) (a left-exact functor) to the exact sequence and localize to get: 0 --> S^{-1} Hom(M,N) --> S^{-1} N^n --> S^{-1} N^m Instead, you could localize and apply Hom, to get something similar (looking like the right side of the proposition). There are morphisms from the first exact sequence to the second, and by the five lemma we get what we want. Here's why finite presentation is necessary. Let A=N= the integers, S the powers of 2, and let M be a countable direct sum of copies of Z. Then the left side is S^{-1} of a countable _product_ of Z's; elements of this have bounded denominator. The right side is a countable _product_ of S^{-1} Z's, which includes terms with unbounded denominators. Precisely: the term (1, 1/2, 1/4, 1/8, ...) is in the right side but not the left. 1.4 Change of multiplicative subset (1.4.3) saturated multiplicative set S: can't make it any bigger, i.e. if a divides any element of S, then a is in S 1.5 Change of ring (i.e. Base change). (1.5.7) Commutativity of base change: If you do two base changes in a row, it's the same as doing one big one. (1.5.8) I'm not sure of the point of this: A is a subring of B. For every minimal prime p of A, there is a minimal prime q of B with p = A \cap q. (I didn't go through the argument.) Why do we care? If we have Spec B --> Spec A in this case, the minimal primes of A correspond to the minimal primes of B. 1.6 Identification of a module M_f with an inductive limit. 1.7 Support of a module (1.7.1) "est A tout entier" -- is all of A. Supp(M) is closed if M is finitely generated (AM Ex. 19), but isn't in general. Example? (1.7.4) If M isn't a f.g. A-module, then Supp(M) isn't necessarily the +++ I've forgotten what would go here! (1.7.5) If M is a finitely generated A-module ("finite type" in EGA's language), Supp(M) (the set of primes p such that M_p is non-zero) satisfies Supp(M \otimes_A N) = Supp(M) \cap Supp(N). Because localization commutes with tensor products, we always have an inclusion of the left into the right. But equality doesn't hold in general. Here's Brian's counterexample if M isn't finitely generated. Let (A,m) be an arbitrary local ring, N = A/m = residue field, M a non-zero module. Then Supp(M) \cap Supp(N) = {m}. It suffices to find such an M with M (x) N = M/m vanishing. This is certainly not possible if A is artinian, but if m is not a minimal prime (i.e., dimA > 0), simply let M = A_p for a minimal prime p of A. 2. Irreducible and Noetherian spaces. 2.1 Irreducible spaces (2.1.2) So far, there can be more than one generic point for each closed subset. (2.1.3) Kolmogoroff space = for any two points, there's an open containing one but not the other. Johan called this "sober" in his course. Hartshorne Ex. II.3.17: Kol space +Noeth space = Zariski space. Quasicompact = subcover property. If X is quasicompact, each nonempty closed subset contains a minimal non-empty closed subset. 2.2 Noetherian spaces (2.2.1) This probably doesn't matter, but I wanted to make explicit what I think the "maximal condition" is: Given any subsets indexed by some totally ordered set (of _any_ cardinality) stabilizes. Perhaps by Zorn this is the same as just countable totally ordered set. It's been a long time since I've thought about Zorn-type things. (2.2.2) Noetherian induction. If anyone feels like filling in details of (2.2.4) or (2.2.5), let me know. 3. Inverse on sheaves. 3.1 Sheaves with values in a category. (3.1.2) Axiom F for faisceau = sheaf (3.1.6) I'd expressed pleasure that EGA said it would abstain from the point of view "etale spaces of a presheaf" in which sheaves are constructed. (See, for example, Hartshorne exercise II.1.13.) Matthew makes the good point that the purpose of this construction is that it represents a sheaf by an object in the category of topological spaces. Thus it is the first (and easiest) "representability theorem" (and thus is interesting, despite being much maligned). Furthermore, it seems to me that (3.8) is essentially this! Am I being an idiot? 