**The second post is the September 10, 2010 version here.** This fortnight’s reading is until the end of 4.1, although I’ve included 4.2 in case you want to read ahead.

Corrections promised after the last post are included.

**General comments.** Sheaves are fundamental, and are harder than we (as teachers) remember. It is tempting to get through them as quickly as possible to get to the real stuff. This is dangerous — later in the material, it becomes clear that many students don’t think fluently about them. There is another advantage of taking time to learn about sheaves: students who leave a class before getting deeply into schemes take something valuable away with them. (Similarly, if they stick with the class a little longer, they begin to absorb the valuable lesson that a geometric space is well thought of as a topological space with a sheaf of functions, and that we really understand spaces via their functions.)

**For learners.** Read 3.1 to 4.1. Take your time to absorb the idea of sheaves. As with category theory, it takes a little time to learn how to think in this way. The real test of your understanding is your ability to solve problems.

If you would like a “problem set”, here is hard one: 3.2.F (morphisms glue), 3.2.H (pushforward sheaf), 3.3.B (sheaf Hom), 3.3.E (cokernel presheaf), 3.4.E (isomorphisms are determined by stalks), 3.4.J (construction of sheafification by compatible germs), 3.4.O (on surjection of sheaves), 3.5.D (stalkification is exact), 3.5.E (taking global sections is left-exact), 3.5.I (tensor products), 3.6.B (adjunction of inverse image and direct image), 3.6.G (support), 3.7.B (on sheaves on a base). But you should do some others as well (and perhaps instead). If you can do almost all of these, plus some more you find interesting, you are in excellent shape.

And if you have even a little experience with manifolds, try some of the problems in 4.1.

Something else you should do: decide which exercises you like best, and which you like least, and why. Don’t just be told what to do — develop your own taste.

**For experts.**

Here are things that are tricky the first time you learn them: how even to think about a sheaf, a notion which contains so much information; the fact that you can check various things on stalk; sheafification, and how to work with it; the inverse image sheaf; sheaves on a base.

I first try to convince them that they already know the idea of a sheaf, in that they know how continuous (or differentiable, etc.) functions behave.

Some smaller less important comments and questions.

(a) One axiom sometimes used in the definition of a sheaf is that F(\emptyset) is the final object in the category. This follows from the other axioms, once the empty product is defined appropriately.

(b) What is reasonable notation for a skyscraper sheaf (say with set S, at point p in the space X)? I’m using S_p, but this conflicts with stalk notation.

(c) I don’t like the phrase “constant sheaf”, because sections are just locally constant. I thus prefer “locally constant sheaf”. I realize that this conflicts with the sheaf of sections of a finite etale cover, but this doesn’t bother me; both are locally constant sheaves, just in different topologies. Is it so terrible to use the phrase “locally constant sheaf” instead of the conventional “constant” sheaf?

(d) The cokernel sheaf is often just defined. But one needs to know that it deserves this name, i.e. satisfies the universal property of the cokernel. This is a bit annoying, but it is also a bit enlightening. Similarly for the image sheaf.

(e) I have no great insightful explanation of why the inverse image (defined as a colimit) should be adjoint to pushforward.

(f) The Grothendieck quote at the start of Ch. 5 is Colin McLarty’s translation.

(Note to self: 14 comments read. No loose ends.)

September 10, 2010 at 4:48 pm

Sorry, but I think it’s got to be “constant sheaf” for a uh… constant sheaf. You can say “constant presheaf with value S”, and “constant sheaf associated to S” to indicate that you know the constant sheaf does not have the same value on every open, whereas this does hold for the constant presheaf.

On the stacks project blog we had a discussion of the skyscraper sheaves (n a very general setting). Notations you could use: p_*S if you think of a point p also as a map, or the more elaborate i_{p, *}S where i_p : {p} —> X denotes the inclusion of the singleton space {p} in X.

About inverse image: The presheaf inverse image (defined using colimits as in 3.6.2 is the adjoint of pushforward on presheaves. And this is some incredible generality for “presheaves of sets on categories” due to Kan according to Artin’s notes on Grothendieck topologies.

