*(Update May 15, 2013: at this point, to my amazement, all of my questions have been answered, although more good answers to the rest would be most welcome. I have learned a great deal of neat stuff I should have known long before by asking these questions!)*

A new version is now posted at **the usual place** (the Feb. 25, 2013 version). There are many small improvements and patches, but no important changes. Because I want to give you something new to look at, here is a newly added diagram of chapter dependencies. I was a little surprised by what it showed.

The “to-do” list of things to be worked on is now at about 100 items. With the exception of formatting, figures, the bibliography, and the index, I continue to be very interested in hearing of any suggestions or corrections you might have, no matter how small. We are clearly nearing the endgame.

**I now want to ask advice on a number of issues all at once. ** These are mostly small things, and are along the lines of “what is the best reference to point learners to on this topic”. Some of the questions are “unimportant”, in the sense that I doubt they will affect my exposition (although they may be important in some larger sense). For those questions relating to some particular part of the notes, I will give the section number. **Please feel free to respond by email or in the comments.** Here we go!

—

Peter Johnson strongly preferred using “fibre” instead of “fiber”. Does anyone else feel strongly?

1.4.1 Latex question: how do you get the \varprojlim subscript in the right place? *(answer here)*

1.6.12 (unimportant) Do right-exact functors always commute with colimits? (For example, commutes with direct sums, which is what we use, but that is easy to check directly.) *(answer here)*

5.4.M Can anyone get this exercise (that, basically, says that normality descends under finite field extensions)? I think it should be gettable, but not easy, but I’ve had clues that this is harder than I thought. I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) *(two positive response received so far, including this one)*

5.4.N is not a unique factorization domain, but its Spec can be covered with 2 (distinguished) affine subsets, each of which are Specs of UFD’s. Is there some good reference for this? (Presumably it becomes a UFD upon inverting either 2 or 3, but I can’t see why this is the case. And of course I don’t just want to know what is true; I’d like a reference for *why* it is true.) Added later: I should also have added, is there a well-loved reference that shows that the class group of is ? *(answer here) *

6.3.K A compact complex variety can have only one algebraic structure. What is a reference? (A number of sources mention this fact, but I want an actual proof.) On a related point, in 10.3: A variety over is proper if and only if it is compact in the “usual” topology. What is a reference? * (answer here and here)*

6.7 In this section, I mention the Schubert cell decomposition of the Grassmannian. The key idea is that any -dimensional subspace of (where is a field; and say , …, is the standard basis of ) has a canonical basis, where the first to appear in each basis element appears with coefficient 1, and that appears in no other basis element, and that special for that basis element is “to the right” of the of the previous one. Is there a standard name for this? (Normal form? Row-reduced echelon form?) Is there a good (fairly standard) reference for it? (Perhaps this gets too far into how linear algebra is taught in different countries, and I should just not give a reference, and instead give it as an exercise.) *(answer here, although I’m also happy to get more references)*

8.4.H Interesting fact: I almost wanted to say that effective Cartier divisors are the same as codimension 1 regular embeddings. But I could only show this in the locally Noetherian situation (or more generally, when the structure sheaf is coherent). The reason for the problem is that the definition of effective Cartier divisor is in terms of open subsets (for good reason), while the definition of regular embedding is in terms of stalks (for good reason), and getting from the latter to the former requires Nakayama. If you think I’m not giving the right definition of one of these two notions, please let me know! *(see here for an interesting follow-up, thanks to Laurent Moret-Bailly)*

9.1.7 Peter Johnson did not like my use of the phrase “open subfunctor” in 9.1.7. Is anyone else bothered? How seriously? *(current plan after discussing with Peter: leave as is)*

9.4.E Can anyone get this exercise (that, basically, says the product of integral varieties over an algebraically closed field is also an integral variety)? I think it should be gettable, but no one I know has gotten it (possibly because I haven’t asked it in homework sets). I want to be sure I have the level correctly gauged. (Feel free to respond by email if you get stuck.) *(two positive response received so far, including from Gyujin Oh)*

