A new version of the notes is available here. (It, and all older versions, are available at the usual place.) The main change from the previous version: The index is potentially done.
Now the editing I will do will be primarily in response to comments and suggestions from others. I am very interested in any suggestions and corrections you may have (including things you told me before that I have forgotten). The only things I’m not interested in are: latex issues (margins, etc.; but typos and errors in spacing are fair game) and ugliness of figures (but content of figures is fair game).
Math 216: Foundations of Algebraic Geometry 
July 31, 2023 at 3:01 pm
In 12.2.L, the bold sentence seems to be missing a verb, “we define”, maybe.
I’d just like to say how much I’ve got out of your notes and the AGITTOC lectures – your time and generosity is greatly appreciated 🙂
August 2, 2023 at 1:32 pm
Yes, you are right!
And thanks for your kind words!
July 31, 2023 at 3:07 pm
Very minor: in the paragraph following 12.3.A, there’s an extraneous “it”.
August 2, 2023 at 1:27 pm
Thanks, that will be fixed in the next version. You are also reading a section that is much simpler than before (I think/hope), so I hope you like it. But there is also a higher chance of errors or infelicities, so please let me know if you are not happy.
The proof of Krull’s Principal Ideal Theorem is the same, and I continue to be unhappy that I have not internalized it. But the rest has become much nicer (after I’ve talked with people, read some Hochster and Qing Liu and some others, and imbibed more folklore).
August 12, 2023 at 3:09 pm
Yes, indeed, I’ve made much more progress with this version. Actually the proof of Krull’s PI theorem pulled a lot of things together for me during a long walk yesterday, and among other things made me appreciate Nakayama a bit more.
A very small typo in they proof at the top of p344: in the third line, you’re using Exercise 12.3.B, not 12.3.4.1.
August 1, 2023 at 1:33 am
After 11.3.13 Proposition, “intersection of any two affine open subsets of A” should be “of X”?
August 2, 2023 at 1:24 pm
Yes, thanks — this will be fixed in the next version!
August 7, 2023 at 8:20 pm
Page 34 last full paragraph 1st sentence: “The constructive definition ⊗…” should be “The constructive definition *of* ⊗…”, I t hink?
August 8, 2023 at 8:43 pm
Thanks, fixed!
August 7, 2023 at 9:58 pm
Also, p. 47 display before 1.5.H – way too much space around the single equation in the display.
August 8, 2023 at 8:44 pm
This one is an artifact of latex — the large table on the next page causes it. So I’m not worrying about it at this stage (although I might make it a floating table, so text can flow around it).
August 12, 2023 at 3:10 pm
Yes, indeed, I’ve made much more progress with this version. Actually the proof of Krull’s PI theorem pulled a lot of things together for me during a long walk yesterday, and among other things made me appreciate Nakayama a bit more.
A very small typo in they proof at the top of p344: in the third line, you’re using Exercise 12.3.B, not 12.3.4.1.
August 28, 2023 at 4:20 pm
Thanks, now fixed! And I also find that if I can understand things while walking around, I am very happy with my understanding.
August 18, 2023 at 5:22 pm
Hi Ravi,
Thank you for the updated index. Some minor things: in Definition 2.7.6 on p.93, should the general definition of the support be the set of points for which the stalk is not the *terminal* object, as opposed to the initial object? Something similar pops up in Exercise 2.7.H on the following page, where in the problem statement of part (a) we should be showing that the stalk (i_*F)_q for q outside Z is the singleton, as opposed to showing that it is the empty set, I think? Lastly, for notational consistency the adjoint pair referenced in Exercise 2.7.F on p.93 might instead be (\alpha^{-1}, \alpha_*), as opposed to (\alpha^*, \alpha_*)?
December 4, 2023 at 1:53 pm
There is also a stray fragment “irr” at the end of 3.6.3.
January 29, 2024 at 6:58 pm
Thanks, fixed!
