Thanks to Bhargav and some editing by yours truly we now have a section on formal glueing in the stacks project. In fact it is in a new chapter entitled “More Algebra”. The main results are Proposition Tag 05ER, Theorem Tag 05ES, and Proposition Tag 05ET (look up tags here). The original more self-contained version can be found on Bhargav’s home page.

What can you do with this? Well, the simplest application is perhaps the following. Suppose that you have a curve C over a field k and a closed point p ∈ C. Denote D the spectrum of the completion of the local ring of C at p, and denote D* the punctured spectrum. Then there exists an equivalence of categories between quasi-coherent sheaves on C and triples (F_U, F_D, φ) where F_U is a quasi-coherent sheaf on on U = C – {p} and F_D is a quasi-coherent sheaf on D and φ : F_U|_{D*} —> F_D|_{D*} is an isomorphism of quasi-coherent sheaves on D*.

An interesting special case occurs when considering vector bundles with trivial determinant, i.e., finite locally free sheaves with trivial determinant. Namely, in this case the sheaves F_U and F_D are automatically free(!) and we can think of φ as an invertible matrix with coefficients in O(D^*). In other words, the set of isomorphism classes of vector bundles of rank n with trivial determinant on C is given by the double coset space

SL_n(O(U)) \ SL_n(O(D*)) / SL_n(O(D))

Another interesting application concerns the study of “models” of schemes over C. Namely, instead of considering quasi-coherent sheaves we could consider triples (X_U, X_D, φ) where X_U is a scheme over U, and so on. In this generality it is probably not the case that such triples correspond to schemes over C (counter example anybody?). But if X_U, resp. X_D is affine over U, resp. D or if they are endowed with compatible (via φ) relatively ample invertible sheaves, then the result above implies in a straightforward manner that the triple (X_U, X_D, φ) arises from a scheme X over C.

Here is a list of algebra projects that I eventually want to have written up for the stacks project. This list is a bit random, and I will edit it every now and then to add more items. Hopefully I’ll be able to take some off the list every now and then also. If you are interested in helping out with any of these, then it may be a good idea to email me so we can coordinate. It is not necessary that the first draft be complete, just having some kind of text with a few definitions, some lemmas, etc is already a good thing to have.

G: A bit about Galois groups of fields. (Also the infinite case.)

I: A bit about inertia and decomposition groups. (Not just local fields.)

Pi: A bit about “Galois groups of rings”, i.e., a bit about finite etale extensions of rings and how this is related to (unramified) Galois groups. (Lenstra’s notes.)

BR: Write about Brauer groups of fields. My favorite exposition of this material is a set of lectures by Serre in Seminaire Cartan, Applications algebriques de la cohomologie des groupes. II: theorie des algebres simples, exp. n. 5, 6, 7. (Search for Serre on Numdam.)

HO: Write up Gabbers proof of Br = Br’ for affine schemes, see Hoobler’s paper on this topic. This also leads to some nice material about K-theory of rings.

CI: Write about complete intersection rings. Introduce the notion of a complete intersection ring (for a Noetherian local ring using its completion and the Cohen structure theorem), and prove that if A —> B is a flat local homomorphism of Noetherian local rings, then B is CI if and only if A and B/m_AB are CI. This is a result of Avramov. Use it to show that the localization of a CI ring is CI.

BP: Bass’ result “Big projective modules are free”.

UFD: Regular rings are UFDs and related material.

P: Write about p-bases.

E: Write about excellent and quasi-excellent rings.

GND: General Neron desingularization.

JH: Artin’s “Joins of henselian rings”. You can generalize the main algebraic trick in this paper a bit. I don’t quite remember how or what though. Anybody?

DC: More introductory material on (unbounded) derived categories. Currently the focus in the discussion of derived categories (in the chapter on homology) is to quickly get to a point where you can start using them.

D(R): It would be useful to have a preliminary discussion of the derived category of the category of modules over a ring (before actually introducing it in general perhaps? not sure).

D: Duality (in algebra). Matlis duality. Local cohomology. Dualizing complexes. Finiteness theorem.

L: Definition and basic properties of the cotangent complex (not the Netherlander complex, but the full one, in the setting of ring maps).

HH: Introduction to Hochschild homology.

HA: Introduction to Hopf algebras, modules, comodules, etc.

Meta 0: Find examples and counter examples illustrating the results in the algebra chapter.

Meta 1: Clean up the beginning of the algebra chapter and put in some really basic stuff.

Meta 2: Find a more reasonable organization of the algebra chapter which however does not lead to vicious circles.

ZZ: and so on.