10.3.9 Is there an example of a non-smooth group variety over a field , i.e. a finite type reduced group scheme over that is not smooth? Translation: is there a group variety that is not an algebraic group? *(answer: yes! example here)*

11.3.13 Over an algebraically closed field, every smooth hypersurface of degree at least in is not uniruled. What is a good reference? (I know why it is true! As with many of these questions, I’d like to know where to point people to.)* (answer here)*

13.8 I mention Tate’s theory of non-archimedean analytic geometry. Is there a “right” source to point the interested reader (who is just starting out) to? *(possible answer here)*

19.9.B In (19.9.7.2), we have , and the discussion is away from characteristic . I want to say that the normalization factor is because of characteristic , but I couldn’t convince myself that this was true. Presumably it is. Is there a good reference? (Remark for comparison: one can also write in terms of : . Here the prime factors of are and ; but the reason for the is not characteristic .) * (answer: yes, see here)*

20.2.H Suppose is a complex elliptic curve. Then is always or . It is if there is a nontrivial endomorphism from to itself (i.e. not just multiplication by followed by translation); the additional class comes from the graph of this endomorphism. Is there a reference for this fact that I can/should direct learners to? (answer: **yes**, see **here**)

21.5.9 Is there a good reference for the Lefschetz principle? (Examples currently mentioned: Kodaira vanishing in characteristic 0; and non-jumping of hodge numbers in characteristic 0.) *(good answer here)*

21.7.8 (not needed) It is a nontrivial fact that irreducible smooth projective curves of

genus have finite automorphism groups. I know three arguments: using the Neron-Severi theorem (and the Hodge index theorem) (see Hartshorne V.1.9, for example); the fact that the automorphisms are reduced and form a scheme (too hard); and by action on Weierstrass points. I am surprised that this is so hard. (Note: I know that the idea can be quickly outlined to someone learning. But I want an easy complete rigorous proof. As long as I am asking, I also want someone to give me a Tesla Roadster.)

21.7.9 Smooth curves in positive characteristic can have way more than automorphisms. Is there a “best” reference?

28.1.L Is there a canonical reference for Tsen’s theorem, that any proper flat morphism to a curve, whose geometric fibers are isomorphic to is a Zariski -bundle? Follow-up question (posted March 5), in response to David Speyer’s comment **here**: Does anyone have a (loved) reference for the fact that the universal plane conic (over the space of smooth plane conics) is not a -bundle? (See David Speyer’s comment for a little more detail.) * (possible answer to the first question here; answer to the second question here)*

29.3.B I currently define node only in the case of a variety over an algebraically closed field, in which case I say that it is something formally isomorphic to . I gesture toward the definition in other cases. For example, if is *not* algebraically closed, I define it as , where is a quadratic with no repeated roots. I want to say that if is reducible, then this is said to be a *split node*, and otherwise it is a *non-split node*. I’d thought this was standard notation, but google suggests otherwise. Does anyone have strong feelings about this?

29.5 (This is a follow-up to discussion in the **27th post**.) I am reluctant to introduce new terminology in a well-established field, but there is a notion that I think deserves a name. Suppose is a proper morphism. (For the technically-minded, it is likely that “finitely presented” should also be added, but I will play it safe, and not include this.) Then I want to say that is [something] if the natural map is an isomorphism. **Not EinStein **suggested the name *-connected*, and I quite like this — it suggests that this notion is even stronger than connected, and suggests in what way it is stronger. Another possibility is -isomorphic (which I suggested, but which I currently like less well). Opinions? (Are you offended by giving this a new name? Or do you like one of these suggestions? Or do you have another idea?)

30.3.4 Is there a canonical (“introductory”) reference for (which will require an introduction to derived categories)? (Brian Conrad’s book** Grothendieck duality and base change** perhaps?) *(possible answer here)*

February 26, 2013 at 3:58 pm

30.3.4 Kashiwara/Schapira’s Sheaves on Manifolds develops the formalism of derived categories and constructs for manifolds, which is perhaps a good place to get a feel for Verdier duality in a setting free from the complexities of scheme theory.

