A revised version is now posted at **the usual place** (the March 25, 2012 version). We have reached the end of the second quarter of our academic year, so I want to pause and look back on where we are, and fill in those who are just watching the notes evolve. (The course webpage is **here**.)

If we continue at the current breakneck pace, we will finish all the central material I have claimed can be covered in a single-year course. We may not succeed, but it will not be because the goal is impossible. (Instead: I have some material still to think through and prepare, and I may not manage it to my satisfaction.) I am well aware that I have 30 weeks to work with (longer than the academic year at most universities), and the people in the class are not typical, in many ways.

More precisely: in the notes, we’ve reached elliptic curves (we will begin the next quarter showing that they are group schemes). I consider everything up to 21.8 to be in very good shape. There are things that still need fixing, but I have an explicit finite list, which is large, but shrinking. I have no sections that (in my mind) need serious revision before 21.9.

Here are some rambling thoughts, both large and small, in the order in which they appear in the text. Before I begin, I should say that there are many many improvements, due to people in my class, but also a large number of sending emails from elsewhere on the globe, and also posting here. *I want to repeatedly thank you for the huge number of comments you have sent in.*

The section on valuative criteria (13.5) is now in potential “final form”. In other words, it is now self-contained, and open for criticism. I state the criteria (6 in total: valuative criteria for separatedness, universal closure, and properness, each in “DVR” and “general” versions), but do not prove them. I sketch the proof of the valuative criterion for separatedness in the DVR case (I basically give the proof). This is based on the discussion in the post on valuative criteria **here**. Please feel free to complain! (Any attempt to give a complete proof of the valuative criterion of properness ended up being longer than I wanted to include at this point.)

Fun fact (14.5.B): suppose you have a short exact sequence of quasicoherent sheaves. If the first and third are locally free, then so is the second. If the second and third are locally free *and of finite rank*, then so is the first. I had wondered about a counterexample if the “finite rank” hypotheses were removed. Daniel Litt has given me one, and posted it **here**. (Perhaps this or something like it is in the literature? Perhaps this should be added to the **stacks project**?)

I am mildly curious about the following (cf. 16.4). (Not curious enough that I’ve given it any thought, but curious enough that I’m hopeful someone has a very fast answer.) If is a graded ring, and is a graded -module, if is finite type, is the corresponding quasicoherent sheaf finite type? And similarly for coherence? Presumably yes. *(Update June 29, 2012: Fred Rohrer has explained this now, see below.)*

The way in which I first discuss pullbacks has evolved (17.3); three different approaches all come into it (the affine-local picture; the universal property; and the “inverse image then tensor with structure sheaf” definition). (Feedback I’d earlier gotten: one expert prefers a more general approach, doing things for ringed spaces; two learners found the exercises surprisingly straightforward. So far I’m sticking with straightforward over general.)

The notion “generated by global sections” is slightly awkward, especially when relativized. I’m using the terms “globally generated” (16.3), “finitely globally generated” (16.3), and “relatively globally generated” (18.3.7). If this potentially bothers you, please complain. Ideally make a counteroffer, or at least an argument.

Relative Proj is now done differently (see 18.2). I am now quite happy with the approach, because I have (sadly) given up on dealing with any universal property, as without it, the construction is very easy (when done in the right way). If anyone reads it, please let me know what you think, and tell me what is still confusing. (Summary of feedback to date: people find this an uninspiring topic, but the exercises are gettable.)

In Exercise 18.3.B, we show that the composition of projective morphisms is projective *if the final target is quasicompact*. (That wacky hypothesis is part of the sign that the notion of projective notion is not great.) I am curious: does anyone know a counterexample without the quasicompactness hypothesis? This isn’t important (it will undoubtedly never come up for me in real life). [Update August 21, 2012: **I’ve now asked it on mathoverflow.**]

*(Update March 27, 2012: there were many typos in the Chow’s Lemma section, so a revised version is now here.)* In 20.8, I prove the following form of Chow’s Lemma: if is proper, and is Noetherian, then there exists surjective and projective, with also projective, and with an isomorphism on a dense open subset of . I want to include all other versions that reasonable people (or even reasonably unreasonable people) might reasonably use — with references, but most likely without proofs. The versions I can think of are: (i) weaken “proper” to “finite type and separated”, and weaken the conclusion to “ is quasiprojective” (rather than projective), and (ii) a generalization where is replaced by a Noetherian scheme, and (iii) = (ii)+(i) (EGA II.5.6.1). If is reduced, or irreducible, or integral, then we can obviously take to be as well. EGA II.5.6 has a variant where the target is quasicompact and separated, with a finite number of irreducible components. Are there any other variants I should care about?

March 27, 2012 at 11:03 am

Just before equation (23.4.11.2) the superscripts are mingled: complex conjugation should interchange the indices, not create two equal indices.

March 27, 2012 at 7:49 pm

You’re absolutely right, now fixed (in the next version to come out). Thanks for catching that!