More later.

September 14, 2010 at 9:14 am

I like the two possibilities of the skyscraper sheaves. I think I’ll go with the second one (at some later edit). About the constant sheaf — I’d feared and expected this answer. And on the inverse image: I buy/understand the argument, but feel like I don’t fully understand it in my gut; but perhaps I’m asking too much.

September 28, 2010 at 11:52 am

Dear Ravi,

I second Johan; it has to be “constant” sheaf.

Cheers,

Matt

January 4, 2011 at 1:44 pm

Okay, the constant sheaf it will be…

September 11, 2010 at 7:29 am

Minor typo: on p. 60, 3.2.5, has reference to unlinked section “germofdiffun”.

[Thanks, fixed (in next version)! — R]September 11, 2010 at 7:23 pm

A bevy of minor errata (up to about page 70). More later.

A general comment : it would be nice to have the bibliography. Several times I clicked on a bibliographic entry and ended up at the end of the book…

3.2.3 : You are defining the stalk of a presheaf, not a sheaf.

3.2.5 : You refer to a nonexistant section germofdifffun

3.2.6 : In the identity axiom, it would make sense to include the phrase “for all $i$”.

p. 61, line 4 : I don’t think there should be a comma after experts.

3.2.F : It would be more precise to say “…we associate the set of continuous maps of $U$ to $Y$…”.

3.2.G : Like in my previous comment, I’d replace “maps $s$ to $Y$” with “maps $s : U \rightarrow Y$”

3.3.B : In the 2nd line of the third paragraph, the phrase “is the is” is a typo.

3.3.D : The hint in this exercise looks like it is trying to define the restriction map for the cokernel instead of the kernel.

3.3.I : This is phrased in a funny way, as you assert that $\mathcal{F}$ is not a sheaf during the setup, and then the problem is to prove that $\mathcal{F}$ is not a sheaf.

3.4.D : This is phrased in a slightly imprecise way. It might be easier to parse if you spelled it out : if $\phi_1$ and $\phi_2$ are morphisms from $\mathcal{F}$ to $\mathcal{G}$ that induce the same maps on each stalk, then $\phi_1 = \phi_2$.

3.4.3.1 : The spacing around this diagram seems a bit excessive.

3.4.6 : The display is confusing. First, it seems jarring to have “for all $x \in U$, there exists some $x \in V \subset U$” — this makes it sound like “for all $x$, there exists some $x$”. Second, it seems nonstandard to have two “:” symbols in a set definition. I suppose that one reads them as “such that”, but I think the tradition is to have only one (at the very least, I did a double-take and had to think a bit to figure out what was going on).

September 12, 2010 at 3:22 am

Ha! I didn’t realize the pdf had internal links until you just mentioned clicking on a reference! I am so used to having clickable entries highlighted by color… Great!

September 12, 2010 at 1:22 pm

Thanks! With the possible exception of the comments below, I’ve made all the changes.

Bibliography: Sorry, I forgot it, and have put it back — just click on the same link; the version with bibliography is there. (As you might guess, for each post, I take a big pdf, then remove the middle chunk of “work in progress”.)

3.4.3.1 That’s latex’s fault, and I’m not worrying about formatting issues right now (as I may want to change conventions later). Sorry!

3.4.6 I’ve now turned most of the offending symbols into words, and I hope it scans better now.

September 12, 2010 at 3:50 pm

Dear professor Vakil,

I am a student from Korea who is planning to study algebraic geometry with the book ‘Hartshorne’ and this lecture note.

I found some typos on the first note, but since it seemed to me that numerous comments on it might be enough to check all the typos, I did not post any comments on it. Let me tell you what I think remaining typos are in this version.

2.2.18. The the ith -> The ith

2.2.21. 3rd line, an essential away -> an essential way

Warning below 2.3.C., even if A is an integral domain, for that injection, S should not contain 0.

2.4.1 line -5, g_j = F(m) g_j -> g_j = F(m) g_i

2.5 line 1, an very useful -> a very useful

2.6.H (b), except for the parenthesis part, (b) is the same as (a).

