The eleventh post is the February 19 version here. *(Update February 24: a revised version is now posted, see this comment.) * The next post should appear on March 12. As always, I am behind on responding to emails, and to comments here. Comments from many people are incorporated (including now some from Chris Davis at UC Irvine).

**For learners.**

Read 15.2 and skim 15.3. Read 16.1-16.3, and look at 16.4 if you feel like it. Read 17.1-17.5. I’ve included 17.6 in case you are curious.

15.2 relates invertible sheaves to Weil divisors.

15.3 discusses effective Cartier divisors.

Chapter 16 is on quasicoherent sheaves on projective A-schemes, and graded modules. I find it surprisingly tricky, which means that it is likely harder than it needs to be, so advice is appreciated.

I intend to re-do 16.3, and give a different exposition of Serre’s theorem, and in particular introduce the notion of ampleness. I field-tested the exposition this quarter, but haven’t put it in. I hope to soon — sorry! *(Update Feb. 24: now done!)* I want 16.1-3 to be as readable as possible, and I’d like to leave the harder stuff that we won’t use for 16.4, which is optional. *(Update Feb. 24: this is now easier in the new version, but not field-tested.)*

I’ve not really edited chapter 17 since last year, so I hope it’s okay.

17.2 and 17.3 establish the basics of pushforwards and pullbacks.

17.4 is an essential application: invertible sheaves and maps to projective space.

17.5 is an essential consequence of 17.4: the “curve-to-projective extension theorem”.

17.6 (starred) discusses the Grassmannian as a moduli space. I haven’t looked at it in a while, but would be interested in hearing feedback on how accessible it is, and how to make it more accessible.

Exercises to do: there are necessarily a lot.

15.2.C, 15.2.E, 15.2.F, 15.2.H, 15.2.I, 15.2.J, 15.2.Q, 15.3.B, 16.1.A, 16.1.C, 161.E (which may be hard), 16.2.A, 16.2.B, 16.3.B, 16.3.E (Serre’s theorem — this exposition will be changed though), 17.2.A, 17.2.C, 17.3.A, 17.3.E, 17.3.G, 17.4.A, 17.4.G, 17.5.B. As always, it hurt my soul to omit some exercises from that list, so please try others that catch your eye. (In the Feb. 24 version, replace the 16.3 problems with some sampling of *16.3.C, 16.3.F, 16.3.H, 16.3.J, 16.3.K, 16.3.L, 16.4.C.)*

If you are looking for exercises to give you practice with explicit examples, try 15.2.A, 15.2.D, 15.2.G, 15.2.I, 15.2.N (and lots more in 15.2), 16.2.A, 17.4.B (and lots more in 17.4), 17.4.H (fun), 17.5.A.

**For experts.**

15.2. The link between line bundles and Weil divisors is nonobvious and hard and takes some getting used to. I find this easy to forget.

15.3: Inflammatory comment: I’ve found very little use for Cartier divisors in this first course. *Effective* Cartier divisors are useful, but even they are secondary. I prefer to just stick to line bundles (invertible sheaves), and to get at them using Weil divisors. I hope I have provoked people to try to convince me otherwise.

16: I would have thought that graded modules and quasicoherent sheaves would be easy (because they are so classical), but I find them harder than the foundational formal generalities of scheme theory. Graded modules are not my friend.

16.3 will be rewritten (hopefully soon). Things like ampleness can be done in a great deal of generality, at great expense. I will do ampleness in only moderate generality, where it isn’t much work, and which suffices for all applications I’ve seen in talks. *(Update Feb. 24: now done.)*

Important Question: what is the “right” notion for the relative version of global generation? (“Relatively globally generated” sounds awkward.) Of basepointfreeness? (How should that last word be hyphenated?) Some will interpret this as “what is the current convention” — I’m hoping for something broader, including “this variant would be completely unconfusing, and might change the convention”.

16.4: the adjointness between functors between graded modules and quasicoherent sheaves is surprisingly annoying, and not so necessary for where we are going. Hence I’ve starred it, so people don’t think it is necessary. (My philosophy on exposition of these foundations: The straight road is not so bad, even though it passes near treacherous hills.) This has the advantage of putting the most annoying parts of 16 into an optional section, so most people can ignore it on first reading. No one liked this section when I taught from it. Interesting fact: it seems surprisingly hard for a beginner to show that every closed subscheme of a projective scheme is cut out by homogeneous equations. *(Update Feb. 24: this exposition is now improved.)*

17.2 and 17.3: I do pushforwards and pullbacks relying heavily on affine open sets. A lot of things are quite easy then. An interesting complication arises because pushforwards of quasicoherents are not necessarily quasicoherent, while pullbacks always are; I’ve chosen not to work in the larger category of O-modules, and to stick to affines. I think there may be errors around exercise 17.3.B, and have to dig up my (very large) to do list.

