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.]