February 26, 2013 at 4:15 pm

Thanks! An excellent suggestion! Plus the book is already in the bibliography.

February 27, 2013 at 12:20 am

You should be able to correct the subscript using

`\limits`

; thus,`\varprojlim\limits_{\mathscr{I}}`

createsFebruary 27, 2013 at 6:10 am

Excellent — thank you!

March 11, 2013 at 7:37 am

By the way,

\nolimitshas the reverse effect. I find this useful in typesetting exterior products; compare and .March 11, 2013 at 8:55 am

This is very good to know (both for this project, and in general). I will (non-systematically) fix this as well.

February 27, 2013 at 11:11 am

5.4.N — Doesn’t this follow from the idea in 14.2.8, i.e. the class group of the localization away from one prime is the quotient of the whole class group by the subgroup generated by that prime. If readers are familiar with the fact that every fractional ideal of a Dedekind domain can be generated by two elements, then I suspect there is a cheap proof.

6.3.K — Section 19 of GAGA for the uniqueness of algebraic structures, and Section 7 for compactness in the usual topology versus properness.

March 1, 2013 at 10:45 am

Thanks Sam! About 6.3.K — I should have properly checked in GAGA. What a remarkable paper!

About 5.4.N — you’re absolutely right; the main issue to check is that the primes and are not principal, which you can do using the usual norm in the complex numbers (there are a only a few elements of the ring of small norm), using the fact that the class group of is .

This leads me to another question I should have asked: what is a (good, loved) reference to point people to for this fact? Presumably many of the most-loved references about finiteness of class numbers (using the “geometry of numbers”) have this as an example. I’ve now added this to the list of questions above.

March 1, 2013 at 11:02 am

Well I first learned how to compute class numbers of imaginary quadratic fields using Minkowski from Artin’s textbook on Algebra. But I’d imagine the technique is in almost any algebraic number theory text. A free reference is William Stein’s book. He does the example of which works out almost the same way, in Section 7.3.

March 1, 2013 at 11:49 am

Thanks again Sam! Artin’s Algebra is certainly rightfully well-loved, and he does this precise example explicitly in Theorem 7.9 (p 416-7); that specific reference is mainly for my own benefit. It is also great that William Stein has that book online.

March 7, 2013 at 9:51 pm

A follow-up about GAGA: if you want to see how the argument goes, I encourage you to check out Pieter Belmans’

GAGA in 24+epsilon tweets.April 20, 2013 at 6:51 pm

The last section of the first chapter of Mumford’s Red Book also shows that compactness is equivalent to completeness.

April 26, 2013 at 12:28 pm

Thanks Charles! I’ll check it out, and likely add a reference.

February 28, 2013 at 9:34 am

The answer to the unimportant 1.6.12 is No. Examples can be found at

http://mathoverflow.net/questions/93716.March 1, 2013 at 10:42 am

Thank you! Although not needed for the discussion in the notes, it is something I have been wondering about (a lot!) for a long time. I think that it is a very natural question (given, for example, the facts shown in 1.6.12), and I am heartened that others found it natural too (and that Charles Rezk and others have such clean answers).

March 1, 2013 at 1:24 pm

What I had sent about open subfunctors (in a long list of comments) was:

[[

Where did you get the name “open subfunctor” from? Is NOT in general a

subfunctor – 1.7.1, all from Ch. 0(!) of [EGA I] (1971 ed.). Should use

the more careful terminology of 4.5.3. But I believe they uselessly

complicated things by focusing so much (e.g. mid-1.7.8) on subfunctors.

]]

The main question is: Is it OK if your open subfunctors (in Notes, 9.1.7,

16.7) are not subfunctors, as defined (in an unsophisticated way) in EGA?

Both of these are relations between two functors. I like Ravi’s definition

but think the conflict should be briefly noted.

On fibre/fibre, I did not express a strong opinion, but merely sent:

[[

Reconsider why you write “fiber” (preferred in U.S.) vs the universally

acceptable form “fibre”, often used — Hartshorne, Stacks, …

Could be globally changed in seconds — alter “fibered” first.