Projects which are done (of course exposition can always be improved upon…):

FG: Formal glueing. Bhargav sent me a write-up. See also his home page. Show that if A is a Noetherian ring and f ∈ A then A is somehow gotten by glueing A_f and A^* along (A^*)_f. Really what I mean is the corresponding result for the categories of modules. See Section 4.6 of this paper. You can find this in a bunch of locations in the literature for example, M. Artin, Algebraization of formal moduli II. Existence of modifications, Annals of Math. 91 (1970), pp. 88–135. OR D. Ferrand, M. Raynaud, Fibres formelles d’un anneau local noethérien, Annals Sci. Ecole Norm. Sup. (4) 3 (1970), pp. 295–311; especially: Appendix 308–311. OR L. Moret-Bailly, Un probleme de descente, Bull. Soc. Math. France 124 (1996), pp. 559–585.

A while back I changed all the occurences of “étale” into “etale”. Then yesterday, when Emmanuel Kowalski sent me some corrections to French spelling, I asked him to make a patch changing back “etale” into “éale” and overnight he emailed me one. To see the results for yourself, take a look at the chapter on Étale Cohomology. Better right?

But now that we have this I remember why I made the change originally. It was because having all the french accents in the scanned copies of EGA makes it basically impossible to quickly search for words in the text. For some reason this really annoys me (it is of course a failure of the software I am using and not of EGA).

As a funny consequence now searching for “etale” or “étale” in the dvis and pdfs of the stacks project fails too. I tested using xdvi, xpdf, and okular. Can somebody who uses acrobat reader see if it does work with that? Also, I assume that it does work if you have a french keyboard?

PS: A workaround is to search for “tale” and not “etale”.

Let R be a ring such that for every x in R either x or 1 – x is invertible. Then I claim that R is a local ring. Take some time to think this through…

Brian Conrad complained here that the statement above is not true because the zero ring is not a local ring. I agree with him. The same mistake was made in the stacks project! Argh!

Fixing it led me to review the definition of a locally ringed topos. I want the definition of a locally ringed topos (see Definition Tag 04EU) when applied to a ringed space to produce a locally ringed space. Hence I decided to add a condition that guarantees that 1 is “nowhere” 0 on a locally ringed topos. Any complaints?

Note that Exercise 13.9 of Exposee IV in SGA IV suffers from the same confusion too (although, of course, I may be misreading it). I also haven’t read Hakim’s thesis which SGA tells you to do (my bad). Have you?

Writing the previous post clarified my thinking and it allowed me to understand Mittag-Leffler modules better. Namely, condition (*) implies that a countably generated Mittag-Leffler module over an Artinian local ring R is a direct sum of finite R-modules. Hence an indecomposable, countably generated, not finitely generated R-module is not Mittag-Leffler.

An explicit example of this phenomenon is the following. Say R = k[a, b] where k is a field and a, b are elements with a^2 = ab = b^2 = 0 in R. Let M be the R-module generated by elements e_0, e_1, e_2, … subject to the relations b e_i = a e_{i + 1} for i ≥ 0. Then M is indecomposable as an R-module (nice exercise), hence not Mittag-Leffler. Now consider the R-algebra S = R[t]/(at – b). Then S ≅ M as R-modules via the map which sends e_i to t^i. Hence S is not Mittag-Leffler as an R-module.

Let’s return to the question I posed at the end of the previous post. Let R be a ring and S an R-algebra of finite presentation. In the Raynaud-Gruson paper they show that, if S is also flat over R, then the condition that S be Mittag-Leffler as an R-module is roughly a condition on the topology of the map Spec(S) —> Spec(R), namely of being “pure” which I will discuss in a future post. The simple example above shows that we cannot expect a similar result in the non-flat case. Thus, whereas I had at first thought that the Mittag-Leffler condition on S as an R-module would be a “mild” condition, now I think it is a very strong condition, and almost never satisfied in practice unless S is flat over R.

What is a Mittag-Leffler module? Let R be a ring and let M be an R-module. Write M = colim_i M_i as a directed colimit of finitely presented R-modules. (This is always possible.) Pick any R-module N. Then consider the inverse system (Hom_R(M_i, N))_i. We say M is Mittag-Leffler if this inverse system is a Mittag-Leffler system for any N. It turns out that this condition is independent of the choices made, see Proposition Tag 059E.

A prototypical example of a Mittag-Leffler module is an arbitrary direct sum of finitely presented modules. Some examples of non-Mittag-Leffler modules are: Q as Z-module, k[x, 1/x] as k[x]-module, k[x, y, t]/(xt – y) as k[x,y]-module, and ∏_n k[[x]]/(x^n) as k[[x]]-module.