January 29, 2024 at 12:06 pm
I agree with all three of your points. (i) change made. (ii) I decided to make this about a sheaf of groups, so that I could then say it is just the one-element group. Reason: the phrase “the singleton” will confuse most readers (and not to their advantage); more sophisticated readers like yourself will not be bothered. (iii) In the version on my computer this is already fixed, so someone may have pointed it out to me recently.
February 4, 2024 at 7:08 am
I realize that the issue of support of sheaves of things that are not groups is completely irrelevant to anything later in the notes, and I was only distracting and confusing people (and myself), so I have largely removed it, and concentrated on what matters. The revised version is here:
Click to access p97-982024-02-04.pdf
February 4, 2024 at 4:25 pm
This is better. Needs a quick cleanup though – 2.7.8 says “a sheaf of groups G of sets”.
February 4, 2024 at 4:56 pm
Thanks, fixed!
August 18, 2023 at 8:23 pm
Just noticed this: on p. 93, end of line -10, “with to restriction to” => “with restriction to”.
August 28, 2023 at 4:26 pm
Thanks, fixed! And I’ll soon deal with those that you emailed me about as well.
August 21, 2023 at 9:56 pm
p. 179: line 3, “classification”, “finitely” and “modules” are all misspelled; line 4, missing semicolon between “principal ideal domains” and “unique factorization of ideals”.
August 22, 2023 at 8:30 pm
Whoops, never mind – that’s the old edition. In the new one the sentence with all the typos has been deleted. :P.
August 28, 2023 at 4:29 pm
I’m glad to know that I caught at least some of these…
August 28, 2023 at 6:04 am
On page 261, I have repeatedly been bothered by the sketch of the quadric surface. The drawing does not seem to match the defining equation wz-xy. See for comparison Hartshorne’s “Algebraic Geometry” p. 14.
All the best,
Clemens
January 30, 2024 at 1:20 am
Never mind, I feel like I finally wrapped my head around it.
January 30, 2024 at 9:49 am
I was actually thinking about this in the last few days. I wasn’t immediately sure what to do, because the equations are in projective coordinates, and the picture is in affine coordinates. You are right that the “affine chart” pictured is in slightly unusual coordinates, although I also want to have the two rulings very visible as they are in the picture.
(Incidentally, I’ve redone the figure so that all the figures are done with the same style, line width, font, etc. But the resulting handdrawn picture still looks too imperfect.)
Would the following caption solve the issue for you?
Old caption:
“One of the two rulings on the quadric surface V(wz− xy) ⊂ P3 . One ruling contains the line V (w, x), and the other contains the line V (w, y).”
Proposed new caption “One of the two rulings on the quadric surface S=V(wz− xy) ⊂ P3 . One ruling contains the line V (w, x), and the other contains the line V (w, y).
(The quadric surface sketched is actually a^2+b^2=c^2+1. Can you see why this an affine chart for S? What are a, b, and c in terms of w, x, y, and z?)”
What do you think?
February 5, 2024 at 11:30 pm
This sounds much clearer to me. Maybe you should mention that this chart only exists in characteristic not equal 2. Depending on how clear you want to make it, you could also add a reference to the diagonalization of quadratic forms (Exercise 5.4.J.).
February 6, 2024 at 9:45 am
I’ve added a reference to 5.4.J. I decided not to mention the characteristic, since this is a sketch, and already we’ve just sketched it in the real numbers, so it is only an approximation of truth [I was going to say “reality”].
September 11, 2023 at 7:47 am
Hey, Ravi. It seems like exercise 14.2.I overlaps a bit with exercise 14.2.B
February 3, 2024 at 5:24 pm
Good point! Now edited.
September 15, 2023 at 3:39 am
Dear Ravi, two remarks:
1) In the discussion of 9.4.3. (pp. 264-265) you repeatedly write \pi^{-1} instead of (presumably) \pi^\sharp when talking about pulling back functions.