]]

On Ex. 5.4.M: Before, I did not notice anything unusual. Now I see there is

a red herring in the hints — the final object is clearly an integral domain

but you DON’T need to prove it is a field. It is: Any non-zerodivisor in a

f.d. K-algebra, K a field, is a root of some p in K[x] with nonzero constant

term, so has an inverse. Normality is for the main part, via earlier hints.

March 1, 2013 at 8:47 pm

Thank you Peter!

About “open subfunctor” and “fiber/fibre”: I didn’t mean to misrepresent you. But I am very open to changing terminology if there is sufficient enthusiasm.

About 5.4.M: thanks for this; I’ll take a closer look (as this is one of the exercises that needs more tender loving care). Should I take your comment as meaning that you are happy with it, modulo this red herring?

My list of “things to do” includes a reasonable number from your (emailed) comments that I’ve marked but not yet acted on, so thanks once again for these.

March 3, 2013 at 11:55 am

(First of all, thank you for writing and publicising these notes! They’re by far the clearest exposition of basic algebraic geometry I’ve seen – at least as far as I’ve read.)

Fibre vs. fiber: I’m used to both spellings of this word, and I doubt it will impair understanding. The only thing it might damage is the number of hits you get when someone searches online for “fibre product”, but I think most large search engines are sophisticated enough nowadays to conduct all searches modulo alternative spellings and misspellings.

Open subfunctors: if these aren’t in general subfunctors, I’d appreciate a remark on this in the notes. You don’t have to change the terminology to avoid the clash if you think it’s good terminology, but acknowledging the clash would be very helpful.

A comment of my own: there seems to be a notational clash in 3.1.A and B, where the map you usually call ‘pi’ accidentally gets called ‘f’ a couple of times.

March 3, 2013 at 2:47 pm

Thank you Billy!

About fiber vs. fibre: at the very least, it seems worth mentioning the alternate spelling, and I now do (in the version to go public next). Your point about open subfunctors is a good one too; I’ll hold off making any edits to give others a chance to weight in. And thanks for catching the 3.1.A an B issues. By coincidence I caught the 3.1.A ones a couple of days ago, but missed 3.1.B. As you might guess, at some point I decided to standardize notation to always use , which required changing some s to s, and I knew full well I’d mess up a number of times. Now I mess up three times fewer, thanks to you!

(The convention of has led me to have ring morphisms , which has given a number of people ulcers…)

April 22, 2013 at 3:27 pm

Hello again! I found a couple more minor mistakes/inconsistencies now that I’ve got a little bit further. I’ve been working from a slightly old copy, so some of these may not apply any more, but I think they’re all current:

1. Section 9.2.F has the ring morphism going A to B, rather than B to A. Just an inconsistency.

2. 10.2.E has another diagram that needs f changing to pi that you might have missed. (I swear I saw another one in chapter 6 at some point, but I can’t find it now – perhaps you got there before I got here…)

3. In 12.1.G, you write “cotangent space” where I think you mean “tangent space”.

April 26, 2013 at 12:35 pm

Thanks again Billy! I’ve fixed all three. In 10.2.E, if I understand you correctly, it is the equation display right after the exercise, where I had a where I should have had a . But if there are even more out there, just let me know!

March 5, 2013 at 7:56 am

6.7 That’s what I always thought row reduced echelon form was. Wikipedia agrees, as does Bretscher’s Linear Algebra text (which I have used recently, so I have it available). Any good linear algebra text should cover this.

10.3.9 Let be an imperfect field of characteristic , with . Let be the closed sub-group-scheme of . (To check that this is a sub-group-scheme, use the functor of points: If is a -algebra with and , then .) Then is reduced (but not geometrically reduced) and is not smooth (or even regular).

13.8 Brian Conrad’s

lecturesfrom the 2007 Arizona Winter school? Or maybe first point them to Chapter 5 of Silverman’s Advanced Topics in the Arithmetic of Elliptic Curves, so they can see one example worked before they try to learn the general theory?28.1.L In the blog post, you forget to say that is a curve! (It’s correct in the notes.) By the way, another counterexample with a noncurve base is to take to be the locus of smooth conics and to be the universal conic over it.