Thank you for offering this wonderful lecture note. I will study its detail and I hope I can give you some feedback every time.

September 14, 2010 at 9:20 am

Thanks, these are all great (and all fixed)! I hope people reading the current version aren’t thrown by the 2.6.H(b) error; that’s a surprisingly useful exercise.

September 12, 2010 at 5:38 pm

OK, some more small comments, this time from page 70 until the end of 4.1.

3.4.N : Since c obviously implies b, isn’t it enough to prove a==>c? Plus I have to admit that I couldn’t figure out how to make your hint work for a==>b…

p. 70, right after 3.4.O : I’m far from an expert on these kinds of things (and thus about to expose my ignorance), but are you really sure that 3.4.N works for sheaves with values in ANY category? At least to make the proof you sketched in 3.4.N of a==>c work, you need to have enough information to construct an “indicator sheaf” with values in that category. For this, it appears to me that you need eg an initial object (like the empty set) plus a free object on a 1-element set (which also requires a forgetful functor from your category to the category of sets).

For 3.4.O, it looks like you need a final object in your category so you can define a skyscraper sheaf, but I guess it is implicit in the definition of a sheaf that the target category has a final object.

3.5.E : Maybe more precise to say “left-exactness of the section functor” since the exercise deals with more than just global sections.

3.6.3 : It seems to me that the phrase “stalk of $\mathcal{G}_p$” would be better written “stalk $\mathcal{G}_p$”. Also, on the second line the phrase “stalk $\mathcal{G}$ to a set $\mathcal{S}$” should probably be “maps from the stalk $\mathcal{G}_p$ to a set $\mathcal{S}$”.

3.6.E : Isn’t this a special case of 3.6.C?

Sentence before 3.7.B : Should read “I next claim that if $B$ is in our base..”.

p. 76, next to last line : “you use by induction” should read “you cannot use induction”.

September 14, 2010 at 9:31 am

Thanks again! I’ve implemented everything but 3.4.N (which I have to think about — but I won’t forget!).

p. 70, I’ve added the weasel word “reasonable”.

3.6.E also follows from the remark before 3.6.C, so I’ve cut (which will change the names of some later exercises in later versions — caution!).

September 15, 2010 at 10:08 am

I’ve now also removed the hint for 3.4.N that didn’t help you, especially as it didn’t stop you from doing the problem.

September 13, 2010 at 1:18 pm

For exercise 3.2.B part (b), it is asked to show that the collection of holomorphic functions admitting a holomorphic square root on C with the usual topology, forms a presheaf but not a sheaf.

[I assume that the “usual topology” means what you later call the “classical topology”, aka the analytic topology.]

It is obvious this is a presheaf, but I believe that you must have meant to say C\{0}, the nonzero complex numbers, as otherwise it is a sheaf.

As stated, there is no obstruction to gluing since the only possible obstructions are branch points (and there cannot be any, since every point in the cover of C is contained in an open neighborhood of C on which our function is holomorphic).

September 14, 2010 at 9:23 am

This is why I like this exercise! I think it is correct as stated, and you’ve correctly described the crux of the matter — C\{0} is an open subset of C, so failure of gluability for *this* open subset makes this fail to be a sheaf. (Please let me know if I should say more.)

September 16, 2010 at 3:27 pm

Ah, yes, you’re right of course.

My remark above is then the observation that the global sections obey the gluing property, but sections on a proper subset do not. (Which I find interesting.)

Also, it occurred to me after posting my first comment that the question’s wording is mildly ambiguous: I originally interpreted the presheaf in question to have F(U) be the collection of entire functions which possess a holomorphic square root on U. But it is also possible to interpret the wording as saying “holomorphic on U with a holomorphic square root on U” which would include lots of globally meromorphic functions. (Of course, this doesn’t affect the result.)

—

Miscellaneously, “surprising” is misspelled in the first line of chapter 3 in the notes, and in the second line of 3.6.H(d), you have an “oh” where you wanted “\oh”. You might also mention how to pronounce f_!, since you did so for O_X.

September 17, 2010 at 6:51 am

Thanks again. I’ve fixed everything after the “—” too.