Why is this notion important? It turns out that an R-module P is projective if and only if P is (a) flat, (b) a direct sum of countably generated modules, and (c) Mittag-Leffler, see Theorem Tag 059Z. This characterization is a key step in the proof of descent of projectivity. For us this characterization is also important because it turns out that if R —> S is a finitely presented ring map, which is flat and “pure” (I hope to discuss this notion in a future post), then S is Mittag-Leffler as an R-module and hence projective as an R-module. This result is a key lemma in Raynaud-Gruson.

Let me say a bit about the structure of countably generated Mittag-Leffler R-module M. First, you can write M as the colimit of a system

M_1 —> M_2 —> M_3 —> M_4 —> …

with each M_n finitely presented (see Lemma Tag 059W and the proof of Lemma Tag 0597). Another application of the Mittag-Leffler condition, using N = ∏ M_i and using that the system is countable, gives for each n an m ≥ n and a map φ : M —> M_m such that M_n —> M —> M_m is the transition map M_n —> M_m. In other words, there exists a self map ψ : M —> M which factors through a finitely presented R-module and which equals 1 on the image of M_n in M. Loosely speaking M has a lot of “compact” endomorphisms. Continuing, I think the existence of ψ means that etale locally on R we have a direct sum decomposition M = M_unit ⊕ M_rest with M_unit finitely presented and such that M_n maps into M_unit. Formulated a bit more canonically we get: (*) Given any map F —> M from a finitely presented module F into M there exists etale locally on R a direct sum decomposition M = A ⊕ B with A a finitely presented module such that F —> M factors through A. It seems likely that (*) also implies that M is Mittag-Leffler (but I haven’t checked this).

In the last couple of weeks I have tried, without any success, to understand what it means for a finitely presented R-algebra S to be Mittag-Leffler as an R-module, without assuming S is flat over R. If you know a nice characterization, or if you think there is no nice characterization please email or leave a comment.

[Edit Oct 7, 2010: Some of the above is now in the stacks project, see Lemma Tag 05D2 for the existence of the maps ψ and see Lemma Tag 05D6 for the result on splitting M as a direct sum of finitely presented modules.]

Alex Perry wrote a paper about faithfully flat descent of projectivity for modules, and submitted it to the stacks project. His writeup largely follows the exposition in Raynaud and Gruson’s paper [RG]. But, as Brian Conrad mentioned here, there is an error in [RG]. Alex avoids this error by proving that the Mittag-Leffler condition satisfies flat descent in the presence of flatness.

The result of the title of this blog post is Theorem Tag 05A9 of the stacks project. To get an (almost) self contained proof of the theorem start reading the introductory Section Tag 058B entitled “Faithfully flat descent for projectivity of modules”.

Edit: You can now find Alex’s write-up at arXiv:1011.0038v1 [math.AC].

Very short answer: Download the latest version here.

The short answer: The stacks project is aimed at graduate students and researchers in algebraic geometry. It is both an ever growing reference work and a graduate textbook in progress. It is not a usual textbook, in that it is virtually impossible to read from beginning till end. It does not rely on outside references, but develops everything from scratch. The stacks project resembles an open source software project in the following ways: (1) Everybody can contribute material and make suggestions, but the maintainer makes the final decision, (2) all of its source (LaTeX) files can be downloaded, (3) it is licensed under the GFDL, and (4) we keep a complete history of its development.

In the rest of this post we give the long answer which is aimed at mathematicians especially those working in algebraic geometry.

The stacks project started in 2005. The initial idea was to write an open source introductory text on algebraic stacks. I emailed Bill Fulton et al to ask if they were interested in such a thing, and I got a surprisingly positive response from them, but in the end they decided to finish their book project in the traditional way. At the time I also enthusiastically started an email list. An interesting, prescient email by Kevin Buzzard you can read here. The mailing list did not generate activity (exactly as predicted by Kevin). In May, 2008 I started working on the stacks project in earnest. My short term goal was to write an algebra chapter that I was going to use as lecture notes for my commutative algebra course in the Spring of 2008. As a consequence, I started thinking of the stacks project more as a kind of a foundations for algebraic stacks and less as an introductory text. I was quite happy with the progress I made writing about commutative algebra, so I started writing about sheaves on spaces, schemes, etc. From there it became clear how to continue.

What is the stacks project now? Let me describe its basic structure briefly.

The stacks project assumes the reader is a mathematician who has at least had advanced undergraduate courses in set theory, point set topology, and basic algebra including the definitions on rings, ideals, modules, a bit of group theory, fields, vector spaces, and some Galois theory (it probably makes sense to add more basic algebra to the stacks project). The chapters Algebra, Topology, Sheaves try to list the basic notions which are supposed to be understood. Starting from this the stacks project builds up theory.