March 5, 2013 at 12:29 pm

Thanks for the multiple great answers! Thanks separately for 6.7, 10.3.9, and 13.8. Regarding 28.1.L: thanks, I’ve patched the missing hypothesis. I also like this example with noncurve base (and meant to include it in the notes — if I didn’t, I will add it). Follow-up question to this example: do you (or anyone else) have a reference where it is shown not to be a -bundle? (One argument I like uses intersection theory: the total space of the universal conic over is a smooth variety, and can be described as a -bundle over . You can thus work out its intersection/cohomology/Chow ring, and then check that every divisor meets the general fiber with even multiplicity 2, and thus that there is not even a rational section to conic bundle.)

April 8, 2013 at 6:05 pm

Just wanted to add: The group above is not regular at . It actually is regular at some of its other closed points, such as , although it is not smooth there. (Proof: Note that . So . Letting be the ring , this shows that is spanned by , and is thus at most one dimensional.)

Until ten minutes ago, I was confused about whether schemes that are reduced but not geometrically reduced could be regular, so I figured I’d come back and add this.

April 10, 2013 at 11:06 am

Thanks David! That’s a worthy example (those points that are regular, reduced, but not geometrically reduced).

March 7, 2013 at 9:30 pm

Regarding the newly added question about 28.1.L: I’ve just realized that the things they will already know (covered in the notes already) can be used to solve this — the universal conic over is a -bundle over , and thus its PIcard group is . It suffices to show that any divisor meets a given fiber with even multiplicity, hence not multiplicity 1. This will imply that there are no sections, even over any nonempty open subset over . To do this, we need only do this with a generating set for the Picard group. One is the divisor “point on conic lies on fixed line” (which meets a general fiber with multiplicity 2), and one is the divisor “conic is singular” (which meets a general fiber with multiplicity 0, as it is disjoint). I’ll add this to the notes, probably in a double-starred section.

March 7, 2013 at 10:11 pm

Answer to 11.3.13: In Koll’ar’s “Rational Curves” book, Corollary IV.1.11 states that a proper smooth and separably uniruled variety has no nonzero global -forms for . But hypersurfaces of degree at least (in ) have nonzero sections of the canonical bundle. (When making that change, I will also check to be sure I mentioned that this shows that such hypersurfaces cannot be rational.)

March 11, 2013 at 4:21 am

5.4.M: I didn’t find this problem particularly difficult, even without looking at the hint first. But does l/k need to be a finite extension? It is true that this is used in the hint given, but it seems perfectly possible to avoid it. All I actually ended up using was the fact that (A \otimes l) \cap K(A) = A. Maybe I’m just being dense here.

March 11, 2013 at 10:26 pm

I asked John for a solution, so I could get a sense of how difficult the problem might be, and it was very nice. I’m now happy with this problem’s current statement.

March 11, 2013 at 10:40 pm

28.1.L: Bianca Viray and Tony Varilly-Alvardo both like Gille and Szamuely’s

Central Simple Algebras and Galois Cohomology. (Tsen’s Theorem is Theorem 6.2.8, p.143.) Gille-Szamuely looks like a superbly written and friendly introduction. You may enjoyChenyang Xu’s review on amazon, which is so poetic it is almost haiku. (You might then be curious enough to read all of his reviews…)March 12, 2013 at 4:46 pm

19.9.B: Igor Dolgachev, master of the classical algebraic geometry I love so well, explained that the answer is yes, and gave a reference: Deligne’s article

Courbes elliptiques: formulaire d’après J. Tatein LNM 476 (Modular Functions of One Variable IV). See especially p. 64. (Could the power of 2 in the page number similarly be motivated by some issue in characteristic 2?)Another great fact from Dolgachev: the cubic curve over a field of characteristic 2 has no inflection points. (Fun things for you to ponder: This may seem surprising — the Hessian must vanish somewhere on it. How then can you put it in Weierstrass form?)