September 13, 2010 at 3:13 pm

Another list of (mainly typographical) comments:

-page 43. – In the sixth line there should be ‘(…) colimits are right exact’ instead of ‘(…) colimits are left exact’

-3.2.9. – In the definition of skyscraper sheaf of abelian groups there should be ‘S_{p}(U) = {0} if p\not\in U’ instead of ‘p\in U’

-3.2.E (a) – Does constant presheaf really satisfy the identity axiom? Consider for example section over the empty set and empty cover of empty set – then from identity axiom we obtain that S = \mathcal{F}(\varnothing) has at most one element…

Or should we set \mathcal{F}(U) := S unless U is empty, and define \mathcal{F}(\varnothing) as a singleton?

-3.3.B – The definition of sheaf Hom says: ‘Hom(\mathcal{F},\mathcal{G})(U) := Hom(\mathcal{F}|_{U}, \mathcal{G}|_{U})’. This looks a bit self-referential, maybe it will be better to write ‘Mor(\mathcal{F}|_{U},\mathcal{G}|_{U})’ on the RHS?

-4.2. Example 6, eighth line – there should be ‘Spec \mathbb{F}_{p}[x]’ instead of ‘Spec \mathbb{F}_{p}’

September 15, 2010 at 10:23 am

Thanks! There are some subtle issues that you bring up. In 3.2.E(a), you are right: in my definition, I’ve said that the sections over the empty set are S, which immediately means it can’t be a sheaf. And if you fix it by making the sections over the empty set to be the final object in the category of Sets (any one-element set), then you get the contradiction I was hoping readers would get. So I could define the sections over the empty set to be {e}, and make the question uglier and seemingly more contrived. Or I could leave it as is, and let the reader come to their own conclusions. Or I could take it out of an exercise, and observe that it is not a sheaf for silly reasons (F(\emptyset) is wrong), and if you patch it, it becomes not a sheaf for less silly reasons. I’m leaning toward the latter, because this might be least confusing to the reader.

In 3.3(b): it’s not self-referential as written, but you’re technically right: maps of sets are called Morphisms, not Homomorphisms. And in fact “sheaf Hom” really should be called “sheaf Mor” when you have a sheaf with values not in an additive category. I don’t want readers to get thrown by this, so I’ve edited it a bit.

So thanks for catching these subtle issues…

September 16, 2010 at 8:09 am

3.2.E(a) Whoops! I now see why you asked about the identity axiom — because it is in the question! This is now fixed; I’ve removed the presheaf part from the exercise, and made it text, and hopefully made it correct.

September 17, 2010 at 6:57 am

Whoops again! I now see what you means in 3.3(b) (see my response to Tom below). I’ve now fixed it. Maybe you weren’t bothered at all by “Mor” vs. “Hom” at all, and I should just undo that… but I haven’t yet…

If you think I’m misunderstanding you, please feel free to follow up…

September 14, 2010 at 9:13 am

ad (c): What would you call a sheaf which is only locally a “locally constant sheaf”, i.e. a sheaf of sections of a cover?

September 14, 2010 at 6:16 pm

I see your point: this deserves a name, and the meaning of constant sheaf is that the “espace etale” is constant, and there are important cases in which the espace etale is locally constant. Okay, I’m satisfied with the notion of constant sheaf now.

September 16, 2010 at 8:06 am

One more thing on constant sheaves. Which sounds better (or *is* better)?

(a) constant sheaf associated to Z

(b) constant sheaf corresponding to Z

(c) constant sheaf with values in Z

(d) constant sheaf with value Z.

I’m currently going with (a) because Johan used that in his first comment. He also used (d) but it sounds worse to me. (c) sounds pretty good to me.

March 18, 2011 at 8:58 am

I personally think that (c) sounds best. It emphasizes the interpretation of locally constant functions with values in Z, which seems like the easiest way to understand the constant sheaf.

March 20, 2011 at 12:27 pm

Great, thanks — does anyone else want to weigh in?

September 16, 2010 at 9:54 am

These comments are on the September 10 version of the notes. (I’ve tried to remove those comments that have already been noted above, but I may not have gotten them all.) If it’s relevant, I’m a topologist, not an algebraic geometer.