3.2 Presheaves on a basis. Grothendieck defines a presheaf on a given base of opens. He then gives a criterion F_0 (3.2.2) for this to be the "value" of an actual sheaf on this base. In the rest of this section, when he says (pre)sheaf, he means (pre)sheaf on this base of opens. When he says (pre)sheaf in the "ordinary sense", he means the usual (pre)sheaf. In order to turn a presheaf into a sheaf requires projective limits. I think the reason is that the definition involves a product over an arbitrary index set. +++ Kalle points out: I think there are two steps for getting from a presheaf defined on a basis (i.e. a basis of the topology, not "base" as in "base change") to a sheaf: 1. extend it to an ordinary presheaf 2. sheafify it +++ Projective limits are needed in the first step: to define the sections on an open set, cover it with basis elements and take the projectiver limit over these. This does not yet sheafify it. To get a sheaf, we would have to consider finer and finer covers and take the inductive limit over these refinements. This should give the same result as the usual way of considering infinite products of stalks and sections that locally come from a presheaf section. (3.2.4) If K admits inductive limits (so you can even talk about stalks), and you have a morphism of two "sheaves on a basis", then the induced morphism of stalks can be computed in two ways: either by just working on the basis, or by "finding the actual sheaf" and getting the morphism of stalks there. (3.2.6) The projective limit of sheaves is a sheaf -- no sheafification is required! In other words, if you take projective limits of sections over each open set, you already get a sheaf. +++ Benji reminds me that this is Hartshorne Ex. II.1.12. Kalle explains: This is very natural. The sheaf axiom is given in terms of Hom into the set (or group, ring, etc) of sections. Now a Hom into a projective limit is the same as a compatible set of Hom's into each term, so the sequence of Hom's into projective limit is exact if the sequence for each term is exact. We're then reminded that projective limits are left-exact. 3.3 Glueing sheaves. If you have a cover of X, and sheaves on each element of the cover that glue, then you get a sheaf. Not surprising or hard. You need projective limits in your category to do this; presumably because you build the sheaf up by taking a product. So if X is quasicompact, you don't even need projective limits. (Correct me if you think I'm wrong!) 3.4 Direct images of presheaves. (3.4.2) pushforward is a covariant functor (3.4.3) pushforward of a pushforward is a pushforward (3.4.4) stalk of the pushforward sheaf maps to each of the stalks in the fiber 3.5 Inverse images of presheaves. This is an important section! In reading this, and later parts on inverse images, I realized that I'd had some misperceptions about presheaves. Grothendieck's set up here is very nice; it takes care of several things at once, by doing things in generality. Notice that he doesn't introduce ringed spaces until he has to; he proves/reviews lots of things about sheaves first. Similarly, once he introduces ringed spaces, he doesn't introduce schemes until he has to. This is definitely a good section to read through in detail. Here's the set-up. We have two topological spaces psi: X --> Y with a _presheaf_ G on Y. We'll discuss a _sheaf_ on X that will be the inverse image sheaf. If X --> Y is an isomorphism, this will be sheafification. EGA calls it psi^* G, but I'll call it psi^{-1} G, following Hartshorne (and the Springer version of EGA I), to avoid confusion with a later psi^* G. The inverse image sheaf is a sheaf psi^{-1} G on X along with a (presheaf) morphism rho in Hom_Y(G, psi_* psi^{-1}G). (In general, Grothendieck uses the term Hom_psi(G, F) where F is a sheaf on X to mean Hom_Y(G,psi_* F). We'll see that this is the same as Hom_X(psi^{-1}G, F), so Grothendieck's terminology is suggestive: we can chose to view this morphism downstairs on Y, or upstairs on X.) G' (with this rho) is called an inverse image of G if for all _sheaf_ on X F, the natural morphism Hom_X(G',F) --> Hom_Y(G,psi_* F) is a bijection. (This morphism is obtained as follows: given a map G' --> F on X, we can push it forward to get a map psi_* G' --> psi_* F. Compose this with rho.) The owner of an earlier version of this copy of EGA asks why this is unique, and this isn't clear to me either, although I haven't thought