Please take a look at the CRing Project. Its aim is to be “an open-source textbook on commutative algebra, one simultaneously accessible to beginning undergraduates but advanced enough to cover the foundations needed for a serious study of algebraic geometry”. It just started so this might be perfect if you wanted to jump in and join. Email Akhil Mathew for more information.

Another open source algebra text is Abstract Algebra: Theory and Applications which is run by Tom Judson, Rob Beezer, and Lon Mitchell.

Let me know if learn of others (with relevance to the stacks project).

Maybe you’re thinking about writing something and contributing to the stacks project. Here are some thoughts I have about this.

(1) Of course it would be fantastic to get your lecture notes, preprint, etc as a contribution!

(2) Please make sure you have copyright over the material and that you agree to release it under the GFDL

(3) Right now most of the material in the stacks project is really building theory and not so much giving an overview of how to get from A to B. We try almost everywhere to prove lemmas in the correct generality. I want to have (a) more chapters where the material gets discussed in a way that it becomes usable for those not interested in building foundations, (b) more expository material, and (c) more alternative approaches to theorems and foundations. Thus it is quite possible that your material would be welcome even if it has already been covered in the stacks project.

(4) Whenever anybody contributes a text to the stacks project I get to edit it and decide which pieces to delete or which things to add. This is because right now I am the maintainer of the project. Even if I use only 2 pages out of a 5 page manuscript I would still be ecstatic about it.

(5) Being able to copy and paste things literally from your manuscript (without having to retype them) can be a great help. I have no compulsion about taking things from the literature and writing them up, but I worry about using very recent manuscripts of people. (For example, I don’t scour the arXiv looking for tidbits to copy and paste.)

(6) Anybody can take a copy of the stacks project and run a competing “stacks project”. The license allows that! I doubt this will happen, but it could happen. I would kind of enjoy that, but you might not enjoy Mr X randomly editing a manuscript you worked hard on (of course this might happen if I edit it too).

(7) What is in it for you? Fame! Oh wait, no… basically nothing. I would add you to the list of contributors and mention your name in the logs for any commit which adds your material.

The motivation for writing something should be that you want to explain something more clearly than in the literature, explain a particular technical point, explain something in a new way, prove a lemma you just realized is true but you’ve never seen before, prove a new theorem, or you simply want to wrok through something in order to understand it better. You should publish your write-ups on the web with your name on them (i.e., put it on your web-page, dump it on the arXiv and/or submit it to a journal). But once you’ve written something, you’ve published it, and you’re willing (and able) to share it, then just send it over and I’ll see if (parts of) it can be incorporated.

Let f : X —> S be a quasi-compact morphism of schemes. Then f is universally closed when it satisfies the existence part of the valuative criterion. This is a straightforward application of the following cute fact: If A —> B is a ring map and the image of Spec(B) in Spec(A) is closed under specialization, then it is closed.

It turns out that if S is locally Noetherian and f of finite type, then it suffices to check the existence part of the valuative criterion for discrete valuation rings. Jarod Alper told yesterday how to prove this based on Lemma Tag 05BD which I initially introduced to study impurities (more about this in a future post).

Here is the lemma: Let f : X —> S be quasi-compact. Suppose that g : T —> S is a morphism of schemes, Z a closed subset of X_T and t a point of T not in the image of Z. Then one can find, after shrinking T to a neighborhood of t, a factorization T — a –> T’ — b –> S of g such that b is locally of finite presentation and such that there exists a closed Z’ ⊂ X_{T’} which contains the image of Z and whose image in T’ does not contain a(t).

In particular, if the image of Z in T is not closed and t is a point witnessing the non-closedness of the image, then a(t) is a point of T’ witnessing the non-closedness of the image of Z’. In other words, if f is not universally closed, then there exists a base change which is locally of finite presentation which is not closed. By some straightforward argument we deduce that it suffices to check that f crossed with A^n is closed in order to prove that f is universally closed. This is Lemma Tag 05JX. In particular, if we now assume that f is of finite type and S is locally Noetherian, then it is easy to see that it suffices to check the existence part of the valuative criterion for discrete valuation rings in order to be able to conclude that f is universally closed. See Lemma Tag 05JY for a precise statement.

A key observation is that we do not assume that f is separated. (In the separated case there is a proof of the criterion using Chow’s lemma, see Lemma Tag 0208.) Proving things for non-separated schemes is a testing ground for proving results in the setting of algebraic stacks (since the non Deligne-Mumford ones are rarely separated). Jarod really made his suggestion in the setting of finite type morphisms of  locally Noetherian algebraic stacks and I think the above goes through (mutatis mutandis), although I have not written out all the details (Jarod and I worked it out on the blackboard though).

[A word of caution: Points of an algebraic stack X are defined as equivalence classes of morphisms from spectra of fields. There is a natural topology on the set |X| of points. But it need no longer be true that |X| is a sober topological space; this can already be false for algebraic spaces. Moreover if U —> X is a presentation it need not be the case that you can lift generalizations along the map |U| —> |X|; there is a counter example for algebraic spaces already due to David Rydh I think. I do think we should define closedness of morphisms of algebraic stacks in terms of these topological spaces, but as you can see from the above you have to be very careful when you try to think about what that means.]

Did you know that if R is a ring, M is a finite R-module, and φ : M —> M is a surjective module map, then φ is an isomorphism? Just learned this today. This is Lemma 4.4a in Eisenbud if you want a reference. Or see Lemma Tag 05G8 in the stacks project.

If you know about limit arguments etc, then you immediately see how to prove it for finitely presented modules (reduce to Noetherian case, etc, etc). Thinking about it some more you may come to the conclusion that this is one of those things that is simply not true for finite modules in general. So I enjoy lemmas like this since it feels as if you are getting away with something!

[Edit: Just (6:02 PM) received an alternative proof of this lemma from Thanos D. Papaïoannou which is I would say a more honest and in particular completely standard proof. So now there are two proofs… More anybody?]

Just a quick reminder about references to the stacks project. Please refer to results in the stacks project by their tags. The tags system is explained here and here. To look up a tag you type it into the box on the query page. In a stacks project pdf just click on the name of a lemma to find its tag. Of course you can use the numbering in the current version, but if you want your references to work long term you need to use the tags.

For nerds only (if you are reading this then you are one): if you use hyperref then putting

\href{http://math.columbia.edu/algebraic_geometry/%
stacks-git/locate.php?tag=0123}{0123}

in your latex source makes the tag 0123 in your pdf a hyperlink to the lookup page.

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!