2.6.H: the second part of the theorem is a duplicate of the first part

2.6.10: boil down the statement > boil down to the statement

2.6.10, left-adjoints are RIGHT exact

3.2.9: “subscript of a point” would be clearer as “subscript -_p” or something like that

3.2.12: you haven’t said that O_X is the sheaf of smooth functions

3.3.B: The notation here is quite confusing, since in the same line (the definition of the sheaf Hom(F,G)!) you use Hom(F,G) to mean an abelian group, and a sheaf of abelian groups. as a result “implicit in this is the fact that Hom(F,G) is an abelian group” looks like a typo at first glance

3.3.3: since it’s later important that the cokernel of a map of presheaves is not a sheaf, it might be better to define it explicitly in the text. (removing the possiblity of students worrying whether it’s their definition that’s wrong)

3.4.2: I think U_p is confusing notation for an open neighborhood of p.

3.5.E: it is not clear enough that “exact” in “the global sections functor is not exact” is stronger than “exact” in “Show that [this complex] is exact” (since that complex is only 0 -> A -> B -> C)

3.6.A: I think you mean Exercise 3.6.B (not 3.6.3)

4.1.2: a smooth morphism corresponds to a submersion, not a smooth function

4.1.3: Riemann surface to me means a complex curve with the analytic topology

4.2.1: “in different places” is ambiguous; “in different groups” would be better. (also maybe an example where the values aren’t both 1 would be better, even though this is mathematically irrelevant)

4.2.1: Example 3, “after reading this precise in Chapter 13”

4.2.C: I am suprised you don’t emphasize this exercise more. I would rank it as one of the most important exercises in this chapter (because it is so commonly missed). Similarly for 4.2.N.

4.2.H: the paragraph beginning “In general, inverting zero-divisors”: it’s hard to tell where this fits in the logical flow (it’s a caveat attached to the previous paragraph, but this is not immediately obvious). also, you say things behave oddly, but geometrically there is nothing odd, which could be confusing.

4.2.L: interpret the corresponding map of rings AS given by

September 17, 2010 at 6:55 am

Thanks! It’s relevant that you are topologist in that I don’t want you to be needlessly thrown by the exposition. I’ve made changes in response to everything you’ve said, with the possible exceptions of the following.

3.3.B I’ve just realized what an earlier comment was asking about! I’d changed the tex macro for Sheaf hom in response to a comment of Johan’s, and wanted it to be all in calligraphic font, but it didn’t do that, so the result was nonsense. I’ve fixed that.

3.3.3 I don’t follow you here; do you mean “don’t just say similarly, say what it is”? I’m pretty sure you mean that.

3.4.2 I agree that U_p is imperfect, but I can’t think of anything better (as this open set is supposed to depend on p). (I think s’_p is worse.) If you can think of anything better, please let me know!

4.2.C and 4.2.N: Okay, I’m convinced. I feel that way too; and if a topologist agrees, then I’m happy. They are now labeled “important”.

September 17, 2010 at 4:57 am

Thanks for this week’s update. A typo that hasn’t been mentioned yet is in

3.2.4 where you have \mathcal{O}(U) which should be \mathcal{F}(U) in my opinion.

It was a really helpful typo though, because I stumbled over the section over and over and finally went back to do exercise 2.4.C again. The first time I only checked for commuting diagrams and the universal property, but of course the heart of the matter really lies in the definition of the equivalence relation. Maybe you could point this out to a reader, who reads similarily sloppily, by asking if he can state the relation explicitly (..are in relation if and only if..) and ask him why it is an equivalence relation.

September 17, 2010 at 7:00 am

Thanks, the typos if fixed! And more important: can you say a little more about what would be helpful to say in 2.4.C? I fully agree this is a very central exercise to understand not just in one’s head but also in one’s heart.

September 17, 2010 at 8:32 am

The way I understand your formulation (and I hope I got it right) is that one should show that the quotient set of the disjoint union over all A_i by the equivalence relation, induced by forcing a_i to be equivalent to a_j if there is any f:A_i->A_j such that f(a_i)=a_j, is the colimit of the diagram.