about it much. Suppose there were an H with sigma in Hom_Y(G,psi_* H) with a similar property. Then we have two maps rho: G --> psi_* G' sigma: G --> psi_* H The universal property indicates that there is a unique alpha mapping G' --> H inducing a map from psi_* G' --> psi_* H that completes a commutative triangle, and a unique beta going the other way (with an analogous property). But why should these "compose" to give the identity? +++Andreas fills in the missing link: (Note: this is easy to understand once you draw the figures he describes.) The maps rho, sigma, alpha, beta have the property that psi_* alpha \circ rho = sigma psi_* beta \circ sigma = rho (that's how alpha and beta are constructed using the universal property). Hence we have psi_* (alpha \circ beta) \circ sigma = psi_* alpha \circ psi_* beta \circ sigma = psi_* alpha \circ rho = sigma, but on the other hand psi_* id \circ sigma = sigma, so by the universal property again, alpha \circ beta must be the identity. (In short, this is - just as Grothendieck says - nothing but the "usual trick" to show uniqueness for objects defined by a universal property.) Assuming that we have a unique preimage sheaf, then maps from psi^{-1} G --> F correspond to maps from G to psi_* F, an adjointness property. If we have a map above u, call the map below "u^{flat}". If we have a map below, call the map above "u^{sharp}". (3.5.4) psi^{-1} is a covariant functor. (We'll see later that in nice circumstances such as any we're concerned with, it's comprehensible, and exact.) 3.6 Simple sheaves and locally simple sheaves. It's unclear to me what this section is doing here. It should be after 3.7 I think. Nothing exciting here: simple sheaves are what I think of as "locally constant sheaves" (often called, by tragic abuse of terminology, "constant sheaves") -- the sheafification of the constant presheaf. +++ But clearly there are locally simple sheaves that are not simple: two points with discrete topology, sections over the two points are two different groups. For an example with a different flavour: you can have monodromy too. 3.7 Inverse images of presheaves of groups or rings. Now we return to the subject of 3.5, where we have sheaves of sets (or sets with additional structure, e.g. rings). Here's a construction of the inverse image sheaf. First get a presheaf: for each open set U in X, take the inductive (=direct) limit of values on open sets containing psi(U). (This should remind you of the stalk of a point.) In particular, each section of X can be thought of as a section over an open set containing psi(X). Then sheafify. Sheafification is necessary, as Johan pointed out to me: let Y be one point, and G any presheaf on Y. Then this construction gives you first the constant presheaf on X, which you then have to sheafify (if X is 2 points for example). (3.7.1) has another construction that I don't like so much. Essentially it avoids the sheafification step by doing it all at once. Just so you can read it, here is the second sentence. En effet, pour tout ouvert U subset X, definnissons G^{-1}(U) comme suit: une element s' de G^{-1}(U) est une famille (s'_x)_{x \in V} ou s'_x \in G_{\psi(x)} pour tout x \in U, et ou, pour tout x \in U, la condition suivante est remplie: il existe un voisinage ouvert V de psi(x) dans Y, un voisinage W \subset \psi^{-1}(V) cap U de x et un element s \in G(V) tels que s'_z = s_{\psi(z)} pour tout z \in W. On verifie immediatement que U --> G^{-1}(U) satisfait bient aux axiomes des faisceaux. With this other construction, though, you get some things quickly. For example, the stalk at a point x \in X is canonically isomorphic to the stalk at a point psi(x) \in X. (If you think about it the right way, you essentially get that they are the direct limits of the same things.) Based on the results of 3.5, you'd only know that there was a map from one stalk to the rest, but I can't remember offhand which way it would go. Because of this stalk issue, and because the stalk of a presheaf is the stalk of the associated sheaf, psi^{-1} is an exact functor from the category of sheaves on Y to the category of sheaves on X. +++ 3.8 Sheaves of pseudo-discrete spaces. This is important for formal