April 5, 2012 at 10:30 am

Dear Ravi,

Have you given any thought to the idea of publishing these notes somewhere?

April 8, 2012 at 5:13 pm

Dear Rex,

I’ve thought about it a vague sense, but have carefully avoided giving it any thought, as it’s more important (given the extreme finiteness of my time) to try to iron out all the bugs and make the notes as useful as possible. I could certainly imagine it happening, but so far I’ve declined to let publishers consider it or send it out for review. I also like the idea of mathematics that is as freely available and widely distributed as possible.

April 18, 2012 at 6:40 am

Ravi,

For when you do get around to thinking about this: I would like to make the point that many of us find bound books more accessible than pdfs or printouts. Witness the sales of Hatcher’s book on Algebraic Topology, in spite of the fact that it is freely and legally available online on the author’s webpage.

Best,

Charles

April 18, 2012 at 7:14 pm

Thanks (to Rex too) for the thoughts! I also prefer solid versions, and also I’d like to have a “frozen” version that people use to talk to each other (or refer to) without having to worry about comparing versions.

One more point I wanted to make, motivated by Rex’s comment: before any “freezing” of the document takes place, I want to take full advantage of my access to so many thoughtful people via this site (and email). It is possible for writing today to be quite a different process than it was even a couple of decades ago. Traditionally, someone writes something (a book, an article), and can send it to a few experts, who give comments. In the case of these course notes, expert comments have certainly been very helpful (from Brian Conrad, Matthew Emerton, Johan de Jong, David Speyer, Brian Osserman, Martin Olsson, Allen Knutson, S’andor Kov’acs, …), and have had a large impact on how things are explained, and on some larger architectural points. And expert advice is coming at a different point in the process than usual — not at the end, but in the middle. An even more dramatic impact has come from people using the notes in the way they are meant to be used — to learn and understand the foundations of the subject. The advice that audience focuses on things that really matter, and on the things that experts will naturally notice less. There are issues that are really quite tricky when you first learn them, that become second nature later on. The best people to give advice on identifying and circumventing the rocky bits are people who are dealing with them the first time. (There is also an intermediate group — a number of people such as Charles who are in the process of becoming experts.)

April 12, 2012 at 6:27 am

Going through the exercises I noticed that 11.2.D and 11.1.M are the same.

April 12, 2012 at 8:56 am

Good point, thanks! That is now fixed. In doing that, I realized that the discussion of where two morphisms agree (in 11.1) is best moved to the start of 11.2, as that is where it thematically fits.

June 29, 2012 at 4:41 pm

Fred Rohrer at Tubingen answered something I was wondering about, and asked about! I’m very pleased. His email is excerpted below, with his permission. (ps I owe responses to a number of comments. Things have been slow as I put out many fires left burning for some time due to the end of a busy quarter.)

dear ravi

in the 21st post in your blog about your algebraic geometry notes you asked whether certain finiteness properties are preserved when passing from a graded module to the associated quasicoherent sheaf on a projective scheme. the answer is yes. to my knowledge this results do not appear in the literature, but i will sketch the (easy) proof below. (i happened to think about this questions in the setting of toric schemes, i.e. for more general graduations, where the results are still true under some mild assumptions, e.g. simpliciality.)

i hope you find the time to let me know if this satisfies your curiosity.

best regards,

fred

*********************************************************

In the following, “graded” always means “Z-graded”.

Let S be a positively graded ring that is generated by finitely many elements of degree 1. Let X=Proj(S). We denote by T the functor from the category of graded S-modules to the category of quasicoherent O_X-modules that maps a graded module to its associated quasicoherent O_X-module. Then , T is exact and commutes with inductive limits.

PROPOSITION

T preserves the properties of being of finite type and of being of finite presentation.

Proof: We show only the finite presentation case, the other one being proven analogously. Let M be a graded S-module of finite presentation. Then we can find a graded finite presentation of M, i.e., an exact sequence of graded S-modules of the form F_1–>F_0–>M–>0, where F_0 and F_1 are finite direct sums of graded modules of the form S(n) for some integers n. (Here, (n) denotes shifting by n.) Now, let f be an element of S of degree 1. We apply homogeneous localisation at f to the above exact sequence. This functor being exact we get an exact sequence of S_(f)-modules of the form (F_1)_(f)–>(F_0)_(f)–>M_(f)–>0. As homogeneous localisation commutes with direct sums, the S_(f)-modules (F_i)_(f) are finite direct sums of S_(f)-modules of the form S(n)_(f), and this is precisely the n-component of the graded ring S_f. But by [EGA II.5.7] we know that the S_(f)-module S(n)_f is isomorphic to S_(f) and thus free. Therefore, our exact sequence is a finite pre

sentation of the S_(f)-module M_(f). This implies that T(M) restricted to D_+(f) is of finite presentation, and doing this for every f in S_1 we get our claim. qed.