2) In the discussion immediately after Theorem 18.1.2. (cohomology of line bundles on projective space) you mention that its top cohomology group is generated by the homogeneous degree-m Laurent polynomials in (x_0 … x_n)^{-1} k[x_0 … x_n]. After following the calculations in section 18.3. I feel like it should be the module of homogeneous degree-m Laurent polynomials, in which every variable appears in negative degree, i.e. the degree-m part of (x_0 … x_n)^{-1} k[x_0^{-1} … x_n^{-1}].
September 25, 2023 at 6:50 am
Another thing I noticed: In the proof of 6.21. (pp. 190-191) you invoke exercise 6.6.C. I assume that there should be the hypothesis that A is Noetherian as otherwise q need not be finitely generated.
January 29, 2024 at 6:07 pm
That’s a good point. I had intended it, but the vague discussion made it unclear, so I’ve now stated it explicitly.
January 29, 2024 at 6:16 pm
1) Thanks; I caught three instances which I fixed.
2) I agree again, thanks! Now fixed.
September 15, 2023 at 11:07 pm
On p. 92 you say ‘the pullback of the “space of sections” is the “space of sections” of the pullback’ – by pullback here do you mean inverse image sheaf?
September 15, 2023 at 11:09 pm
I guess the same goes for the mention of pullback at the end of 2.2.11, p. 75?
February 4, 2024 at 6:50 am
Yes to both. I’ve clarified both.
I think of inverse image as a kind of pullback, and I find “pullback” to be a more elegant (or at least visually appealing) descriptor.
But I guess it is good intellectual hygiene to linguistically distinguish the two notions especially since we are defining one (and getting at its properties) in terms of the other — the pullback is (at least in one incarnation) defined in terms of the inverse image.
September 25, 2023 at 6:00 am
pg 43, 1.4.8 the definition of filtered set says that any two elements should have a lower bound z, but z should be upper bound when contrasted with the filtered category definition. Otherwise filtered set is not a filtered as a category.
February 3, 2024 at 5:24 pm
I *think* that’s okay; the way I considered posets as categories makes these agree. If $x \geq y$ in a poset, then there is a morphism $x \rightarrow y$ in the corresponding category. (I think of the “arrow symbol” going in the same direction.) Please let me know if I’ve managed to get myself confused though (as this is the sort of mistake I tend to make)!
September 25, 2023 at 10:25 pm
On p. 94, Exercise 2.7.H(a), I believe the stalk (i⁎𝓕)_q should be 𝓕(∅), not ∅.
September 25, 2023 at 10:39 pm
(which of course means that (i⁎𝓕)_q is a singleton set, the terminal object of Set)
January 29, 2024 at 6:23 pm
Agreed! I’ve done something slightly different to try to lesson confusion for readers (see my response to Daniel Rostamloo’s comment #7 above).
September 26, 2023 at 8:10 pm
p. 93 2.7.6: You define the support of a section for sheaves of abelian groups, and then define the support of a sheaf for the more general scenario of sheaves of *sets*. Then you observe that the support of a sheaf can be characterized in terms of the supports of its sections, but this doesn’t really make sense for sheaves of sets, because the notion of support of a section doesn’t really extend to sheaves of sets. This characterization works for sheaves of abelian groups, but it’s not until the next paragraph that you actually explain how to generalize the definition of support of a sheaf in terms of the initial object of the corresponding category. So some rearrangement would probably help here.
February 4, 2024 at 7:09 am
Retyped from elsewhere because it is worth saying twice: I realize that the issue of support of sheaves of things that are not groups is completely irrelevant to anything later in the notes, and I was only distracting and confusing people (and myself), so I have largely removed it, and concentrated on what matters. The revised version is here:
Click to access p97-982024-02-04.pdf
September 30, 2023 at 8:24 pm
Small typo in exercise 8.4.D. A space missing between “Suppose” and “\pi”.
November 18, 2023 at 10:26 am
Was rereading the group schemes chapter 6.6 and there seems to be some inconsistency with notation. In the definition of group objects the map X x X x X -> X is given by (m, id) in the associativity axiom but the map Z x X -> X x X in the identity axiom is given by (e,id). These are doing the same thing — prescribing a map by maps on the factors. So the notation should be the same.