schemes (I.10). Suppose you have a space that is locally quasicompact (e.g. schemes), i.e. every point has an open quasicompact neighbourhood. Suppose you have a sheaf of sets on this space. (It's easy to add more structure, e.g. rings, later.) Here's a simple example to follow through: the simple (i.e. locally trivial) sheaf on countably many points (with the discrete topology), where sections over a point are the two-element set {A,B}. Then if we endow the sections with the discrete topology, we don't necessarily have a sheaf of topological spaces! (We do if all opens are quasicompact.) In the example above, the problem is that we don't have an isomorphism between global sections over the whole space and the _product_ (in the category of topological spaces) over all the points. (Essentially: the product of the discrete topology _isn't_ the discrete topology.) There's a unique way to topologize the sections so that the sections over any quasicompact have the discrete topology. 4. Ringed spaces. Grothendieck's done what he could with plain sheaves, and now he adds the next layer of complexity. For the most part, things come for free (as you've already seen them in Hartshorne). Life gets more interesting when talking about inverse images. 4.1 Ringed spaces, A-modules, A-algebras. (4.1.1) We start of with ringed spaces (X,A) where A is the structure sheaf, and X is the underlying topological space. We have germs as before. A morphism of ringed spaces (X,A) --> (Y,B) is as you'd expect, except he describes the morphism theta (which looks like an O in our terrible copy), so theta is a psi-morphism, and corresponds to morphisms theta^sharp: psi^{-1} B --> A (remember sharp meant upstairs) and theta^flat: B --> pi_* A (flat meant downstairs). In the second last paragarph of (4.1.1), G should be B, and F should be A a couple of times. (4.1.2) We get an induced ringed space structure for any subset of X. Presumably this is only of interest when the subset is open. Any other interesting examples? Possibilities: closed subspaces, locally closed subspaces. Although it isn't clear to me what's interesting here. (4.1.3) Obvious definitions of A-algebras. (4.1.4) He has graded rings graded by Z. (4.1.5) Important facts about SheafHom. First of all, if you just do Hom on open sets, you get a presheaf, but it actually turns out to be a sheaf. (This was a Hartshorne exercise. If anyone wants practice and wants to figure out details, I'll put them here.) Also, there's a natural map from the stalk of SheafHom to the hom of the stalks. That's basically clear. It's in general neither injective nor surjective. Wedge products are defined. Given F and n, the presheaf whose sections above U are the nth wedge product of the sections of F is separated (i.e. injects into its sheafification); details? Stalks of wedges are naturally isomorphic to wedges of stalks. (4.1.6) Making sense of ideal sheaves times O_X-modules. (4.1.7) Gluing ringed spaces together; similar to gluing sheaves together, not too exciting. 4.2 Direct image of an A-module. Not that exciting. (4.2.1) is somewhat illegible; the second line starts "(X,A) --> (Y,B)" the third line starts "B --> psi_*(A)" the fourth line starts "psi_*(F)" the fifth line after the equation starts "B-module". Pushforward is left-exact because global sections are left exact of course. (4.2.3) Uses the fact that SheafHom didn't require sheafification. 4.3 Inverse image of a B-module. (4.3.1) Things get more interesting here. psi^{-1} was exact, but we want to turn it into an A-module, so we have to tensor it with O_A (over psi^{-1} O_B). This gives us psi^* as a functor from O_B-modules on Y to O_A-modules on X. It's right exact (because tensor product is). Note that if a morphism of schemes is flat, then pullback is _exact_; this is why. Because tensor commutes with localization, the stalks are what you'd expect: pullback the stalk from downstairs, and tensor with the stalk of the structure sheaf upstairs. (4.3.2) "The inductive limit of stalks of sheaves = stalk of the inductive limit of sheaves." Reason: stalks are themselves direct limits, so this basically uses the fact that inductive limits commute. (Assuming that the inductive system