COROLLARY

T preserves the properties of being pseudocoherent and of being coherent.(*)

Proof: By the proposition it suffices to prove the statement about pseudocoherence. Let M be a pseudocoherent graded S-module. Let G be a sub-O_X-module of T(M) of finite type. By [EGA II.2.7.11] there exists a graded sub-S-module N of M such that G=T(N). By (the proof of) [EGA II.2.7.8] there exists a graded sub-S-module N’ of N of finite type such that T(N)=T(N’). As M is pseudocoherent, N’ is of finite presentation. Thus, G=T(N’) is of finite presentation by the proposition, and therefore T(M) is pseudocoherent. qed.

(*) Reminder: “pseudocoherent” := subobjects of finite type are of finite presentation; “coherent” := pseudocoherent and of finite type. (This is Bourbaki terminology and coincides for O_X-modules with the (usual) EGA terminology.)

August 21, 2012 at 9:31 am

This makes me happy. I’ve now added this to the notes. Currently I refer to the url.

June 29, 2012 at 9:41 pm

For someone stupid like me, I have some problems understanding your book:

1st: You talk of coordinates but never define what coordinates are. I have checked many algebraic geometry books and they never define what “coordinates” are. Coordinates of projective space? Yes. But coordinates in general, I dont think you ever define them. Its really frustating to read about change of coordinates or things like that and not knowing what coordinates are. What actually are coordinates? What means coordinate-independent?+

2nd: You never give concrete examples of schemes and worse, you never prove why some common spaces are actually schemes. Its nice to hear that some space is a scheme, but you never actually prove why its “a locally ringed space isomorphic to Spec(A) for some commutative ring A”. You would have to prove 1) Its a locally ringed space 2) Its isomorphic to Spec(A) for some A. And I dont think you ever do that. How should a student know that something is a scheme if you never actually give a concrete example of why some space satisfies the definition?

I hope my question dont come as too dumb but Im actually learning and would like an answer.

July 13, 2012 at 1:21 pm

Dear Emil,

Many thanks for your message.

Regarding “1st”: That’s a good suggestion, to explicitly define the informal notion of “coordinate”. I’ve done this in the new version (to be posted soon). If a ring is generated (over the base field or, for more experienced people, over a base ring) by some elements of the ring, those generators are often called coordinates. (So think “generators” = “coordinates”, and “choice of generators” = “choice of coordinates”.) The motivation is as follows. A choice of r coordinates is a choice of embedding in r-space, and these r generators are the restriction of what people usually call the coordinates on affine r-space. You can see the idea in section 4.2.7, made precise in Exercise 7.2.D (in the current version, not yet posted; likely the references are the same or close in any recent version).

“Projective coordinates” are defined near the start of section 5.5.

I never use the phrase “coordinate-independent”, but I’ll tell you what it means here, because you may find it useful in other circumstances. A construction is coordinate-independent if it does not depend on any choices (of “coordinates”). In other words, if you make the construction using your favorite coordinates, and I make the construction following the recipe with my favorite coordinates, our two constructions are canonically isomorphic. (Useful example in linear algebra: if V is a finite-dimensional vector space, then there is no coordinate-free isomorphism of V with its dual, but there *is* a coordinate-free isomorphism of V with its double dual.) More generally, the word “canonical” should be interpreted as “choice-free”.

Regarding “2nd”, I’m not sure what you mean. There is a section (5.4) called “three examples”. There is no handwaving there. The key exercise to keep in mind when constructing schemes by gluing are Essential Exercise 5.4.A, which builds on Exercise 3.7.D. Projective schemes are explicitly constructed as well (by this means), and are proved to be schemes. Every explicit scheme in the notes is shown to be a scheme; even the Grassmannian is constructed.

One concluding comment, which is perhaps my most important point: this is not the ideal entry into the subject. (My own opinion is that there is no single best entry into the subject; the world is too infinite to be linearly ordered.) This approach will suit only a certain kind of person, and such people are definitely a small (and slightly strange) minority. If it doesn’t suit you, that’s fine; you’re in the majority.

best,

Ravi

July 15, 2012 at 12:16 pm

One comment that may be helpful: It is extremely unusual to prove, directly, that a concrete example is a scheme by showing 1) and then 2). For starters, is always a locally ringed space, as is proved early on, so no one ever bothers to show 1) directly: it is implied by 2).

Moreover, the most common way examples are constructed is (loosely) in two stages. First, you show in general that a certain kind of construction always produces a scheme. Such constructions include , projective space, closed subschemes specified by the vanishing of certain sections,…. In stage two, you apply these constructions to get specific examples that are automatically schemes, because you already showed that the constructions work.

July 15, 2012 at 12:18 pm

Note: everywhere is says “formula does not parse” should be Spec A. [I used correct but somewhat sophistocated latex code that WordPress appears not to understand at this time.]

July 15, 2012 at 2:34 pm

Thanks Charles, for your consistently helpful comments! I’ve changed operatorname to text, so now it *does* parse. I should mention that your latex was quite standard; I’m surprised wordpress didn’t understand it!