March 15, 2013 at 10:21 pm

Dolgachev’s (and Deligne’s) answer leads me to want to know one more thing: I assume the j invariant far predates thinking about elliptic curves in characteristic 2, presumably by more than a century. Why then did the ancients come up with this definition of an invariant, rather than ? (This is from personal curiosity, not for the notes.) If no one knows the answer, I might ask on

MathOverflow, although I can imagine that the question will be closed.March 16, 2013 at 7:14 am

I don’t know any actual history, but I always assumed it was because the -series starts off , with all coefficients integers, this way.

March 22, 2013 at 1:20 pm

Excellent point. And Akshay Venkatesh explained to me that your answer and Dolgachev’s are linked, via the Tate curve.

March 12, 2013 at 10:17 pm

21.5.9: Perhaps

this mathoverflow responseby Martin Brandenburg is the best answer.March 16, 2013 at 10:09 am

Dear Ravi, it’s not true that the set of regular points in a noetherian scheme is open as you state on page 354 line 9 of FOAGfeb2513. The theorem 24.4 in Matsumura actually gives a sufficient condition for the openness of the regular locus.

For further results in this direction see Stacks, tag 07P6. For a counterexample see §5 of the following paper of Nagata.

March 22, 2013 at 9:41 pm

Dear Nuno,

Thank you! I’d completely misunderstood that result of Matsumura, which proves that a

criterionfor openness holds for locally Noetherian schemes, but does not show that the criterion is always satisfied. The situation is much more interesting than that!I have now removed the offending sentence in the notes.

April 20, 2013 at 7:07 pm

Concerning 29.5: I have often thought that this property needs a name. One name I have heard used (on a mathoverflow question or answer, I believe) is “anti-affine,” which I rather like: is

affineif the -algebra structure of tells you everything; it makes sense that should be anti-affine precisely when this morphism tells you “nothing.” It also leads to the statement that “if a morphism is both affine and anti-affine, then it is the identity,” which could be either nice or confusing depending on your point of view.Of the two options you suggest, I definitely agree that -connected works better then -isomorphic.

April 26, 2013 at 12:31 pm

Great. I prefer -connected to anti-affine, so I’ll tentatively leave things as they are. (I also patched up the latex as you requested in a follow-up comment, and deleted the follow-up comment.)

April 28, 2013 at 7:54 am

Just a small correction:

in the hint for exercise 3.2.G, the range of the map defined should be a capital A.

(is there a better place I should send corrections to? should I notify on such small errors?)

Thanks for the great book!

April 28, 2013 at 2:57 pm

Thanks Edo! Now fixed. (Someone else caught this at almost exactly the same time, but I can’t know remember who.) This is a great place to send corrections to. Feel free to email me instead if you feel more comfortable doing that. The advantage of posting here is that others can see the corrections to be made (because it sometimes takes me time to fix things).

And thanks for the kind words!

May 1, 2013 at 12:55 pm

Follow-up to 8.4.H: I

asked about this on mathoverflow, and Laurent Moret-Bailly gave a wonderful example of a codimension 1 regular embedding that is not an effective Cartier divisor (seehere). I find this a great example. And I now have a new entry in my (short) list of “interesting non-Noetherian rings”.May 2, 2013 at 6:14 pm

A few more typos and the like.

1. At the start of 13.1, it’s not clear what V is. (Perhaps Stanford students are expected to have already seen vector bundles, but they were new to me!)

2. In 13.1.H, there’s a typo: ‘ran’ instead of ‘rank’.

3. In the sentence after diagram 23.1.0.2, do you mean “Tor(M, -)” instead of “Tor(-, N)”, and “functoriality” instead of “covariance”? (“Covariance” might just be another way of saying “functoriality” here, but I haven’t heard it.) In the same sentence, the word “homomorphism” (just before the delta) should probably have an “s” at the end.

4. On the third line of page 603, the image of your differential should land in A^{n_{i-1}}, not A^{n_i}. In that same paragraph, the use of the subscript “i” to decorate the element “a_i” is a little confusing.