November 18, 2023 at 11:20 am
I think you mean 7.6 here. Also, I think the problem is that the map Z x X -> X x X in the identity axiom is given by e x id (not (e,id)).
That aside, I agree. I think the first one should be m x id (and similarly id x m for the vertical map).
November 29, 2023 at 10:30 am
That’s right, 7.6, sorry. I think what I’m trying to say is that either the (e, id) or e x id work to denote the maps since what’s going on is clear from context, but that it should be consistent.
February 4, 2024 at 7:31 am
I see your point, and I see that I used 3 different notations for essentially the same thing. I’ve tentatively decided to go with \times, but let me now if you think ordered pairs would be clearer. Revised version is here:
Click to access p2272024-02-04.pdf
February 4, 2024 at 9:35 am
I think it looks good.
February 4, 2024 at 10:06 am
thanks!
December 19, 2023 at 5:24 pm
There is an extraneous remark on p117 in remark 3.6.3. “… as Exercise 10.5.L. irr”. The “irr” is probably a typo from the following section on irreducibility.
January 29, 2024 at 7:06 pm
Thanks, fixed!
January 6, 2024 at 5:26 pm
Just to be consistent with notation, I think the definition of weighted projective spaces in 9.3.8 should be on k[x_0, …, x_n]. Eg. for k[x_0,…, x_n] giving each x_i weight d_i we define an n-dimensional weighted projective space P(d_0, …, d_n).
January 6, 2024 at 5:26 pm
Sorry, definition 9.3.13…
January 29, 2024 at 12:11 pm
Good point – fixed!
January 7, 2024 at 3:03 pm
Steely eyes these few days it seems. In 12.1.3 there is a typo “… a topological space is pure dimensional or equidimensional” there should be a space between “pure dimensional” and “or”.
January 22, 2024 at 5:48 pm
You’re right! Another fault of adding index entries too quickly. Darn. Now fixed.
January 10, 2024 at 1:40 pm
Possibly a typo: the statement in 13.7.4 should be “… for any valuation ring A with *fraction* field K, …” right now it’s *function field*.
January 22, 2024 at 6:41 pm
True! But I also realize I can even delete that entire clause referring to K, so I’ve done so.
January 12, 2024 at 11:06 am
The compositions in exercises 17.3.B and 17.3.C are incorrect. \pi: X\to Y and \rho:Y to Z has composite (\rho \circ \pi):X \to Z, not (\pi\circ rho).
February 4, 2024 at 7:16 am
Good catch! I saw one instance of this problem in each exercise, and fixed them. I hope I didn’t miss any.
January 12, 2024 at 11:54 am
I think it would be better to say “\pi-very ample” instead of “very ample” in exercise 17.3.I.
January 22, 2024 at 5:47 pm
You’re right. I’ve made this change.
January 12, 2024 at 9:10 pm
In theorem 18.7.1, I think (b) should be an identification of *sheaves* R^{0}\pi_{*}\mathscr{F} with \pi_{*}\mathscr{F} instead of R^{0}\pi_{*} (which is a functor).
February 4, 2024 at 7:19 am
Good catch! Actually, I’d prefer it to be an isomorphism of functors, so I’ve now said it that way, without \mathscr{F}.
January 14, 2024 at 10:14 pm
Minor: In section 1.6.13, I think the text should be:
“… and F is (-)\otimes N *for some fixed A-module N*”
right now it reads
“… and F is (-)\otimes N for some fixed N-module*”.
January 22, 2024 at 11:15 am
Thanks, fixed! I am mystified how this error lasted so long.
(Also more generally thanks also very much for the many useful comments you are making.)
January 16, 2024 at 7:45 pm
It seems exercise 22.4.P continues with the notation/setup in exercise 22.4.O, in which case the blowup would be \beta and not \pi.
Right now 22.4.P denotes the pullback of a divisor on X as \pi^{*}D, and not \beta^{*}D.