is filtered (implying: for any two elements, there is a third element that dominates them both), an element of both sides can be interpreted as a section of one of the sheaves in the system, in some neighbourhood.) (4.3.3) Tensor commutes with pullbacks (4.3.5) Here we pull back an ideal sheaf I of the structure sheaf B downstairs. Because psi^{-1} is exact, psi^{-1} I is a subsheaf of psi^{-1} B. When we tensor with the structure sheaf A upstairs, we still get a morphism Psi^* I --> Psi^* B, which he said in (4.3.4) equals A. So we get a subsheaf of A, which he calls (Psi^* I) A, or even I A. You can check that (IJ)A = I (JA) = (IA)(JA), although I didn't. (4.3.6) Inverse image of inverse image is inverse image. 4.4 Relation between direct images and inverse images. (4.4.3) We again have a natural bijection Hom_A(Psi ^* G, F) --> Hom_B(G,Psi_* F). Here's the morphism. We have a natural morphism psi^{-1} G --> Psi^* G = psi^{-1} G \otimes A, given by s --> (s \otimes 1). So if we have a morphism from Psi^* G to F, then composing, we get a morphism from psi^{-1} G to F, which (by definition of psi^{-1}) gives a morphism from G to psi_* F. Conversely, if you have a map from G to psi_* F, this corresponds to a map from psi^{-1}G to F. Tensoring with A, we get a map from psi^* G to F (as F is already an A-module). Did I screw up here? F \otimes_{\psi^* B} A shouldn't necessarily be A... +++Benji writes: Yes, I think you screwed up. Once you have psi^{-1} G --> F , you do not actually tensor with A . Just use the sheafified version of Hom_R( M, N ) = Hom_A( A \otimes M, N ) when A is an R-algebra, M is an R-module, and N is an A-module. If u is the map from M to N then a \otimes x maps to a u(x) . We then get canonical homomorphisms G --> Psi_* Psi^* G and Psi^* Psi_* F --> F, similar to in (3.5.3) and (3.5.4) (4.4.5) We see that this picture behaves well with respect to inductive limits of G's. (4.4.8) A pullback of a pullback is a pullback. 5. Quasi-coherent sheaves and coherent sheaves. 5.1 Quasi-coherent sheaves. (5.1.1) Some notation: O_X^{(I)} is a _direct sum_. +++ Benji has a neat example. \Gamma( U, O_X^{(I)} ) is not the same as \Gamma( U, O_X )^{(I)} . For example, let X be the disjoint union of X_i ( i in I ). Let phi_j : O_X --> O_X^{(I)} be the inclusion in the j'th factor. Let s_j = \phi_j(1) \in \Gamma( X_j, O_X^{(I)} ) . Trivially, s_i and s_j agree on the (empty) overlap, so they glue to give s \in \Gamma( X, O_X^{(I)} ) . This is non-zero in every component. I guess that the point is that direct sum is a special case of direct limits, and so the sheaf direct sum is not the same as the presheaf direct sum. +++ Ravi writes: This is a good example of how the direct limit of sheaves, in the category of presheaves, need not be a sheaf! He defines "generated by global sections". He gives an example (!) of of something that isn't generated by global sections in _any_ neighbourhood of a point. In particular: take the classical topology on R, and take the locally constant = simple sheaf Z. Consider the subsheaf that is the same, except that all sections over connected opens including 0 are 0. Then the _stalk_ at 0 is still Z. (5.1.2) If f:X--> Y is a morphism of ringed spaces, and F is an O_X module generated by global sections, then the canonical morphism f^* f_* F --> F of (4.4.3) is surjective. +++ Kalle points out that this is basically immediate. (5.1.3) Quasicoherent O_X-modules are defined! For all points of x, there is a neighbourhood such that the sheaf is the cokernel of some nice homomorphism O_X^{(I)} --> O_X^{(J)} (where I and J are arbitrary index sets). For example, O_X is quasicoherent. (5.1.4) Quasicoherent sheaves pull back. Reason: pullback is a combination of straight pullback (exact) and tensoring with the structure sheaf upstairs (right-exact). 