5. In 23.1.B (1), calling the map a “covariant functor” seems a little confusing. It seems cleaner to do away with the resolutions before talking about functoriality, so that we can work in the category of (honest) A-modules rather than the rather artificial category of A-modules-with-a-fixed-resolution.

6. In 23.1.B (3), “Tor” should probably be “Tor(M, -)” again – you haven’t yet proven anything about Tor when the first argument isn’t fixed, I don’t think.

7. In fact, in 23.1.B, functoriality never seems to be explicitly proven. (Perhaps it’s almost too silly to write down, though. (2) shows that resolutions are irrelevant, and earlier paragraphs show that a map N –> N’ uniquely determines a map Tor_i(M, N) –> Tor_i(M, N’); now, if I’m not mistaken, functoriality seems to follow directly from the fact that, given modules-with-resolutions and maps N –> N’ and N’ –> N” between them, the composite of their lifts is a lift of their composite. But maybe it’s at least worth a mention?)

May 3, 2013 at 12:36 pm

Thanks Billy! These comments are particularly helpful. Here are some responses and follow-ups.

1. This is an important point I want to get right. I would expect the reader not to have seen vector bundles before (and in any case I want to write for that majority of readers). So I want to get across the notion in a way that is easy to read, and convinces the reader that this is a natural notion. (The notion of “fibration” comes up in a similar way, and I’ve been similarly unsuccessful in introducing the idea.) I wanted the first paragraph to include a couple of examples to keep in mind (mobius strip, and tangent bundle), and I wanted the second paragraph to be a formal definition. Can I ask what might help? Is the issue that it isn’t clear what kind of object is supposed to be? Would saying that is a map *of manifolds* solve the problem? I’d really appreciate advice on this.

2. Fixed.

3. Fixed. (“Covariance” meant “functoriality” here, but saying “functoriality” as you advise is better.)

4. Fixed. (And the subscript on the is more than a little confusing! I’ve changed it to a .)

5. If I understand you correctly, here I think this is a feature rather than a bug. I deliberately did this in a way dependent on the resolution, because it ends up being easy; and then somehow surprisingly after the fact, we show independence of the resolution. In other words, I can’t see how to show independence of the resolution at this stage without actually working harder than I currently do. (I’m happy to discuss this more, and even to be convinced!)

Or perhaps you mean: I should write more explicitly

“We get a covariant functor (from the category of -modules with resolution to the category of -modules), independent of the lift .” Perhaps I should explicitly apologize for the temporary awkward notation, to make clear that it really is temporary.

Could you please say more?

6. What I’d done was needlessly imprecise. Now fixed as you advise.

7. I’d like to understand better what you mean before trying to make any changes. (This may relate to #5.) You could mean that I should be clear why (1) is a functor. Or you could mean that I should be clear that

because(1) gives a functor, (3) does too. Can you say more?May 5, 2013 at 7:34 pm

Hi Ravi,

1. A few things made me think that this section was written for people who knew what a vector bundle was already. The main offenders are: (a) it’s not explicitly stated even that V is a manifold (though maybe it can be inferred), (b) the phrases “tangent bundle” and “line bundle” (in the second sentence of 13.1) haven’t been defined yet.

(My intuition, for what it’s worth, goes along the lines of “V is a family of vector spaces continuously parametrised by points of M” or “above each x, there is a vector space, and these vector spaces are glued into a manifold V so that, as x varies, the vector space above x varies continuously”. These are very non-rigorous, but they’re what I managed to pick up through conversations. Perhaps it would be useful if you could find a way to state this intuitive idea semi-rigorously before giving the examples.)

5, 7. Sorry for being unclear. I’ve thought about it a little more, and I think I’ve decided my suggestion to get rid of the resolutions before mentioning functoriality isn’t sensible. But I think the role of functoriality in this section is still a little unclear, so let me try again with my remaining suggestions:

(a) The object in (1) is a straightforward *module map*, not a covariant functor.

(b) It would be useful if you could prove (or sketch, or leave as an exercise) the functoriality of Tor(M, -) for modules-with-resolutions, and show the canonical isomorphism of Tor modules, in separate sections.