The condition above is not yet an equivalence relation because it does not satisfy symmetry. So the “real” equivalence relation is: a_i equivalent to a_j if there are f:A_i->A_k, g:A_j->A_k such that f(a_i)=g(a_j). Now one still has to check that transitivity holds (there one finally sees where the filtered axioms chip in).

My problem was that when first comming across the exercise I wasn’t too much bothered by the equivalence relation and just checked the diagrams. I have two suggestions what you could do:

A. Just add something like “(Hint: Verify that ~ is indeed an equivalence relation. How can it be stated explicitly?)”

or

B. Explain the relation in more detail, like I have done above, and ask the reader directly to verify that this is an equivalence relation.

Suggestion A would be more compatible with your philosophy that if most people can figure out what is meant, you like to keep it simple. Suggestion B is the more direct approach.

I hope I didn’t write any nonsense, I am starting to get the feeling that I maybe missed the point of the exercise afterall.

September 18, 2010 at 2:02 pm

Hi Felix,

No, you didn’t write any nonsense, and it sounds like you got precisely the point of the exercise. I agree with what you wrote. I went with your A (which is indeed in keeping with the philosophy of the notes), plus I explicitly say what the equivalence relation is, precisely what you said.

September 20, 2010 at 7:33 am

Hi Ravi, just reporting some typos:

first line of 2.2.1: ” a set of maps, or morphisms (or arrows or maps)” I think you want to delete one occurrence of the word “maps” there.

also in 2.2.1, first line of page 18: “there is only on morphism” -> “there is only one morphism”

last bit in first paragraph of 2.3.9: “we now taken” -> “we have now taken”

in 2.5.G, do you really want a forgetful functor S^-1 M -> M? I thought forgetful functors should send an object to “essentially itself”, only viewed in a different category.

the spacing in the sixth complex in 2.6.6 looks a bit weird.

2.6.7: “many subtle facts will be come obvious” -> become

opening paragraph of chapter 3: “the reason for the name is will be…” you don’t want “is” there.

3.1.1.1: I think you want m_p instead of m

and that’s as far as I got yesterday đŸ™‚ (re-reading, since I learnt this part last year)

September 21, 2010 at 3:44 pm

Thanks Amy, all fixed!

October 27, 2010 at 3:24 am

Here is an example of a presheaf which obeys gluing but not identity, and may be more natural to a beginner than the sheaf of sheaves:

The “presheaf of k-cochains”. Let X be a topological space, and let Delta be the k dimensional simplicial complex. For any open set U in X, let C_k(U) be the set of all continuous maps from Delta to U (chains). Let C^k(U) be the abelian group of integral valued functions on C_k(U).

The C^k(U) form a presheaf, by the obvious restriction, but sections do not glue uniquely. This is a pain when attempting to fit the standard exposition of simplicial cohomology into the sheaf theoretic framework.

January 4, 2011 at 1:44 pm

This is enlightening to me. Will it be enlightening to other readers (enough to add length)? Can someone else express an opinion?

December 12, 2010 at 6:23 am

Peter Diao pointed out that the espace etale definition (notably the topology) was wrong. It should now be patched (in the first version appearing after Dec. 12, 2010). Thanks Peter!

February 13, 2011 at 12:07 pm

I’d had a note to myself to give a hint to make the adjunction of pushforward and inverse image as simple an exercise as possible. Dima Trushin sent me a hint that I found better than the one I had intended, and I have now included it. It will be in the next posting (hopefully Feb. 19, 2011), and it is also here (Exercise 3.6.B). I wanted to note here that this hint came from him — thanks Dima!

September 8, 2011 at 12:41 pm

Since January, I’ve intended to post Dima Trushin’s original explanation he sent me. I’m finally doing it: it is here.

March 18, 2011 at 12:01 pm

Dear Ravi,

Is there any particular reason why you use the term “espace etale” and not simply “etale space” (which I assume is what it means)?

