As a kind of secondary goal for the stacks project, I would like the terminology to be as “standard” as possible. What this means exactly may not be clear in all instances, but to start off with I decided to make all definitions logically equivalent to their counterparts in EGA I, II, III, IV. In only one case sofar have I changed the definition: namely David Rydh convinced me that we should change unramified to the notion used in Raynaud’s book on henselian rings (i.e., only require locally of finite type and not require locally of finite presentation).

A good example of the kind of confusion that happens over definitions is the case of rational maps. In EGA I (both the original version and the new edition) a rational map from a scheme X to a scheme Y is defined to be an equivalence class of pairs (U, f) where U is a dense open of X and f : U —> Y is a morphism of schemes. In my opinion this is a very handy notion which in almost all situations does exactly what you want, and is quite easy to explain to students, etc. Next, one defines a rational function on X to be a rational map from X to the affine line. You can also define a sheaf of rational functions on X which is denoted by a calligraphic R.

Next, one can define the sheaf of meromorphic functions. Kleiman has a nice paper “Misconceptions about K_X” which corrects the construction of the sheaf of meromorphic functions on X in EGA IV 20.1. Note how the symbol used here is a K and not an R. Basically one inverts the multiplicative subsheaf of O_X consisting of sections which are nonzero divisors in each stalk. A meromorphic function on X is then defined to be a global section of the sheaf of meromorphic functions. An (easy but not completely trivial) argument shows that a meromorphic function f on X actually gives rise to a regular function on a schematically dense open part of X.

Some people conclude that EGA’s definition of rational functions is wrong and that we should replace the notion of a rational map by something that has a chance of recovering meromorphic functions when applied to rational maps from X to A^1. To do this sometimes people redefine a rational map as an equivalence class of pairs (U, f) where U is a schematically dense open of X…

… but this notion also exists in EGA where these maps are called pseudo-morphisms or strict rational maps from X to Y, see EGA IV 20.2. A pseudo-function is a pseudo morphism from X to A^1. It is not at all clear to me that a pseudo-function is the same thing as a meromorphic function (hopefully Brian Conrad will chime in here and tell us, but the point I am trying to make is that it is not a triviality).

My approach in the stacks project has been to use the notion of rational maps as defined in EGA I (i.e. not pseudo-morphisms). Also we define the sheaf of meromorphic functions as in Kleiman’s paper (i.e. not using pseudo-functions). Only if absolutely necessary will we work through the material in EGA IV about pseudo-morphisms and introduce it.

Of course a definition cannot be wrong. What is great about having good definitions is that they allow you to make very precise statements about the relationships between objects. My tendency is to go with the definitions as stated in EGA; it appears that Grothendieck and Dieudonne tried their best to make sure the definitions are good in the sense above.

Let R be a ring and I an ideal. For an R-module M we define the completion M^* of M to be the limit of the modules M/I^nM. We say M is complete if the natural map M —> M^* is an isomorphism.

Then you ask yourself: Is the completion M^* complete? The answer is no in general, and I just added an example to the chapter on examples in the stacks project.

But… it turns out that if I is a finitely generated ideal in R then M^* is always complete. See the section on completion in the algebra chapter. I’ve found this also on the web in some places… and apparently it occurs first (?) in a paper by Matlis (1978). Any earlier references anybody?

Please take a look at Ravi’s blog about his Math 216 graduate course at Stanford university. Students and others have been chiming in leading to a total of 265 comments in 6 months. A group of people (mainly graduate students?) are working through the material as it gets updated on Ravi’s blog, and these people provide most of the which are helpful and constructive comments on the blog. Moreover, even though Ravi is not actually teaching his course this year, the blog gives one a sense of activity much like for a real course. Of course, since I am teaching my algebraic geometry course this year based on Ravi’s lecture notes, I may be more inclined to say so than others.

You can download the latest version of Ravi’s notes here. Let me give you a bit of my own preliminary impression of these notes; you can read Ravi’s philosophy behind them on his blog and in the introduction to the notes.

As everybody who has taught an algebraic geometry course knows it is virtually impossible to feel satisfied with the end result. In my experience it actually works well when younger people teach it because they have a fresh take on it, want to get to some particular material that is important to them and they are less likely to get stuck in details. I personally never teach algebraic geometry the same way twice, and I usually end up covering a fair amount of material despite feeling like I did not at the end of it.

One of the pleasing aspects of teaching the material out of Ravi’s notes is that I do not have to organize the material as much as I usually do. Mostly I am happy with the order in which things get done, although I moved the material on quasi-coherent O_X modules and on morphisms of schemes earlier in my lectures. Also, in hindsight, I should probably have skipped chapter 2 (category theory) and jumped straight to the chapter on sheaves. A key feature of Ravi’s notes is that more than 75% of the proofs of lemmas, propositions, and theorems are left as exercises. As lecture notes often Ravi explains why things are true, with lots of examples, rather than providing a formal proof. Results from previous exercises are used throughout the text, not always with explicit references (especially in the exercises themselves of course). When lecturing it sometimes made me wonder to what extend I’ve really built up the theory from scratch (which is the stated goal of the course). Of course here you can rely on outside references and ask that students read those, ask that the students do lots of exercise, and so on. One of the standing assumptions underlying the setup is that students will work hard on their own to understand the material. Moreover, I think no matter how you teach algebraic geometry you cannot build it up completely from scratch in your lectures, i.e., the students are always going to have to do a lot themselves, and maybe by building it into the course material they are more likely to do it?