17.4: I hope I’ve foregrounded this idea enough, that in effect is the interpretation of the functor of points of projective space. Question: what is the difference between a linear system and a linear series? Because I don’t know the difference, I’ve stuck to the first. Is there any reason to consider linear systems over schemes that are not over some field? One could of course — but has anyone ever used this notion? (And if not, I’m not going to contort myself to make a definition no one uses.)

17.5 I hope the reader realizes this Curve-to-projective extension theorem is very important!

17.6 is on the representability of the Grassmannian functor. I haven’t had a chance to look at this again before posting, but feel free to comment (i.e. criticize). *(Update Mar. 10 2011: I’ve taken a look, and I’m happy with it. It will be slightly tweaked in the next version, coming out around March 12.)* I’d like to point out that in any introductory algebraic geometry book I know about, the Grassmannian is never defined rigorously. (Update March 8, 2011: Ulrich Gortz and Torsten Wedhorn do it properly; see my response to Torsten’s comment too.) I’d like this section to be completely rigorous, yet not too forbidding. Note that the Plucker equations are *not* used to show the Grassmannian is projective, and if you use them, life seems to be harder. *(March 10, 2011: I also vaguely remember someone saying that the valuative criterion for properness is necessary to show that the Grassmannian is projective, but as you’ll see, that’s not the case.)*

**For everyone.**

I will try to reserve “pullback” of a sheaf for pullback of a quasicoherent sheaf (f^*), and “inverse image” for f^{-1}.

February 20, 2011 at 1:24 pm

I haven’t read through the section carefully, but I was looking at the last exercise on Grassmannian Bundles. Could you say a few words on why you introduce them?

Thanks!

February 22, 2011 at 1:34 pm

They won’t be used later in the notes, so I doubled-starred 17.6.G. But they can be useful in thinking about geometric problems. You might imagine situations where you have a vector bundle of given rank, and you might care about the set/space of rank k subbundles. Or similarly, you could imagine having a projective bundle, and being interested in the space of all P^k’s in it. Any example I give will necessarily be too specific, but I’ll give one anyway (off the top of my head, without much thought). Consider the space of quadrics in CP^3 (quadratic forms in 4 variables up to scalars, which forms a P^9). Most quadrics are smooth, and thus have two rulings. This suggests that you have a double cover of P^9 (branched over the discriminant locus). The cheapest way of making this precise (if you have this tool ready to go in your toolkit) is to use the corresponding Grassmannian bundle. Whoops, I just realized that this is a trivial Grassmanian bundle, and thus a bad example. But it’s a good example to see in general, so I won’t delete it.

How about this: if you have a foliation of an n-dimensional manifold by k-dimensional submanifolds, you have an induced section of the space of k-dimensional subspaces of the n-dimensional tangent space. You could imagine that this could reasonably come up.

In short, I just wanted to provide an off-the-shelf tool that can turn up in talks or in research, but I didn’t want it to distract people.

(If anyone else has a better example from their own work, please help me out!)

February 23, 2011 at 6:50 am

Thanks! The second example has convinced me that I am interested.

On an unrelated note, in the past I have though about taking a flag variety, the tangent space of it, and then considering flags in each fiber to make flag vars over every point. Etc, etc. I think it is a funny idea.

February 21, 2011 at 1:51 pm

I realize that chapter 12 was posted some time ago but I had some questions about 12.2.G.

Assuming that k is perfect, it is not hard to show that any dimension n irreducible variety is birational to a hypersurface in A^{n+1}_k. However I am not sure how to do the exercise as stated. It seems false just by dimension considerations – hyperplanes have dimension n-1.

I would appreciate any pointers.

Thanks.

February 22, 2011 at 1:26 pm

I indeed meant A^n to be A^{n+1}, and hyperplanes should have been hypersurfaces. Thanks for catching that! I think k needn’t be perfect for the result to hold.

March 8, 2011 at 10:27 am

Update (after I responded to Torsten Wedhorn’s comment below) I retract my statement that k needn’t be perfect! And in fact I removed this exercise.