(The reason I ask for this is as follows. To prove functoriality, it seems that we need to do two (very easy) things. Firstly, remark that homology is functorial, which I don’t think you do (maybe this should happen in 1.6.D?). Secondly, prove that, if we’re given modules + maps + resolutions + lifts , , , , then the composite is indeed a lift of the composite . But in (2), you prove separately that “the composite is homotopic to the identity” – a fact that should already follow easily from (the second half of) the above proof of functoriality. This was the original source of my confusion – in (2) some obvious facts are stated explicitly, but in (1) some less obvious facts are omitted without mention!)

This may come down to a personal preference, of course. Anyway, let me know if anything I’ve said is still unclear! (I should also work out how to input LaTeX on wordpress – I hope everything is readable.)

May 29, 2013 at 1:59 pm

Thanks Billy! As I mentioned in email, 1 is now taken care of. About 5, 7:

(a) it is indeed a module map, but it matters that it is a functor; but I now mention the functor part in (3).

(b) Functoriality is now asked in 1.6.d. I don’t think that the composite being homotopic to the identity follows from functoriality.

I’ve now tighted up this section a bit.

About latex on wordpress — it is remarkably easy: just write [dollar sign] latex [the latex things you want to say] [dollar sign]. Feel free to try it out if you want — I can delete your comment after. I’ll edit some of your above comment to show you what it looks like.

May 10, 2013 at 2:03 am

Just two small typos in the chapter on spectral sequences:

just after 1.7.G, it says:

“But it is well-defined modulo the differentials of the (r − 1)-closed (p + 1, q + 1)-strips,” – I think it should be (p-1,q+1)- strips.

In definition 1.7.11: The definition says Y=d(S+S)/S where I believe it should be Y=(d(S)+S)/S.

I hope I didnt make a mistake..

Edo

May 17, 2013 at 10:50 am

I think you are right on both counts. (Someone please correct us if not!) And with 1.7.11: I see where I made the error (in April 2012, while correcting another error caught by Yuncheng Lin), so I’m thus sure that it is an error.

May 15, 2013 at 11:34 am

20.2.H:

Burt Totarohas sent me an answer to this, and I have added this to the notes. I post his answer here (with his permission and my thanks):I suggest the reference:

D. Mumford. Abelian varieties. Hindustan Book Agency (2008).

Namely, Application III in section 21 of Mumford’s book

says that an ample line bundle A on an abelian variety X

over any field determines an involution

x |-> x’ on the Q-algebra End^0(X) := End(X) tensor_Z Q,

the “Rosati involution”. The choice of A also determines an

identification

NS^0(X) = {u in End^0(X): u’ = u}.

(Here I’m writing the formula exactly in Mumford; the superscript

0 means to tensor with the rationals.)

It follows that (given a choice of ample line bundles on X and Y)

we have a direct-sum decomposition

NS^0(X x Y) = NS^0(X) + Hom^0(X,Y) + NS^0(Y),

by viewing End^0(X x Y) as “block matrices”. In the case where

X is a complex elliptic curve and Y=X, this gives the formula

you wanted: NS^0(X) always has rank 1,

while Hom^0(X,X)=End^0(X) can have dimension 1 or 2 as a Q-vector

space.

It’s more than you need as a reference, but it’s fun to go a little

further than Mumford did and think what is a more “natural” description

of NS^0(X), without first having to choose an ample line bundle on X.

This is easy to work out (e.g., by untangling the definition

of the Rosati involution), and I’m sure it is well known

to the experts. The result is that there is a natural identification

NS^0(X) = {u in Hom^0(X,X^): u = u’},

where X^ is the dual abelian variety, and a homomorphism u: X -> Y

of abelian varieties has a dual map u’: Y^ – > X^. (So a homomorphism

X -> X^ is formally like a “bilinear form” on X, and we’re saying that

NS^0(X) is the space of “symmetric bilinear forms”.)

It follows that we have a natural direct-sum decomposition

NS^0(X x Y) = NS^0(X) + Hom^0(X, Y^) + NS^0(Y).