March 20, 2011 at 12:30 pm

Purely convention. There’s a possible reason, although I should admit I’m not sure how I feel about it: I’ve interpreted the accent on the last e in ‘etal’e as making the meaning “the ‘spread-out’ space”. I should also say that I’m not happy about the fact that etale is coming up in terminology in two ways are not the same (even if they are philosophically related).

March 21, 2011 at 3:07 am

Wouldn’t it make sense to refer to the “espace etale” simply as the space of sections of F? This is what the espace etale is engineered to be, after all; I could not imagine a more appropriate name. It also avoids the issue of the word “etale” arising in two different contexts.

March 21, 2011 at 7:44 am

This is an excellent suggestion, and I like it a lot. I intend to make this change, unless someone jumps in and argues otherwise, or has an even better idea. (Of course, I’d say “this is usually known as the espace etale”.)

Update May 11, 2011: change now made.

March 18, 2011 at 12:53 pm

Dear Ravi,

You mention in one paragraph of 3.3.1 that the natural map of the sheaf of differentiable functions into the sheaf of continuous functions is a “forgetful map”. This interpretation seems strange to me. I would say that this is not so much a forgetful map (we aren’t forgetting anything) as it is simply an inclusion (of a smaller set of functions into a bigger set).

March 20, 2011 at 12:32 pm

It’s forgetful in the sense that we are forgetting the differentiable structure, and considering it as only a continuous function. This is admittedly not an overwhelmingly convincing argument.

March 20, 2011 at 12:38 pm

I guess my pseudo-objection is that the term “forgetful” is typically used when we have some collection of data and we define a map by throwing out some of the data. In this case, if we didn’t know that these sheaves were being given to us as sheaves of functions (i.e. replace the sheaf of differentiable functions with a sheaf of sets isomorphic to it, and do the same for continuous functions), then we would probably not think that there was any forgetting going on.

March 20, 2011 at 12:42 pm

That’s very reasonable, and indeed I took this as your original objection. I’m not sold on this terminology, although I haven’t felt moved to word it differently (as I have with other conventional word choices I’m not thrilled by). If there’s a strong consensus toward changing this, I’d certainly consider it.

March 19, 2011 at 2:14 am

Dear Ravi,

It might be helpful to mention that the conclusion of exercise 3.4.D holds even when F is not a sheaf. The fact that a morphism F —> G is determined by what it does on stalks only uses the sheaf conditions on G.

So this exercise can be used to show the universal property of sheafification on the next page.

March 20, 2011 at 12:37 pm

That’s a nice point. I’ve made that change. (In fact, one needs only for G to be a separated sheaf, because that’s all that is needed in 3.4.A, but this is less important.)

March 20, 2011 at 7:40 pm

In fact, I think that the hypotheses of a number of exercises at the beginning of section 3.4 can be weakened. The payoff for doing this comes when you are asked to prove that a sheafification of an image presheaf is an image sheaf in Ex. 3.5.C; the exercises as originally stated don’t quite apply to the situation of that problem, but the versions with weakened hypotheses do.

March 21, 2011 at 7:41 am

Thanks — can you say which exercises and which hypotheses? (You may have already told me all this in the comments you’ve sent by email, which I’ve started to go through.)

March 21, 2011 at 9:18 am

3.4.E can be weakened to: F a sheaf, G separated. But this doesn’t make any later exercises simpler.

If you weaken 3.4.D to F arbitrary and G separated, you can weaken 3.4.N to: For any morphism of presheaves, if all maps of sections are injective, then all stalk maps are injective. Furthermore, if F is a separated presheaf and G is any presheaf, then all stalks being injective implies phi is a monomorphism in the category of presheaves.

This version of Ex. 3.4.N greatly shortens the solution of Ex. 3.5.C.

Finally, (b) => (a) of Ex. 3.4.O can be weakened to: If phi: F -> G is any morphism of presheaves in which all stalk maps are surjective, then for any separated presheaf H and morphisms psi_1,psi_2: G->H for which psi_1 phi = psi_2 phi, then psi_1=psi_2. But I don’t think this makes any later exercises simpler.

March 21, 2011 at 3:13 pm

Also, for Ex. 3.3.H, you can get away with F being a sheaf, but G only separated. I don’t think this helps later exercises, though.