February 22, 2011 at 9:56 am

I’m editing the ampleness section in the hopes of (re)posting it soon, and have a reference question for experts, for an important aside. I am sure that on every quasicompact quasiseparated scheme, every ideal sheaf is generated by its finite type subideal sheaves. Can someone give me a reference? (Or tell me that it is false, or at least not clearly true to experts?) My ideal reference would be to a tag from the Stacks Project, or EGA, but I’ll take what I can get.

March 1, 2011 at 9:23 pm

Ravi, on any qcqs scheme, any qcoh sheaf (let alone qcoh ideal sheaf) is the direct limit of its finite type qcoh subsheaves. This is EGA I Corollary 9.4.9. You may not believe me if you look it up (since there are extra hypotheses there), so let me explain.

The point is that if you read the errata in EGA (which I highly recommend before reading anything: always check the errata and transcribe them into the entire volume first) you’ll see that in the errata at the end of IV_1 they systematically and thoroughly explain how to fix up lots of statements of earlier stuff to replace “either locally noetherian or qc/separated” with “qcqs”. (Yeah, I know, locally noetherian schemes are not qc. Quomodocumque.)

The point is that the proofs actually work in that generality with virtually no change provided one makes suitable use of stuff in IV_1 (as they exhaustively explain). In particular, they show exactly what to change in statements of results in EGA I section 9 (and elsewhere) to make the proofs go through basically verbatim in the qcqs case (e.g., use IV_1, 1.2.6–1.2.8 to eliminate the need for Corollary 9.2.2 in EGA I when considering qcqs morphisms). For instance, 9.4.7(b) in EGA I should have “quasi-separatedness” for X, which ultimately makes 9.4.9 in EGA I be OK in the qcqs case.

Alternatively, you could look at the “new” Springer edition of EGA I. In there they did a massive rewrite and inserted the qcqs formalism right from the start (along with other hyper-generalities which I think detracted from the pleasant style of the original…but my recollection is that they put in the flag schemes, so you may like it). The exhaustion by finite type qcoh subsheaves is in there somewhere.

April 24, 2011 at 11:17 am

Thanks! Now added (and also Torsten’s wonderful book is also added). Thanks in particular for the detailed explanation of how to patch the original EGA.

February 23, 2011 at 10:51 am

Unless I am much mistaken, the top short exact sequence on page 309 is split. Thus, if the sequence right below it were obtained by tensoring, as the text claims, it would be exact, not just right exact.

February 24, 2011 at 11:48 am

You are right! The sequence isn’t obtained by tensoring, and the Feb. 24 revision (see above) now patches this (but leaves to the reader to see why the result is left-exact — I hope this becomes clear after a little bit of thought, but let me know otherwise).

February 24, 2011 at 11:42 am

Update: I’ve now re-done the section on Serre’s Theorem A, by adding in ampleness (in the absolute sense, and over a ring). This resulted in more length, but gave me a chance to introduce an important notion. The new version is dated February 24, posted in the usual place. Any comments on the ample section (or the resulting changes to the section relating graded modules and quasicoherent sheaves) are of course very welcome. (A key new player is the following result. If you have a finite type sheaf F and a line bundle L on a scheme X, and s is a section of L, then any section of F on D(s) may be interpreted as a global section of F \otimes L^n divided by s^n (for some n). After moving this into the notes, I realized that many years ago Jarod Alper had made a case for this being central, and only now do I understand why.)

February 26, 2011 at 6:50 am

Hi Ravi,

A few comments/questions about this great section of the text.

After defining linear systems, you could state a “coordinate free” version of Theorem 17.4.1:

Maps X \rightarrow P(V^*) are in bijection with base-point free linear systems (L, V, \lambda) on X.

[Done, at some point in the past. It is Exercise 17.4.I. I’ve just added a link to it from the statement of Theorem 17.4.1. — Rps all italicized comments were added March 29 2012]

(In general, algebraic geometers like to work with coordinates, but maybe every once in a while you can please the representation theorists.)

[Actually, algebraic geometers prefer to work in a choice-free way too! I often try to use coordinates in the notes just to make people new to the area feel comfortable. — R]Also, maybe you could give an example of a non-complete base-point free linear system.

[This one I decided not to do, in that I hope it isn’t hard for the reader to come up with one. But if this is really a cause for confusion, I may still do so. — R.]Finally, I have a question about Theorem 17.5.1. Your comment about the relationship between this theorem and valuative criterion for properness is a bit cryptic to me. Perhaps you could spell this out in more detail. Also, does Theorem 17.5.1 hold for all proper schemes Y? (I once asked a Math Overflow question about this.)

