(i) reference question (pathological non-Noetherian examples), and (ii) the Jordan-Holder package

Update to the notes: I will probably post a new version of The Rising Sea notes after our quarter ends. By the end of two quarters of the three-quarter sequence this year, we will reach the end of the chapter on curves, and there are only around ten known issues in the notes I want to fix up until that point. (But there are a triple-digit number of issues afterwards, which I hope to make serious progress on next quarter!)

(i) I’d like to give explicit references to examples of what behavior can go seriously wrong in the non-Noetherian setting, and long ago I’d scribbled some notes, but now I can’t remember what the actual references were to. Can anyone point me to where these are in the literature (because I’m sure I’ve seen them)? I had thought they were in the stacks project, but couldn’t easily find them there. The right person to ask is the incomparable Johan de Jong, and I can email him later, but I thought I may as well ask here first.

(a) There is a projective flat morphism where the fiber dimension is not locally constant.

(b) There is a finite flat morphism where the degree of the fiber is not locally constant.

(c) There is a projective flat morphism where the fibers are curves, and the arithmetic genus is not locally constant.

(d) There is a projective morphism for which the pushforward of coherent sheaves are not always coherent.

Behind these might be the famous “five-pound” counter-examples in the stacks project (aka tag 05LB, and now you can never forget its tag), based on a ring R with an ideal I such that R/I is flat but not projective. It is easy to describe R and I: R are the infinitely differentiable functions on the real numbers, and I are those functions that vanish in an open neighborhood of zero.

These examples are really unimportant in general in some sense, but I like having a strong sense of where the boundaries of civilization are, and what kinds of monsters actually live beyond those boundaries.

(ii) In return, I’ll give a sample new brief section. I realize now that I’ve been teaching and understanding the Jordan-Holder theorem wrong in group theory (way back in “introduction to group theory”). I’m now going to teach it in a way that naturally leads you to the notion of length, and simple objects in more general categories (without using those words), in a more natural way. One day I would like to write it up as a “bedtime story” in the style of what I did for exact sequences here. But I here is the write-up in a short section in the current version of The Rising Sea. I want it to be friendly and easy (for someone reading The Rising Sea, not someone seeing group theory the first time).

jordanholdermar1123Download

Incidentally, we traditionally teach/present the groups/rings/fields/modules class/material in the order “groups, then rings, then perhaps some modules or fields, then move from there”. I think I will next try to do it: “you know linear algebra, so let’s define a field, and vector space together, then start the course with abelian groups and their actions (leading to quotients etc., ie experience with the abelian category package extending vector spaces), then rings, then a touch of modules (and quotients etc), then UFDs, PIDs, Smith normal form from which we get classification of finitely generated abelian groups. Only then, to general groups, actions, quotients (with added weirdness). I would have to not give up any “canonical” material in the class (Math 120 at Stanford) in order to set them up for the next class, but I think I could do it. Unsurprisingly, I like Paolo Aluffi‘s approach (his homepage has a picture of the tree in which he lives); he told me last week he also does largely follows this path, although he actually does rings first. (It seems logically harder than doing abelian groups first, but conceptually I think he is right that it is easier, because people will already have an excellent intuition for the integers to build on.) He does this in his book Algebra: Notes from the Underground, which is in keeping with the excellent philosophy of his book Algebra: Chapter 0.

 

A fun example from Daniel Litt: slightly unusual behavior of infinite rank vector bundles

Suppose is a short exact sequence of quasicoherent sheaves.     If two out of three of are locally free, what can we say about the third?  

If and are locally free, then need not be.    This is a fundamental example.

If and are locally free, then is too.    This is important and not too hard.

If and are locally free and finite rank, then is too.  This is also useful.

But if and are locally free, and infinite rank, then need not be locally free.  The purpose of this post is to give Daniel Litt‘s explanation to me of a simple example of this.  I thought this example was not right to include in the Rising Sea (it would be distracting), but I wanted to preserve it for posterity (and for myself).

I will describe a surjective map of free modules whose kernel is not locally free. Viewed geometrically, this provides a surjective map of trivial (infinite-dimensional) vector bundles whose kernel is not a vector bundle. Of course, the kernel is projective.