As we build theory we adhere to the following basic rules: (1) Every mathematical result in the stacks project has a proof, (2) no outside results are used in proofs, (3) there are no forward references, (4) every time a previous result is used, it is explicitly referenced, (5) every statement explicitly states all of its assumptions, (6) we try to prove things in the correct generality, e.g., we do not add hypotheses for the purpose of simplifying the proof, (7) we avoid long proofs by explicitly introducing lemmas for intermediate results, (8) every result is moved to its natural location, for example a result on modules and rings is put in the algebra chapter.

There is a lot more that can be said about each of these rules; this is just a rough outline. A consequence of the rules above is that while reading a proof in the stacks project one can find  the basic results a lemma depends on by clicking on the hyperlinks.

You may wonder what level of generality we have in the stacks project. The answer is that on schemes the level of generality of the stacks project is similar to that of EGA. For the material on sites and sheaves, the level of generality is comparable to SGA4. The material on algebraic spaces is more general than that in Knutson’s monograph as we allow our algebraic spaces to be non-quasi-separated. For the same reason, our material on algebraic stacks is more general than that in the book by Laumon-Bailly.

How far have we gotten so far? At the moment of writing this we have quite a lot of algebra and a lot about schemes, morphisms of schemes, descent, sheaves and sites, some results on cohomology, etc. We have results on algebraic spaces; here some of the material is a bit closer to current research (as we are studying “arbitrary” algebraic spaces — see above). Finally, altogether we have developed enough theory to state the definition of an algebraic stack, and prove some basic results on properties of algebraic stacks and morphisms, and it would be relatively painless to write more general results on types of algebraic stacks, and types of morphisms (this will be done in the future).  What is currently missing is a discussion of several fundamental results by Artin, for example 1 and 4 from the list here. Currently, I am working on flattening stratifications, in order to write about Hilbert functors, in order to do 4 mentioned above.

How are you supposed to use the stacks project? My hope is that users will look for and find basic results on algebra/schemes/algebraic spaces/algebraic stacks in the stacks project and learn by locally reading. One problem is that the stacks project is already large enough so that finding a particular result may not be easy (there is an index of definitions). Hopefully, google will eventually get better at picking out spots in pdfs; also putting pointers from the blog to results in the stacks project seems to help.

How can you help? As you read the material you may find typos, small errors, annoyances, lemmas stated in an incomprehensible fashion, missing results, missing or incomplete proofs, have requests for additional results, etc. Whenever this happens we ask that you write a quick email to stacks.project@gmail.com. As another rule to add to the above we have: (0) If a (fixable) mistake is found then it gets fixed quickly.

Of course there are other ways you can help. There is a todo-list. If you have written an expository note on some material you might consider adding it to the stacks project (some of the chapters try to be more informal, see for example the chapter on etale morphisms). For another suggestion, read the last paragraph of the email by Kevin Buzzard I mentioned above!

Text editors and TeX have made the task of writing math papers easier (of course it hasn’t always led to better mathematics). It is for example trivial to move around lemmas, proofs, etc, whereas traditionally this was done by cutting and pasting (literally). Since internal referencing is automatic you do not have to worry about numbering. Hyperlinks make it possible to quickly jump back and forth between referent and reference (i.e., when checking all the assumptions are satisfied). All of these technological advances make it easy to write very long mathematical texts. In the stacks project we try to keep the basic layout of the LaTeX source files as simple as possible, see coding style. The idea is that this makes it easy for outsiders to quickly make a local edit, and that we can add layers by writing software that parses the text and adds additional features. Examples are the tags system (which provides stable references) and a python script to check for forward references (and coding style violations).

Technological innovations will eventually improve the layout and presentation of the stacks project, e.g., by rendering LaTeX source directly on web pages, but this will not improve the mathematical exposition. So the contributions to the stacks project that really count are the mathematical ones.

What can we expect from the stacks project in the future? I’m not sure, we’ll see, but… it also depends on you!

Here are some questions of the form: I don’t know this, do you? These questions showed up in the stacks project, but we did not take a lot of time to find them in the literature or try to solve them. Maybe you know how to do some of these or a reference? Here they are:

1. Does there exist a scheme which is connected, all of whose local rings are domains, but which is not irreducible?
2. Let X be a scheme over a field k. Let k ⊂ K be an extension of fields. Let T be a connected component of X_K. Is the image of T in X_k a connected component of X?
3. Is it true that a Noetherian ring all of whose local rings are Japanese is Japanese?
4. If f : Z —> X is a closed immersion of schemes, then is f_* exact on the category of abelian sheaves on (Sch/Z)_{fppf}? I think probably not in general, but I don’t have an example. Do you?
5. If X is an algebraic space (as defined in the stacks project, so not necessarily quasi-separated) does X satisfy the sheaf condition for fpqc coverings?