[See Brian’s comment below. I’ve added a bit in the notes saying that the result holds for proper Y, but not giving any explanation. — R]I can’t resist one last question. This section contains lots of important geometry. Is there any way that it can come earlier in your text?

[Good question. I think much of the geometry here relies on things introduced here. But one can get to these notions faster, and in earlier versions of the course I did that. I later found that given the nature of the year-long course, doing that was suboptimal, because it caused some things to be introduced without motivation. But I am hoping that people giving classes will judiciously decide what to include based on their students, the nature of their class, the way their mind works, and so forth — for example, I prefer to have the end of a class have some sort of punchline. — R.]March 13, 2011 at 8:01 am

Joel, concerning 17.5.1, the valuative criterion solves the version of the problem after composing with Spec(O_{C,p}) –> C, so one is reduced to the following general (and useful!) question. Let X and Y be schemes over a base S and x a point of X. Let f:Spec(O_{X,x}) –> Y be an S-morphism. Does there exist an open U around x in X such that f extends to an S-morphism U –> Y? (This would be sufficient for the case of application since there one can detect equality of maps using the generic point of the unique irreducible component though p.) The answer is affirmative if Y is locally of finite presentation over S (and is a toy version of “spreading out” results which are very useful for reducing certain local or global construction problems to variants after passing to a limit object which may have nicer structure).

To prove this formulation, we lose nothing by working over an open affine in S around the image of x and replacing Y with an open affine around f(x) and also X with an open affine around x. Then it is a question of algebra: if A and B are algebras over a ring R with B of finite presentation and if B –> S^{-1}A is an R-algebra map with S a multiplicative set of A (such as complement of a prime ideal) then does this extend to an R-algebra map B –> A_a for some a in S? That in turn is immediate from consideration of a finite presentation of B over R and the identification of S^{-1}A as a suitable limit of A_a’s.

March 29, 2012 at 4:50 pm

I have finally responded to Joel’s comments (in italics, in the middle of his comments).

February 26, 2011 at 6:15 pm

Minor typo: in excercise 15.2.E either you do not really want to reference 15.2.2.1 or I am missing something.

March 3, 2011 at 9:26 am

I’ve reworded this a touch to try to make it a bit clearer. What I mean is that given an element of K(X)^*, you get a rational section of O(D) via (15.2.2.1). I fear this is still confusing; if I can say more, please let me know (and if you have a suggestion, please let me know that too!).

February 28, 2011 at 11:36 am

Dear Ravi,

I just stumbled upon your great notes. I like in particular how many notions are motivated. I also have some remarks:

I think that one needs some hypothesis in Exercise 12.2.G: Either k has to be perfect or, more generally, the variety should be geometrically reduced. Otherwise there is the following counterexample: Let K be a finite field extension of k that cannot be generated by one element (in particular it is not a separable extension). Then Spec(K) is a variety in your sense but it is by definition not birational to a hypersurface in A^1.

Every quasi-coherent module over a quasi-compact quasi-separated scheme is indeed the filtered inductive limit of its quasi-coherent submodules of finite type (without “quasi-coherence” hypothesis I would not know of such result and would doubt it very much – even for ideals). This can be found in EGA I (new edition as Springer book), (6.9.9). I can’t resist to give as a second reference Corollary 10.50 of http://www.algebraic-geometry.de. There one also finds a rigorous scheme-theoretic definition of the Grassmannian…

Best, Torsten

March 8, 2011 at 10:26 am

Thanks Torsten!

On 12.2.G — you are completely right. I’ve decided to remove this exercise, as it isn’t doing what I wanted.

Thanks too for the references.

I was very pleased to see the table of contents to your book — I will order it later today! I like the ordering of the topics, and I suspect (given your comments above, and who you are) that I will like the presentation. In particular, I have updated my comments on not knowing a good first source for the Grassmannian!

March 3, 2011 at 12:40 pm

In the proof of 16.3.10, specifically (c’) => (b), I don’t understand the use of closed points. On the one hand you seem to be assuming the closed points are dense, which need not hold for quasicompact schemes (e.g., consider Spec of a local ring). On the other hand, I don’t see why you could not just replace closed points by arbitrary points of the scheme. (I’m more certain of this last in the first paragraph of (c’) => (b) than in the third paragraph.)