Is it a good idea to have many different algebraic geometry texts? Tentatively, I would say more is better. I have personally found Ravi’s notes useful in the following way: if you can find what you’re looking for in Ravi’s notes (e.g. by googling) then you’ll quickly find pointers unencumbered by details or generalities.

Overall I am very happy with my course and the notes so far. One of my questions is how much commutative algebra I will cover teaching the course in this way (traditionally at Columbia we teach a first semester of commutative algebra and then a second semester on schemes — in one semester focused entirely on commutative algebra you can cover quite a bit). I’ll report on this in another post about Ravi’s notes at the end of the next semester, so stay tuned.

Here is a list of projects that make sense as parts of the stacks project. (For a list of algebra projects, see this post.) 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. Moreover, we can have several chapters about the same topic, of different levels of generality (the reason this works well is that we can use references to the same foundational material in both, so the amount of duplicated material can be limited).

• If X is a separated scheme of finite type over a field k and dim(X) ≤ 1 then X has an ample invertible sheaf, i.e., X is quasi-projective over k.
• If f : X —> S is a proper morphism of finite presentation all of whose fibres have dimension ≤ 1, then etale locally on S the morphism f is quasi-projective. This also works for morphisms of algebraic spaces.
• Local duality; see also the corresponding algebra project.
• Cheap relative duality for projective morphisms. Start with P^n over a (Noetherian) ring and deduce as much as possible from that.
• More on divisors and invertible sheaves, Picard groups, etc.
• Serre duality on projective varieties.
• Classification of curves.
• Quot and Hilbert schemes.
• Linear algebraic groups.
• Geometric invariant theory. I think that a rearrangement of the material in the first few chapters of Mumford’s book might be helpful. In particular some of the material is very general, but other parts do not work in the same generality. Note that we already have the start of a chapter discussing the myriad possible notions of a quotient, see groupoids-quotients.pdf.
• Resolution of two dimensional schemes.
• Semi-stable reduction theorem for curves. (Is there any way to do this without using resolution of singularities of two dimensional schemes or geometric invariant theory?)
• Abstract deformation theory a la Schlessinger (but maybe with a bit of groupoids thrown in).
• Deformation theory applied to specific cases: zero-dimensional schemes, singularities, curves, abelian varieties, polarized projective varieties, coherent sheaves on schemes, objects in the derived category, etc.
• Brauer groups of schemes.
• The stack of curves and pointed curves, including Kontsevich moduli stacks in positive characteristic are algebraic stacks.
• The stack of polarized projective varieties is an algebraic stack.
• The moduli stack of polarized abelian schemes is an algebraic stack.
• The stacks of polarized K3 surfaces.
• Alterations and smoothness (as an application of moduli stacks of curves above).
• Add more here as needed.

Just a short update. The semester is in full swing here at Columbia University and there are a lot of things to do (including writing letters of recommendation), so I have had less time to work on the stacks project. I hope/expect to get back to it soon.

Currently, I am still working through the details of the paper by Raynaud and Gruson. I found a (repairable) error in the proof of the main geometric result (existence of devissage; last sentence of the proof of Proposition 1.2.3). It is a small error, but it really is an error and you have to slightly change the set-up in order to fix it. Of course I may be wrong, but I do not think so (for those of you who are taking a look at the paper: try to imagine what it would mean to replace the sentence mentioned above by a fully written out argument, checking all the details). In addition to this, I’m having trouble finding simplifications for almost any of the arguments, as each of the later results in the paper uses the earlier results, in other words, I haven’t been able to split off some parts as independent from the rest.

I am going to finish writing it all up, as soon as I have more time. But for the moment this experience is teaching me a lesson. Namely, I started working through the details of Raynaud-Gruson as I wanted to have a very general result on flattening stratifications. I was eager to do this, as I wanted to discuss Hilbert schemes/spaces/stacks in the “correct” generality. And this in turn I wanted to do because I want to explain the proof of Artin’s result that a stack X in groupoids over (Sch) whose diagonal is representable by algebraic spaces such that there exists a surjective, flat, finitely presented morphism U —> X where U is a scheme is an algebraic stack. Looking back what I should have done is write a chapter on Hilbert schemes/spaces parameterizing finite closed sub schemes/spaces/stacks (maybe even restricting the discussion to the representable separated case). This is much easier, is quite interesting in its own right, and is sufficient for the application in the proof of Artin’s theorem.

On the upside, I have learned a lot more about flatness in the effort to get this material written out fully!

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.