Let be a projective module over a ring which is not locally free.  (Such things exist, see for example the Stacks Project tag 05WG.  To be explicit, there and is the ideal (where there are 1’s).)

Then is a direct summand of a free module, say .  Writing

gives as a direct summand of a free module with free complement. Then there is a short exact sequence

exhibiting as the kernel of a surjective map of free modules, where the map is the projection.

 

Eric Larson’s proof of the Cohomology and Base Change Theorem (and Grauert’s Theorem)

Update July 19, 2022: this later post contains a draft chapter incorporating this argument, so you can see it in context.

I want to share a proof of the Cohomology and Base Change theorem that really makes it much much more clearer to me than it had been before.  There might be some uncertainty as to its origin, so I’ll give my take on it.  I heard it from Eric Larson a couple of years ago.    We had discussed the fact that it should be some easy-to-understand fact about maps of free modules over Noetherian rings.   (Many people have known this — Max Lieblich and I had discussed it before; it is alluded to in Nitsure’s excellent presentation in FGA Explained; and I am certain similar discussions have happened many times in the past.)

Eric went home and figured it out, and the next day sent me a short and sweet argument in a pdf file (see below).   He had thought he had heard it before, but the only plausible sources he could have heard it from (Joe Harris, or the argument in Eisenbud and Harris) don’t have the argument.    But he had certainly heard an argument for Grauert’s Theorem (where the base is reduced).  So until I learn otherwise, I suspect he just thought he had heard a full argument of Cohomology and Base Change, and then tried to reconstruct how it should go, and just figured it out.   In other words, for now, I believe the argument is original to him.  But I would have expected that such a simple argument (especially with its central insight) would have been independently discovered earlier, perhaps many times. However, I am not aware of it in the literature anywhere.  Can anyone tell me where it has appeared before?  It is such a simple argument that, if I had not been consciously looking for it earlier, I would have believed that I had surely known it myself. So if someone says “I knew that, I just never wrote it down”, I’m not going to value that too highly. On the other hand, there are a number of great explanations of things in algebraic geometry that Dennis Gaitsgory told undergraduates when he was at Harvard; and Nitsure’s argument is presumably written down somewhere;  so I know there are things I haven’t seen. 

(Incidentally, Ogus told me a fantastically elegant proof of the local criterion for flatness, which was in his first published paper I think, that I love.)

Of course, I’ve digested Eric’s argument into The Rising Sea (and I really wish I hadn’t moved around part of a chapter a couple of years ago, leaving the manuscript in some disarray; I hope to post the new version before too too long).  And I’ll eventually post my take on Ogus’ argument here, and I’m also digesting it into the Rising Sea.

One of Eric’s insights, for me, is this.      When we generalize the notion of “finite-dimensional vector space” to families, we get finite-rank vector bundles.  These have constant rank, but for a coherent sheaf to be a (finite rank) vector bundle, we need more than it be constant rank. Eric suggests that to generalize the notion of “map of finite-dimemnsional vector spaces” in a particularly good way, we want more than a map of vector bundles (as coherent sheaves) — and we want more than that it is of “constant rank”.  It is this stronger condition about maps of vector bundles that gives this strong condition of cohomology commuting with base change, which can be applied in different settings (including Grauert, and the Cohomology and Base Change Theorem).

Here is Eric’s argument.

 

the Kunneth formula for quasicoherent sheaves on varieties

Although it won’t be evident from this site, I’ve been gradually editing the notes, mainly in response to suggestions and corrections you have sent me or told me. Because things are (very slightly) rearranged, there are lots of temporarily broken links, so I’ve not posted a version in a long time. I think I would like that to change. There are lots of little things I want to tweak — for example, adding Eric Larson’s “proof from the book” of Cohomology and Base Change, and Arthur Ogus’ proof (also “from the book”) of the local criterion for flatness.

