Just a brief note about the results produced by looking up a tag on the query page. If the tag you’re looking up points to a lemma, proposition, theorem, remark, example, exercise, situation, or definition, then the search will also return the latex code of the corresponding environment. Here is are two examples to see what it looks like. Enjoy!

Of course eventually we’ll want to render the latex directly (I prefer not do this until there is a good way to render diagrams coded with the xypic package). But in some sense seeing the latex source is better: it tells you the latex label for the result, and it allows you to copy and edit the latex source directly and email me your improvement. It would be trivial to have the page also print the proof (in case the tag points to a lemma, proposition, or theorem). Let me know if you think this is a good idea.

Denote Sch the site of schemes over Q endowed with the fppf topology. Let F = Q(x_1, x_2, x_3, …) be the purely transcendental extension of Q generated by countably many elements. Let X = Spec(F). Let G = Z[X] be the free abelian sheaf on (the sheaf represented by) X. This sheaf has the following amusing property: If k is a field then

• G(Spec(k)) = 0 if trdeg(k/Q) finite, and
• G(Spec(k)) is not zero else.

The reason is that Mor_{Sch}(Spec(k), X) = ∅ if the transcendece degree of k is finite.

Here is another amusing abelian sheaf H. For any scheme S in Sch let I_S be the category of arrows f : T —> S where T is a nonempty connected scheme which is locally of finite type over some field of finite transcendence degree over Q. (Yes, this is a bit contrived.) A morphism (f : T —> S) —> (f’ : T’ —> S) in I_S is a morphism a : T —> T’ such that f = f’ o a. Define H(S) to be the set of maps σ: Ob(I_S) —> Z such that σ(f : T —> S) = σ(f’ : T’ —> S) if there is a morphism between f and f’ in I_S. In other words, σ is constant on “connected components” of Ob(I_S). In the case that Ob(I_S) = ∅ we set H(S) = 0. I claim that H is a sheaf (see remark below). Then H has the following property: if k is a field then

• H(Spec(k)) = Z if trdeg(k/Q) is finite, and
• H(Spec(k)) = 0 else.

The reason is that if there exists a morphism T —> Spec(k) with T nonempty and locally of finite type over a field of finite transcendence degree over Q, then k has finite transcendence degree over Q.

Remark: Suppose Sch’ ⊂ Sch is a full subcategory consisting of locally Noetherian schemes such that if T is in Sch’ and T’ —> T is locally of finite type, then T’ is in Sch’. Then Sch’ is also a site (with fppf topology) and the inclusion functor u : Sch’ —> Sch is cocontinuous. This gives rise to a morphism of topoi g : Sh(Sch’) —> Sh(Sch), see the chapter on Sites and Sheaves in the stacks project. Warning: this morphism of topoi is in the “wrong” direction. The sheaf H above is the sheaf g_*Z when we take Sch’ the category of schemes which are locally of finite type over a field of finite transcendence degree over Q. (Note that in our example Sch’ does not have all fibre products, but that doesn’t matter.)

Conclusion: The category of all schemes (over a given base) is too large to expect (fppf) sheaves to exhibit any kind of “coherent” behaviour as the input ranges over spectra of fields.

Let S be a scheme. Suppose that G is a sheaf of groups on (Sch/S)_{fppf}. What kind of conditions guarantee that any fppf torsor is actually an étale torsor? I know that if G is representable by an affine smooth group scheme this is OK. So this suggests looking at formally smooth sheaves G. Is there a counter example?

PS: What about the sheaf of the preceding blog post? Here G is formally étale.

[Edit Tuesday January 04, 2011. Bhargav just send me the following example of a formally smooth G with an fppf torsor which is not an etale torsor. Let S = Spec(k) where k is separably closed but not algebraically closed of characteristic p > 0. Let F be the sheaf which for a k-algebra R gives

F(Spec(R)) = {r in R | r^{p^n} = 0 for some n > 0}.

In other words F is the colimit of the sheaves alpha_{p^n} of p^n roots of zero. Allowing arbitrary p-power roots of 0 gives formal smoothness. The injective map alpha_p -> F gives fppf F-torsors that are non-trivial because H^1_{fppf}(Spec(k), alpha_p) -> H^1_{fppf}(Spec(k), F) is injective. And H^1_{fppf}(k, alpha_p) is non-zero by non-perfectness of k. But H^1_{etale}(Spec(k), F) = 0 since S has no connected etale covers.]

Max Lieblich asked if one could find an abelian sheaf G on the category of schemes in the étale topology such that

1. G(X) = G(X_{red}),
2. G(X) = 0 when X has only one point,
3. G is not zero, and
4. G is limit preserving.

I’ll tell you why he asked in a minute, but first let me tell you an example: Let A be an abelian group. Let F be the presheaf on the category of schemes which associates to a scheme X the group of constructible functions a : |X| —> A modulo locally constant functions. Let G be the sheafification of F in the étale topology. Then G works. (For more details, look at the the section entitled “Sheaves and constructible functions” in the chapter “Examples” of the stacks project.)

Why did this come up? Consider the stack [Spec(Z)/G] classifying étale G-torsors. Then the morphism f : Spec(Z) —> [Spec(Z)/G] is an equivalence of categories of sections over the spectrum of any field, f is formally étale, and the stack [Spec(Z)/G] is limit preserving, but f is not an equivalence (as G is not zero). This answers a question posed by Dan Abramovich.

There exists a flat proper morphism f : X —> S all of whose geometric fibres are connected nodal curves such that f is not of finite presentation. An explicit example can be found in the examples chapter of the stacks project. Once you’ve understood why the example works, you easily see that you can even make an example where all the fibres are stable curves, S is connected, and the genus of the fibres jumps!

But let me go out on a limb here and make a wild guess: If you assume that there exists an integer g > 1 such that f is flat, proper, and all fibres are stable curves of genus g, then f is of finite presentation.

Why do I think this is true? I think it is true by analogy with the following results: A finite flat module need not be projective. A finite flat module over a local ring is free. Thus given a finite flat module over a scheme S then you get a well defined rank function. Then the module is finite locally free if and only if the rank function is locally constant in the Zariski toplogy (yet another characterization of finite projective modules, see Bourbaki, Commutative Algebra, Chapter II, Theorem 1).

I also think the following may be true: Given an integer d >= 0. If R —> A is a finite type, flat ring map all of whose geometric fibres are smooth and irreducible of dimension d, then R —> A is of finite presentation. (Irreducible implies nonempty. For this one I actually have a pretty good idea for how to prove it.)

Don’t do this at home kids!

What I mean by the last sentence is that, if you are doing actual moduli, you should just assume that X —> S is of finite presentation. In regards to this, note that if my wild guess is correct, then Definition 1.1 of Deligne-Mumford is the correct one.  Thanks to Michael Thaddeus for pointing out that Deligne and Mumford only assume proper + flat + conditions on fibres.

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.