January 29, 2024 at 12:47 pm
If I understand you correctly, \pi should be \beta throughout 22.4.P. And then although you didn’t say it, also in the next section, on the change of the canonical line bundle under blow-ups, \pi should be \beta throughout for consistency as well. I’ve made all these changes.
January 17, 2024 at 7:45 pm
The additive functor in half-exact Nakayama (24.7.3) should be denoted T, following the notation in the latter part of the theorem statement.
January 22, 2024 at 6:54 pm
Thanks, fixed! There were a couple of other stray phi’s in the proof as well, that I also corrected to T’s.
January 22, 2024 at 11:17 am
Thanks, fixed! That happened when making the index, so I am deeply afraid that I introduced more errors like this elsewhere…
February 3, 2024 at 8:29 pm
In the proof of 27.8.1 (just before Exercise 27.8.A), I think it would be best to denote H.C as something other than k, which is the field in the statement of Castelnuovo’s Criterion.
February 3, 2024 at 9:08 pm
(27.8 should be 28.7 in the above, I think.)
Good point. I’ve changed k to d. I think the only changes are in Step 1 in the proof (but if you notice that I am wrong, please let me know!).
February 3, 2024 at 9:15 pm
Right, indeed. I think it looks good otherwise.
February 3, 2024 at 9:17 pm
Really small, but in 29.2.F G should be a quaiscoherent *sheaf* in the singular.
February 4, 2024 at 7:21 am
Thanks, fixed!
October 9, 2023 at 8:40 am
p.351, Exercise 12.4.C: the hint provided refers to “m-n” and “irreducible varieties,” although the exercise is a reduction of the proof of Theorem 12.4.1 (not Corollary 12.4.2). In particular, the hint is the exact same as in older versions of the notes, before Theorem 12.4.1 was added.
October 19, 2023 at 8:56 pm
Another thing I noticed:
A lot of the references in Chapter 15 are to things not yet covered/proven. For instance, the discussion in 15.2.1 cites Exercise 16.1.B, and the proof of 15.2.2 cites the notation of Exercise 15.4.Y. There are also a few exercises that use the fact that Picard group of P^n_k is Z (such as 15.2.D and 15.2.M) before it is proven in 15.4.H. (I imagine this arose because the order of exercises/exposition got shuffled in the newer version of the notes)
February 10, 2024 at 6:41 pm
You have hit on something important that has managed to survive a long time! You are right, and this is worrying. (Back story: the connection between line bundles and Weil divisors is hard but very rewarding. I realized that a lot of things with line bundles could be done before then, and I moved as much as possible before taking the big hard step, to show how simple the rest of it was. This has had unintended consequences, as so many things do. I have fixed a lot of them, indeed as many as I could think of, but I know/fear/suspect that I’ve missed a number. The changes are over many sections, so rather than just posting a large number of sections, I’ll just post the revised version before long.
In short: I’ve tried to remove forward-references. There are a lot of things that, as you noticed, come from the fact that we know the Picard group of projective space, which we *only* get to know after the hard work of understanding line bundles in terms of Weil divisors. So I now do as much as I can before then; then have a section “Hard but important: Line bundles and Weil divisors”, then a section “The pay-off: Many fun examples” in which we lead off with fun stuff with projective space.
If you have more examples of forward references and have some time, please do let me know about them. Or if you prefer, wait a bit (maybe a week?) for the new version, and take a look at it.
February 10, 2024 at 8:41 pm
The other main example I noticed of forward references comes in Chapter 21, where the notion of the ramification locus occurs (first in 21.4.A) before the definition of being unramified/ramified and the ramification locus definition (in 21.7). Again this seems to be a result of the order of sections being swapped in different versions.
February 17, 2024 at 2:24 pm
That’s a good point; I have now separated out that definition.
February 4, 2024 at 8:16 am
I see what you mean. I’ve decided to remove the hint after trying to figure out what kind of hint would most help. I don’t know whether that was a good idea.