5.2 Sheaves of finite type. (5.2.1) O_X-modules of finite type are defined: for each point x of X, there is a neighbourhood where F is generated by a finite number of sections. Loosely speaking: finite type := "locally finitely generated". For example, O_X is finite type. Tensor products (being right-exact) and quotients of finite type are also finite type. Note that finite type does not imply quasicoherent! As a counterexample, take the example of (5.1.1); here the structure sheaf is taken to be the simple sheaf Z. Then look at the quotient of O_X by the sheaf described there. It's generated by one element, hence of finite type. But one can check that it isn't quasicoherent. I haven't actually thought about this, but it looks easy. (5.2.2) If F is a f.t. O_X-module, and s_i (1 <= i <= n) are some sections of F over U containing a point x, generating F over U, then there is a neighbourhood V in U containing x, such that the stalks of the s_i generated the stalks of F for each point in V. He refers to FAC for this. +++ Benji writes: There are more details in my edition, and it is easy. Use the fact that F is f.t.: shrinking U , if necessary, we can assume that F | U is generated by finitely many sections t_j in F(U) . Since the s_i generate the stalk at x , express each (t_j)_x in terms of the (s_i)_x . By the nature of direct limits, these expressions are valid in some open neighborhood V of x ( V contained in U ). +++ Ravi: It looks to me as though we can take V=U. I didn't read (5.2.3)-(5.2.4) closely, and I will. (5.2.5) finite presentation is defined: for each point x in X, there is a neighbourhood where F is a cokernel of some O_X^p --> O_X^q. He says p and q should be positive; presumably he meant that they should be non-negative just so the definition makes sense, and nothing more subtle. Hence f.pr. ==> f.t. + q.coherent, and f.pr. behaves well with respect to pullback, for the same reasons as before. (5.2.6) If F is a f.pr. O_X-module, then we have even more information about SheafHom: the stalk of SheafHom is naturally isomorphic to the Hom of the stalks. (5.2.7) If two f.pr. O_X-modules have isomorphic stalks (at x), then there is actually a neighbourhood of x where they are isomorphic. 5.3 Coherent sheaves. +++ Brian comments: I should have made clear that the point is that it is "wrong" to think that the *definition* of coherence is as in Hartshorne. Instead, just as one can develop (as in Atiyah-MacDonald) the notion of "noetherian module over a ring" and then define a notherian ring to be a ring which is noetherian as a module over itself (and then go on to prove all kinds of theorems about noetherian modules over *noetherian* rings), the notion of coherence is an intrinsic sheaf-theoretic concept which, in the case of locally notherian schemes, happens to be equivalent to "over an affine is a finite module". That is, for any sheaf of rings A and any A-module F, one defines "coherent A-module", proves basic nonsense about this definition (with no assumptions on A), then define A to be coherent if it is coherent as an A-module, and then prove more interesting theorems about coherent (= "finite") modules over *coherent* sheaves of rings. The main point is then that the structure sheaves on locally noetherian schemes, complex analytic spaces, and locally noetherian formal schemes are ALL coherent sheaves of rings (and in the affine scheme and affine formal scheme cases is *equivalent* to a concept defined in terms of finite modules). Thus, the more abstract notion of coherence as in FAC is a better point of view than the defn as in Hartshorne (which he makes applicable to both schemes and formal schemes, but in a sort of ad hoc manner). (5.3.1) Definition of coherent O_X-module: a) f.t., and b) for _every_ open U every hom u: O_X^n --> F over U, the kernel is of f.t. as well. Notice that this is a local definition. Note that O_X need not be coherent. +++ Easy counterexample from Matthew: Choose your favourite ring A which contains a finitely generated ideal I which is not finitely presented as an A-module, and let X be the point {x} with structure sheaf given by A. Benji recalls that the sheaf of C^infinity functions on a manifold is not coherent. +++ Andreas gives another one: Let X be the real line and O_X be the sheaf of continuous functions. Then, consider the morphism u: O_X -> O_X given by multiplication with the function g(x) which is 0 for x<=0 and x for x>=0. The kernel of u is the subsheaf of O_X of all continuous functions f with f(x)=0 for x>=0. This is not of finite type, it is not locally generated by finitely