The tenth post is the January 29 version here. I apologize for taking longer than hoped. I’m only gradually catching up, and I am less far behind than three weeks ago.

The development of quasicoherent sheaves here is more affine-local than in most sources, but I think it makes the arguments much simpler to digest. It is true that quasicoherent and coherent sheaves work in the more general setting of locally ringed spaces, but making the arguments work there require (in my mind) a different flavor of thinking, which is not rewarded later in an algebraic geometry first course. I want to make the central notion of quasicoherent sheaves as digestible as possible.

(I’ve not yet edited the valuative criteria section — or even looked at it.)

For learners.

Read 14.1-14.7 and 15.1. Read 14.8 if that section is relevant for you (and if so, please let me know — I’m curious how many people care, or if this section should just be cut as a distraction).

It is important to develop a good feeling for the ins and outs of quasicoherent and coherent sheaves, so it is important to do exercises, and I’ll suggest more than I usually do.

14.A (transition functions); one of 14.1.B-F (for practice of working with transition functions); 14.1.G (the Picard group), 14.2.C (to ensure you understand this perspective on quasicoherent sheaves), 14.3.B (to make sure you understand distinguished affine bases as a mild generalization of bases), 14.3.D, 14.3.E, 14.3.G, 14.5.A, 14.5.C, 14.5.D, 14.6.G, 14.7.A, 14.7.B, 14.7.C, 14.7.D, 14.7.H, 14.7.I, 14.7.J, 15.1.A, 15.1.C. The listed exercises from 14.3 onwards will be used a lot.

That’s already a lot, but if you’d like some “example” questions, consider 14.1.J (if you are a number theorist), 14.7.F, and 14.7.G. Many of the transition function questions are pretty explicit. 15.1.C is explicit (and absolutely fundamental).

A caution on the definition of coherent sheaf: in many sources, a coherent sheaf is erroneously defined as the same as a finite type sheaf. But this is true for a locally Noetherian scheme. Still, even if you are a Noetherian person, it is worth seeing the difference, as knowing correct hypotheses makes remember proofs easier.

As always, I have some questions for you.

(a) 14.1.B-F were found to be needlessly hard in an earlier version. Are they now do-able?

(b) If you haven’t seen Noetherian modules, then you should try the exercises 14.6.B-E (or at least some subset). And if you do, please let me know how challenging you found them (privately by email, if you prefer).

For experts.

Presumably the dual of a free module is always free, even in the infinite case (14.1.C), but I haven’t thought about this.

Does a vector bundle have “transition functions”, or should it be “transition matrices”?

I want to mention the statement (Defn 14.1.4) that if X is a locally ringed space with O-modules F and G with F \otimes G \cong O, then F is invertible. Because this doesn’t get used, I just want a reference. I’m currently referring to

mathoverflow, but is there a more traditional reference (a book) that someone knows offhand?

On a related note, I’d like to state for the record that I don’t like the phrase “invertible sheaf”, but I guess it is easier to say that “rank 1 locally free sheaf”.

14.3.F: is there an easy example of a quasicompact but not quasiseparated morphism of schemes such that the pushforward of a quasicoherent sheaf is not quasicoherent? This is quite unimportant — I’m just curious.

I did not realize until it was pointed out to me by Kiran Kedlaya (via Andrew Critch’s MathOverflow question) that finitely presented modules are “always finitely presented” — if M is finitely presented, then any surjection has finitely generated kernel. This solves a question I’d wondered about for some time, and makes the notion of a “finitely presented quasicoherent sheaf” a reasonable one. (Kiran has set me straight on many things in the past…)

I introduce Noetherian conditions for modules in 14.6.2. (I could have include these facts where we meet Noetherian rings for the first time, but I prefer to introduce commutative algebra as it is needed, in bite-size chunks.) Is the notion of a Noetherian module ever interesting or useful over a non-Noetherian ring? If there are no reasonably important examples, I want to tell the reader.

What is the “right” reference for a proof of Oka’s theorem (14.6)? I don’t just want a reference to a statement — I want the best proof. (Some possibilities: (i) Several complex variables with connections to algebraic geometry and Lie groups, by Joseph L. Taylor section 9.2. (ii) Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973. (iii) Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.) Update Feb. 5 2011: Brian Conrad suggests Grauert and Remmert’s book (“the second best book ever”) on Coherent Analytic Sheaves, section 2.5. Thanks Brian!

If anyone is used to the notion of “support” of sheaves (14.7.C) in other situations: I define it as the points where the stalk is nonzero. I vaguely remember hearing that in analytic situations, people would take the closure of that locus. Can anyone confirm or deny that rumor?

Fun remark on 14.8: Brian Conrad gave me a short (2/3 page easy) argument that the sheaf of smooth functions is not coherent over itself. He also gave me an example (requiring a reference to Bosch-Lutkebohmert’s “Formal Rigid Geometry I”) of a non-Noetherian (important) ring that is coherent over itself. (Update March 8, 2011: I have posted it here, see this comment.)

I decided not to include this additional material in the double-starred section 14.8, but I’m tempted to include the example of smooth functions, which helps explain why smooth geometry is so different from real-analytic, complex-analytic, algebraic, or rigid-analytic geometry. (But I’m intending to take a hard line on what “extra” material gets included, because there is a potentially infinite amount of extra material.)

Notational changes coming (unless someone kicks and screams)

Update Feb. 10, 2011: I’ve moved my proposed notational changes (“[1:2:3]” and “open embedding”) to the post on Notation, so please comment there. Update Mar. 25, 2012: this change has been implemented some time ago!