PS: Question 5 has a positive answer for quasi-separated spaces, see Lemma Tag 03WB. David Rydh has some more results on this. Maybe the real question here is if one can make some horrible counter example, or whether there is some straightforward argument covering all algebraic spaces that everybody has missed so far.

I think there is some confusion in the literature about what a band is, although likely this is just me. I just googled a bit and found, I think, at least two inequivalent definitions. I have also had at least one very confusing conversation with somebody (can’t remember whom), which I now think is due to us having different definitions. For the stacks project, I would prefer the band of a gerbe to be the finest possible invariant of the gerbe. I think this basically tells us what to do. Please don’t read the rest of this post if you already know how to do this.

First, let us make the following basic observation (and you are going to laugh at me for even pointing this out). For an element g of a group G let inn_g : G —> G be the map x |—> gxg^{-1}. Suppose that G, H are groups and that a, b : G —> H are homomorphisms of groups. Then the following are equivalent

• there exists an element h of H such that a = inn_h o b,
• there exist elements g of G and h of H such that a = inn_h o b o inn_g.

The reason is that b o inn_g = inn_{b(g)} o b and that inn_{hh’} = inn_h o inn_{h’}. If the equivalent conditions hold we say that a, b define the same outer homomorphism from G to H. You can compose outer homomorphisms because if you have a : G —> H and b : H —> F and g, h, h’, f in G, H, H, F, then we have (inn_f o b o inn_{h’}) o (inn_h o a o inn_g) = inn_{fb(h’)b(h)b(a(g))} o b o a. OK, so this gives us the category of exterior groups, sometimes called the category of outer groups. An automorphism in the category of exterior groups is often called an outer automorphism. It is clear how to generalize this to sheaves of groups over a site (you have to localize to get the correct notion of an outer homomorphism of sheaves of groups).

Let C be a site. Consider the fibred category PreBands over C whose category of sections over an object U is the category of exterior sheaves of groups over U, so objects are sheaves of groups on U and morphisms are outer homomorphisms. Stackify PreBands to get the stack of bands Bands over C. A band is then an object of the fibre category of Bands over a final object of C (and slightly more complicated if C does not have a final object).

What does such a band B look like? Let X be a final object of C. Then B is given by a system ({X_i —> X}, G_i, φ_{ij}) where {X_i —> X} is a covering, each G_i sheaf of groups on X_i, each φ_{ij} is an outer isomorphism of G_i|_{X_i \times_X X_j} —> G_j|_{X_i \times_X X_j} satisfying a cocycle condition. To get morphisms of bands ({X_i —> X}, G_i, φ_{ii’}) —> ({Y_j —> X}, H_j, ψ_{jj’}) consider the following two kinds of morphisms of systems

1. one given by a refinement of coverings {Y_j —> X} —> {X_i —> X} (note reversal arrow) where the H_j are the pullbacks of the G_i, and
2. one of the form ({X_i —> X}, G_i, φ_{ii’}) —> ({X_i —> X}, H_i, ψ_{ii’}) given by outer homomorphisms of sheaves of groups G_i —> H_i compatible with φ_{ii’} and ψ_{ii’}.

Then (roughly) you have to invert the ones of the first kind and the ones of the second kind where all the G_i —> H_i are outer isomorphisms to get morphisms of bands.

Finally, given a gerbe \cX over C we get a band B(\cX) by choosing a covering {X_i —> X} and objects x_i over the members of the covering in \cX. The associated band is ({X_i —> X}, Aut(x_i), φ_{ii’}), where the outer isomorphisms φ_{ij} come from the existence of local isomorphisms between the pullbacks of x_i and x_j over X_i \times_X X_j. This band is well defined up to unique isomorphism of bands.

Then, given a fixed band B, we say that a gerb \cX is banded by B if there exists a isomorphism θ: B —> B(\cX) to the band associated to the gerbe \cX. But be careful: If we try to classify all gerbes banded by B we could mean either of the following two things: Classify pairs (\cX, θ) or classify \cX’s such that a θ exists!

[Edit: I stole this description of bands from the paper by Max Lieblich and Brian Osserman, see arXiv:0807.4562. Unfortunately, there is a typo in their description of outer morphisms, they divide out by Aut(G) and Aut(H) but the should have used Inn(G) and Inn(H).]