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 HĂ¶rmander. 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!

January 30, 2011 at 3:46 pm

Dear Ravi,

My view is that when the support of a sheaf is not automatically closed (e.g. because the sheaf is locally f.g.) that support is not such a good notion. (Basically, I think that the support of a section, which is automatically closed, is the more basic concept.) But maybe this is idiosyncratic?

Also, there was

a recent MO question about coherent rings where some examples were given.

Regards,

Matt

February 8, 2011 at 8:00 am

Thanks Matt! I’ve added these to my private notes.

February 2, 2011 at 7:05 am

Ravi, a few more comments. At the start of chapter 14, where you mention that fractional ideals are examples of line bundles, it may be helpful to note somewhere that they have a bit more structure: a trivialization near (really at) the generic point.

[I’ve now added this as a remark: fractional ideals are the same as an invertible sheaf with a trivialization at the generic point. — R, March 29 2012]In 14.1.1, note that the matrix entries must be “nice” functions (as nice as you want the bundle to be, e.g., smooth or real-analytic or C^p, etc.). Likewise, the “section” in 14.1.1 must be of the same specified niceness; this can be confusing to beginners, and it underlies the sense in which a nontrivial holomorphic vector bundle can be topologically trivial (the associated sheaves in the two cases are over different sheaves of rings, etc.). Geometrically, that corresponds to viewing a manifold of some level of “niceness” as a manifold with somewhat less “niceness” (e.g., holomorphic viewed as real-analytic, or smooth viewed as C^3, etc.).

For Exercise 14.1.J, this is really a topic in algebra, not specific to number theory, so perhaps flesh out more that what matters is the Dedekind condition, not so much the case of global fields (though that example is obviously important). In particular, the phrase “ring of integers of global fields” is a bit problematic in positive characteristic since the there’s no canonical point at infinity (so the projective curve plays the role analogous to the ring of integers in a number fields) — perhaps say “rings of $S$-integers” in global fields instead (which is good for geometric intuition: passing to a punctured curve — this also connects up nicely with Remark 14.1.8, since finiteness of the class group in arithmetic cases shows that the failed intuition in 14.1.8 actually does work for rings of integers of number fields and smooth projective curves over *finite* fields).

[I’ve now changed this to “(This discussion applies to any Dedekind domain.)” — R.]At the end of 14.1.9, nonzero locally free sheaves of finite rank are never coherent when the structure sheaf is not coherent over itself.

[Good point. Now fixed. -R]In 14.3.4, note that the distinguished affine base is not stable under fiber products when the scheme is not quasi-separated.

[I think this is okay, with the definition I give. -R]The definition of a Grothendieck topology doesn’t actually require fiber products (there’s a notion of “sieve” used in general — related to the filtering aspect which you noted for the distinguished affine base), though in most interesting cases the category admits fiber products — then the definition agrees with the one your describe.

[I now clarify that I’m following the definition of the stacks project, but that there are other useful definitions in the literature. I don’t want to get into details of sites, as it is distracting to the learner at this point. -R]For “topos”, clarify you mean the category of sheaves of *sets* (not sheaves of abelian groups, etc.).

[Done. -R]On p. 290, your conditions (i), (ii), (iii) are *sufficient* for checking a subcategory is abelian, but (contrary to what is suggested by the wording there) they are not *necessary*.

[Good point — done. -R]This comes up when considering forgetful functors that turn out to be fully faithful, such as from various categories of “filtered” vector spaces (satisfying some strong conditions) to the category of vector spaces. No doubt there are simpler examples that one can give.The sentence at the end of 14.4 may benefit from some clarification, but we can discuss it in person. For the first sentence of 14.5, insert somewhere “which commutes with localization” (e.g., infinite direct products are problematic too, like general Hom’s).

[Clarified. -R]For Exercise 14.5.D, the constructions can be given for an arbitrary O_X-module, and then be proved to be q-coh when F is (which might clarify the analysis a bit).

[Duly noted in the notes. -R.]Near the bottom of p. 296, Oka’s theorem doesn’t require “complex-analytic spaces” in its statement (though they are useful for the inductive structure of the proof): the content is that the sheaf of holomorphic functions on C^n is coherent.[Now clarified. -R.]Finally, in 15.1, when setting up O(1) on projective space, perhaps approach it through the task of solving a concrete “moduli problem”; this would also clarify where the formulas defining O(1) come from and why it is a basic object of interest.

March 30, 2012 at 10:13 am

I’ve finally responded to Brian’s comments (in italics, embedded in his comments). Thanks Brian!

February 2, 2011 at 6:21 am

Ravi, concerning the “finitely presented ==> always finitely presented” issue, the same comes up when defining the concept of (locally) finite presentation for algebras (and so then also morphisms of schemes), especially for fpqc descent in connection with these properties (and stability under composition) as well as relating the module and algebra notions in the case of a module-finite map of rings. See EGA IV_1 section 1.4 or so (after a very long Chapter 0!) for an elegant and systematic discussion of these matters.

February 8, 2011 at 8:03 am

Thanks Brian!