many sections around 0. He then gives a series of facts about them. Does anyone feel like working out details of some of these? (5.3.2) coh ==> f.pr. The converse isn't nec. true (as O_X is always f.pr.). Every _f.t._ submodule of a coh is also coherent. Finite direct sums of coherents are coherent. (5.3.3) If 0 --> F --> G --> H --> 0 is exact, and two of the three are coherent, then the third is as well.wou Would someone like to do the details? Note that this isn't true in Hartshorne's definition of coherent sheaves on a scheme. (5.3.4) Im, Ker and Coker of morphisms of coherents are also coherent. Hence if F and G are f.t. submodules of a coherent module H (hence F and G coherent), then the intersection and sum of F and G are coherent, using the morphism F \oplus G --> H. (5.3.5) Tensor and sheaf hom of coherent modules are also coherent. +++ Benji says: if one is finitely presented and the other is coherent, the tensor product is coherent, as is Hom(f. pr., coh.). (5.3.6) If F is coherent, and I is a coherent sheaf of ideals, then IF is coherent (remember that this meant the image of I \otimes psi^{-1} F in Psi^* F = \oh_A \otimes \psi^{-1} F). This just uses earlier parts of 5.3. If O_X is coherent, then in b) the kernel is even coherent. Important fact: If O_X is coherent, then f.pr. ==> coh. Here's part of the reason why. Already f.pr. ==> f.t., so that's a) of coh. +++ Benji says: For b), if O_X is coherent then so are O_X^p and O_X^q (finite direct sums) and then F is the cokernel of a map of coherents; use (5.3.4). (5.3.8) Prolonging submodules of stalks. If O_X is coherent, and F is a coherent O_X-module, then any finitely generated (=finite type) submodule of a stalk of F extends, in a neighbourhood of the point, to a coherent submodule of F. This isn't proven, just taken from Tohoku. I may say more about (5.3.9)- later. 5.4 Locally free sheaves. Not much new here. (5.4.5) "Furthermore, there is a canonical functorial homomorphism f^*(L^\vee) --> ( f^*(L))^\vee (4.4.6), and when L is loally free, this homomorphism is _bijective_." Here I presume bijective means "an isomorphism". Anyone disagree? +++Benji responds: In this context, you are OK. Really, bijective means one-to-one and onto (injective and surjective). In sets or "algebraic" categories, bijections are isomorphisms, but not necessarily for topological spaces. (5.4.7) He describes the group of invertible sheaves as H^1(U,O_X^*). (5.4.9) If you have an exact sequence of O_X-modules 0 --> E --> G --> F --> 0, where F is locally free, then for any point there is a neighbourhood where G = E \oplus F. Reason: you can locally find a splitting F--> G. (5.4.10) Projection formula. 5.5 Sheaves on a locally ringed space. He's done all that without using locally ringed spaces! Now he finally introduces them. (5.5.1) s'annuler = vanishes Coming in a week and a half: 6. Flatness. 6.1 Flat modules. 6.2 Base change. 6.3 Localization. 6.4 Faithfully flat modules. 6.5 Restriction of scalars. 6.6 Faithfully flat rings. 6.7 Flat morphisms of ringed spaces. 7. Adic rings. 60-78, 18 pages. 7.1 Admissible rings. 7.2 Adic-rings and projective limits. 7.3 Pre-adic noetherian rings. 7.4 Quasi-finite modules on local rings. 7.5 Rings of formally restricted series. From Brian: e.g., the p-adic completion of Z_p[X] rather than the (p,X)-adic completion --- it is "formal power series with coeff ---> 0", which is usually called "restricted power series"; these are the kinds of rings you get when you consider the deformations (Z/p^n)[X] of F_p[X] and try to "pass to the limit". From Benji: I ran into restricted power series when I looked up Hensel's lemma in Bourbaki. Here is my take on why you need them. If f(X) is any formal power series over Z_p in n variables X = (X_1, ..., X_n) then f(x) makes sense whenever each component of x = (x_1, ..., x_n) lies in the maximal ideal. If f(X) is a restricted power series then f(x) makes sense for any x . Also, a restricted power series is a strict generalization of a polynomial: just take a discrete ring. 7.6 Complete rings of fractions. (arise in study of local rings on formal schemes; don't skip) 7.7 Tensorial complete products. 7.8 Topologies on modules of homomorphisms.