March 3, 2011 at 12:53 pm

On second thought, the proof probably stands together as is (I was a bit confused in writing the comment above). But I do hold that the use of closed points, and the resulting need to refer to previous exercises, can be a bit confusing.

March 8, 2011 at 10:32 am

I agree that this Theorem is a biggie. (I think Rob Lazarsfeld calls this the “Cartan-Serre-Grothendieck theorem” in his “Positivity in Algebraic Geometry I”.) I’d be very happy to hear any suggestions on how to improve the exposition. There is a trade-off between offloading part of the problems to earlier exercises and having the entire argument in one place. The (c’) to (b) is a particularly delicate part of the argument, I agree.

I don’t see how to easily remove the reference to closed points in this argument. But at least there is a theme here that I hope is clear (and that I’d be happy to make more clear). Suppose you want to prove something about a quasicompact scheme. Then it suffices to show it in a neighborhood of a closed point.

March 17, 2011 at 9:06 am

Brian Osserman sorted me out on “linear series” and “linear systems”. Linear series should be the vector space of sections, and linear systems are the corresponding families of divisors. I went searching, and found evidence that he is right, from the 1911 Encyclopedia Britannica (click here). I wish I knew who wrote this article! (In the next version of the notes, this terminology will be fixed.)

April 21, 2011 at 1:11 pm

Hi,

I realize this is from sometime ago, but I had a few questions about section 16.3.10 where you introduce ample-ness. At the start of the section the assumption is made that S is finitely generated in degree 1. In this case by 11.3.5 we have that the structure morphism is proper. However you assume properness in each exercise/theorem after that. Is the point that we can remove the condition that S is finitely generated and replace it with proper? Any guidance would be appreciated.

April 23, 2011 at 9:58 am

Good question. The definition of very ampleness I gave applies only when the X is proper. (There are definitions in more general settings which I wanted to avoid.) So you can keep the condition that S is finitely generated, and even generated in degree 1, which can be helpful.

April 24, 2011 at 10:51 am

On March 3, Charles Staats wrote an enligtening email to me:

“There is a point of view on twisting by n, and the relationship between graded modules and quasicoherent sheaves on projective space, that has always made more sense to me than the usual definition (as well as helping me keep track of indices). A brief explication would easily fit in the body of an e-mail, but I have attached it as a pdf instead for the sake of nice-looking notation.”

Updated June 13:

The revised pdf is here. (I’d posted an earlier version that he’d sent me in an email that was intended as part of that e-mail. This is free-standing. Thanks Charles!)

August 16, 2011 at 3:47 pm

Concerning the comment “graded modules are not my friend”: One fact that can be helpful when trying to compute with specific graded modules may be found at

http://mathoverflow.net/questions/72739

The statement is a criterion for (among other things) when becomes sufficiently large that . Although I have only encountered this very recently, my impression is that this criterion tends to be overlooked because of the importance of other results in the paper in which it was introduced.

I will also admit that I am posting this here partly in the hope that the typical reader of this blog will be more interested in finding an elementary proof of this (which is the point of the Mathoverflow question) than the typical Mathoverflow user seems to be.

August 16, 2011 at 3:48 pm

Note: My attempt at doing latex seems to have gone completely off here, for which I apologize.

August 16, 2011 at 3:53 pm

Someone should answer his question on mathoverflow and collect the bounty! (And the latex is fixed, thanks to Yifei Zhu.)

September 10, 2011 at 10:38 am

A friend and I just realized that we do not understand globally-generated-ness. Some other folks’ notes online, as well as 16.3.C.a, lead us to believe that if a sheaf is base point free then each stalk ought to be generated by the image in that stalk of the module of global sections. In particular, the structure sheaf of projective space would not be globally generated under this definition. But it certainly is globally generated under the definition given in 16.3. Am I missing something? If not then it might be good to note that many other people use a different definition, and 16.3.C.a should maybe be reworded. Thanks!

September 10, 2011 at 12:50 pm

Actually, the structure sheaf of projective space is generated under this definition. The key point is that is generated _as an -module_ by the image of the global sections. It is in fact generated by any single section that does not vanish at .

September 29, 2011 at 7:41 am

Charles said it better than I can. Ed, if some wording would help people over this tricky point, please let me know. Or even just a pointer to a sentence where people are likely to have this reasonable confusion (one I myself went through), that would help.

November 28, 2011 at 3:07 pm

Charles Staats has an exposition about Cartier divisors that I like quite a bit more than K^* / O^*. You can find it

here. In case that link ever goes stale,this onewill work.