February 8, 2011 at 4:18 pm

In your Definition 14.6.4 you have defined both “finitely generated” and “finite type” quasicoherent sheaves as the same thing (!).

In 14.7.D there seems to be an isomorphism symbol missing over the last arrow.

February 9, 2011 at 8:37 am

In 14.6.4 — thanks! I meant the second of the three definitions to be “finitely presented” — a quasicoherent sheaf F is *finitely presented* if for every affine open Spec A, F(Spec A) is a finitely presented A-module. That is now fixed.

And you are also right about 14.7.D — also now fixed.

February 8, 2011 at 6:12 pm

In 14.7.I I do not understand the remark “Thus finite type sheaves are locally free on a dense set”. Is the reduced hypothesis still needed here? How does it work?

February 9, 2011 at 8:52 am

Yes, reduced hypotheses are still needed. I’ll explain it in the case where X is integral. The rank is uppersemicontinuous, so it obtains its minimum on a dense open set. (We need the fact that the rank function maps to the nonnegative integers.) I’ve added the integral case to the problem. That integrality hypothesis can be relaxed easily.

I think the following argument works for any reduced X. On X, define open sets U_i inductively as follows. U_{-1} is empty. U_i is the subset of X-closure(U_0 \cup … \cup U_{i-1}) where the rank of F is i. Then inductively, the rank of F is greater than i outside of U_i. Furthermore, any point p of X where F has rank i is either in the closure of U_{i-1}, or is in U_i. I think that does it, but I’ve typed this without a lot of thought, so please let me know if I’ve screwed up.

February 14, 2011 at 9:23 pm

Hi Ravi,

I just want to make a comment on the word `immersion’.

I believe that `f:X\to Y is an immersion’ means `injective and induces an epi. between O_Y\to f_*O_X’. In the differential-geometric context, it happens that the above condition coincides with the daily life definition of an injective immersion (using tangent or cotangent maps).

The usual definition only makes sense in some special cases, while the sheaf-theoretic definition of immersion makes sense for any ringed space.

(Note in differential geometry to say a smooth map is a locally closed immersion is equivalent to say it is an embedding, i.e., an immersion that is homeomorphic onto its image. So an/a open/closed immersion is tautologically an embedding.)

February 22, 2011 at 1:36 pm

Thanks Tony! It sounds like you’re not quite expressing an opinion one way or another, but just telling people things that can help them form their own opinion. I’ll add a link to this from the notation page.

March 8, 2011 at 10:08 am

Here is Brian Conrad’s short argument that the sheaf of smooth functions is not coherent over itself.

March 8, 2011 at 10:22 am

[…] 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 […]

May 29, 2011 at 11:43 am

Hi Ravi

In the paragraph above ex 15.2.S (in the latest notes), you say that

A = k[x,y,z,w_1,…,w_{n-3}]/(x^2+y^2-z^2) is a UFD. However, ex 15.2 shows that the Weil class group of Spec A has order 2 (at least in the case n = 3).

March 25, 2012 at 5:38 pm

Late response: I think this is now okay (although admittedly it’s almost a year after you wrote your comment). I’m not sure at what point this got fixed.

July 14, 2011 at 8:00 am

Exercise 14.7.A (in both the January 29 2011 version and the June 27 2011 version) states that if F is a coherent sheaf and G is a quasicoherent sheaf, then sheaf Hom(F,G) is a quasicoherent sheaf. Does anyone have an example of quasicoherent sheaves F and G on a scheme X such that sheafHom (as O_X-modules) from F to G is *not* quasicoherent?

July 14, 2011 at 10:49 am

Let A be a UFD, and let be a non-zero non-unit. Define A-modules M and N by , and N=A. As you point out in the notes, for sheaf Hom to be quasicoherent, the map must be an isomorphism. However, since , we have a map , and this map is not in the image of .

July 15, 2011 at 9:58 am

Pieter, this is terrific. As with all good examples, once you told it to me, it became completely natural, and I won’t forget it. To avoid distraction, I’ll let be a discrete valuation ring and its uniformizer. I’ll write instead of (because that briefly confused me). Thanks!

May 12, 2012 at 4:21 pm

Dear Ravi,

I am reading your lecture notes recently. Could you help me finding the mistake? Since I don’t use reduced property in the proof.

14.7.J (Using the same notation and Nakayama’s lemma as in the notes) Note that m_1,…m_n generate M as A module, then for any prime P, they generate M\tensor A_P/PA_P as A_P/PA_P vector space.(I doubt this argument is wrong..) By the dimension issue, m_1,…,m_n is actually basis at each stalk. Now if ker \phi is not 0, and (r_1,..,r_n) is in the kernel. Localize at any prime P shows that r_i=0 in A_P for any i, which implies that Ann(r_i) is not contained in any primes, hence must be the entire ring A. Therefore r_i=0 for all i, which implies that ker \phi=0

Thank you!

May 13, 2012 at 8:44 am

I figure it out finally. r_i=0 in A_P/PA_P, not in A_P, which means r_i belongs to P. Then r_i is in nilradical, hence is zero(reduced hypothesis is necessary here).

May 16, 2012 at 1:18 pm

Excellent! Please let me know if you catch anything else.