But what is moving me to write this today is an email from Zihong Chen, asking about the flawed proof of the Kunneth formula in an earlier version of the notes (since removed, but it may have been after my last posting). I’d scribbled down notes on what I should have said, so I thought I should type it properly, and post it here. (I’m not sure if I will add it to the notes. The price is two pages, which is pretty steep at this point. But it fits squarely into the narrative of the Rising Sea.)

kunnethDownload

 

announcement: Jarod Alper on stacks

Happy new year all! Here is how I will celebrate the new year.

Jarod Alper will be teaching a course on “Introduction to stacks and moduli” at the University of Washington, and the course website is here: https://sites.math.washington.edu/~jarod/math582C.html

I’m going to attend, and I think this course could be as influential as Martin Olsson’s course on stacks back in the Beforetimes (which was central to his development of his wonderful book). Jarod is allowing me to advertise it here. He told me: “Right now I’ve posted a very lengthy introduction & motivation, which I’ll spend all of about one lecture on. I will be gradually posting (and revising) the notes during the quarter.”

The class meets Mon/Wed 11:30-12:50 (beginning Mon Jan 4, 2021), Pacific time. On the course website, he writes: “If you would like to participate informally in the class, please send me an email at jarod@uw.edu with (1) your name, (2) email address, (3) affiliation (if any), (4) status (e.g. 3rd yr PhD student, postdoc, …), and (5) a one sentence summary of your background in algebraic geometry.”

For Stanford folks: I am strongly recommending that most of my current students attend this (with some exceptions depending on their interests), and also recommending that many of my “maybe-they-are-my-students” consider taking it.

 

Fun example from Nikolas Kuhn: simplest (?) example of an affine open subset of an affine scheme that is not a principal/distinguished open subset

A standard and very reasonable question people have when first learning about schemes is the following: is it true that every affine open subset of an affine scheme is a distinguished/principal open subset? The answer is (as most of you know) is “no”, so the follow-up question is: “what is a simple example that is rigorously provable to someone at the stage where they ask this question”.

Nikolas Kuhn gave a slick answer to this question. Consider the cuspidal curve in the plane (where is of course a field), which has normalization . Then if you remove , you get an affine open subset (why?), but that set is not even set-theoretically cut out by a single equation (hint: pull the equation back to be a function of , but notice that this equation doesn’t lie in the subring .

Entertaining follow-up question: this affine open set is for some ring . What is ?

 

Principal open sets vs. distinguished open sets

One thought I had this morning: the name “principal open set” seems a better (more descriptive) name than “distinguished open set”. I then googled the phrase to see if it had been used in some different way, and found to my surprise that it actually had been used in precisely in this way; I’m not sure where this usage originated.

But I’m inclined to switch over completely to this phrase. The clearest downside is that the notation refers to “Distinguished”. But I already prefer to think of it as the “Doesn’t-vanish” set.

This might suggest a better name for the following two “topologies”: the topology on consisting of principal/distinguished open subsets; and the “topology” on a scheme where the allowed open subsets are affine open subsets, and the allowed open morphisms are “principal/distinguished” inclusions. I’ve been calling the latter the “Distinguished Affine Topology”, and I’m not sure if I gave a name to the former. Are there better names for these?

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Separately, now that AGITTOC (at least the first incarnation) is over, and I might write more here, as a world-readable (and world-commentable) notebook.

I’ve not posted a new version of the notes in a long time, because parts of it have been “closed for construction”. But I might resume soon. It might be easiest to post just a few chapters from the beginning, and gradually move forward.

As always, there are many comments here that I’ve not responded to. You might be surprised to find that I’ve actually read them, and made changes in response to many, and have intended changes in response to others.

 

recent post accidentally not pinned

Hi all,
you may not have noticed my most recent post because it wasn’t “pinned”. So now it is pinned, and if you go to this site, it should be at the top (I think!).  [This post is now out of date – RV, June 15 2020]

 

pointer to 2017-2018 course

The information about the 2017-2018 course is here (and you can also click on the tab above).

 

April 2015 version

My hope of a new version each month has clearly not worked out! But here is one for April, just before the month ends. As usual, a random selection of corrections and minor improvements have been dealt with, and many more remain to do.

The April version is posted at the usual place (the April 29, 2015 version).