March 24, 2011 at 8:46 am

Thanks! I’ve not said any more in cases where the increased generality has no pay-off. I’ve weakened the hypotheses on 3.4.D. I’ve made the extension to 3.4.N as you describe, in a parenthetical comment afterwards, pointing out that it can be helpful in 3.5.C. I’ve not actually checked this, but I know you’ve sent me a solution to 3.5.C, so I trust you!

March 20, 2011 at 6:42 am

Dear Prof. Vakil,

On page 61, example 3.2.8 should read “U is contained in X”, in the definition of the restriction of a sheaf.

March 20, 2011 at 12:41 pm

Thanks, fixed (in the next version)!

March 26, 2011 at 7:56 am

Also,

In example 3.2.8, I think you F|_U(V) = F(V) for all “open” V contained in U, instead of just for all V contained in U. Bu, I am not sure about this.

March 28, 2011 at 8:19 am

Fair enough. F(V) is only defined when V is open, but it’s good (and cheap) to be clear, so I’ve made the change.

(To everyone: you may notice that I’m responding to some comments quickly, and some old ones remain unanswered. I’m trying to at least stay current in the comments in a number of posts, and keep track of which comments still need replies. But there are a number of posts where I haven’t even started catching up. A number of comments, notably by D.H., Philsang, and others, remain unanswered, and I want to rectify this eventually!)

April 5, 2011 at 3:25 pm

Dear Prof. Vakil,

This is just a trivial question about notation.

(The question has to do with Exercise 3.2H on the Pushforward Sheaf or Direct Image Sheaf).

For any topological space X, let C_X denote the category whose objects are open subsets of X, and whose morphisms are inclusion maps.

Hence, any continuous map f:X—->Y gives us a natural functor f*:C_Y—–>C_X where an open U contained in Y is mapped to f^(-1)(U)= f*(U), and inclusion maps in C_Y are mapped to inclusion maps in C_X.

Thus, if F is a sheaf of sets on X (then F is just a contravariant functor from C_X to the category SET, with some additional structure encompassed by the identity and gluability axioms), exercise 3.2H really asks us to prove that the composite functor Ff*:C_Y—->SET is a sheaf of sets on Y.

I was wondering why the reverse notation f*F is used, instead of Ff*. Again, I apologize for this really silly question.

April 17, 2011 at 11:05 am

That’s an entertaining point — what you say is mathematically completely correct. The reason is just one of perspective. It is conventional not to think of a sheaf as an object in a category, rather than a functor from one category to another. So f_* is a functor from the category of sheaves on X to the category of sheaves on Y, and so one writes f_*(F), just as one always do with functors. So you aren’t “wrong”, just unconventional, in a literal sense. (But for better or for worse, if you wrote that in a mathematical paper, you would be “wrong” because you would be going against convention…)

April 24, 2011 at 11:40 am

Dear Prof. Vakil,

I have a doubt. Let X be a topological space, Z_ the constant sheaf on X associated to Z(the integers), F a sheaf of abelian groups on X. I managed to show that (X, Z_) is naturally a ringed space.

But, how is F naturally a Z_ -module, as in what is the natural module action?

I need to figure out a natural module action for every open U from Z_(U)*F(U)—>F(U), but I have not been able to figure one out yet.

Also, thank you for your previous comment.

April 24, 2011 at 11:50 am

Here is one possible hint. A section of Z over U is a locally constant function. If it is the constant 3, how does it act on an element of F(U)?

If it is the locally constant function that is 2 on one open and closed subset, and 3 on the complementary open and closed subset, how does it act?

An alternative approach: what is the action on the level of stalks?

November 19, 2011 at 11:30 pm

I noticed the following in your definition of sheafification on Pg. 72: When you define the set F^{Sh}(U) explicitly, you introduce the notation s_y, without really saying it stands for. Perhaps it is apparent from context as to what it means, but I was a little confused when I saw it the first time.

December 21, 2011 at 10:48 am

Thanks for pointing that out. To avoid that confusion, I have just added a phrase: “Here means the image of in the stalk at .”