I also find this the worst section of the chapter, but have tried to tweak it a bit. I’ve just taught this section, and I *think* that the finite presentation hypotheses in Theorem 12.4.1 are not used; we only need finite type. A revised version is here:
Click to access p354-3562024-02-04.pdf
I am hoping that the resulting version is now readable. It is strange to me that this fact about fibers being well-behaved over a big open set is the hardest to explain in a chapter about dimension. (I think this fact is often not discussed for some reason. If my memory serves from reading them long ago: It is in Shafarevich, but the proof isn’t quite complete; Mumford’s red book has a great argument which is in a special case but the argument just applies without change.)
If anyone reads the revised version and has suggestions (or just approval), especially (but not necessarily) in the next month, please let me know!
October 16, 2023 at 6:45 pm
Hello Professor Vakil,
On page 183, Ex 6.5.F “thow” seems redundant.
On page 184, Ex 6.5.J “A/m_i” , should it be “A/n_i”?
January 29, 2024 at 7:04 pm
About 6.5.F. Worse than redundant — it seems like nonsense! (You were being kind…) I’ve removed it.
About 6.5.J. I think you’re right; now fixed.
October 24, 2023 at 6:04 pm
p192 Prop 6.6.26, the phrase “For an A-module M” seems redundant.
October 30, 2023 at 2:01 am
P203 7.2.D, right before part (a) \pi^{\ast}\mathcal{F} should be \pi^{\ast}\mathcal{G}
December 31, 2023 at 7:33 pm
Thanks, now fixed!
November 22, 2023 at 10:59 pm
pp. 216, third line after figure 7.1, the map f|_{X_1} is just f beacause there is no need to restrict.
January 22, 2024 at 5:34 pm
Actually, this phrasing was deliberate: we *chose* the open set X_1, and I want to be sure to take the inverse for this particular map.
November 22, 2023 at 11:00 pm
pp.222 fifth last line, “an element X” should be “an object X”.
January 22, 2024 at 5:17 pm
Fixed, thanks! (Assuming that was in the definition of “group object.)
February 4, 2024 at 7:23 am
Yes, thanks, fixed!
October 25, 2023 at 12:22 am
Hello Professor Vakil, thanks for your great book! I’ve found some potential issues in the book, and I hope this could make the book better:
1: Exercise 2.7.F seems hard. Here’s my very long solution after some hard work: https://aprilg.moe/index.php/2023/05/14/ex-the-rising-sea-2-7-f/ Maybe it should be marked as hard?
[As I now mention below: yes, it is surprisingly hard, and I have now marked it as hard. Thanks! – R.]
2: Remark 3.6.3 states that the reader could prove Spec A is disconnected if and only if A is the product of two nonzero rings. What’s the intended method here? The only method I could think of is to use the fact that A is isomorphic to the global sections of the scheme Spec A, which utilizes concepts not covered at this part of the book. Is is doable for first-time readers, or should a notice be added?
[In that remark, I mention that it will come up as an exercise later, and indeed at that point that is how you prove it. It seems hard to prove it directly, although I am not sure I trust my judgement on this — now that I think geometrically, I can’t help but see the geometric (harder) approach. So now I no longer encourage the reader to try to prove it, and instead leave the forward reference, so they know they will later understand why it is true. – R.]
3: 13.5.12 includes ‘We will propelrly define zeroes and poles in 13.5.7’, but the definition is before this section, so past tense should be used.
[I meant to refer to refer forward to 13.5.19 (and now do), which is where zeros and poles are defined in for functions on a Noetherian normal scheme. – R.]
4: 13.5.14 includes ‘Translation: a is multiple of b in A_p if and only if \bar{a} \neq 0 in A/(b)’. I think this should be the opposite.
[You’re right! Now I could say one of two equivalent things. “a is not a multiple of b in A_p if and only if \bar{a} \neq 0 in A/(b)” or “a is a multiple of b in A_p if and only if \bar{a} = 0 in A/(b)”. They are of course logically identical. I’m currently going with the first (even though it sounds worse) because it seems to give the right message to the reader. But let me now what you think. It also sounds like you read and understood my (presentation of the) proof of Algebraic Hartogs’ Lemma, which makes me happy, because I wasn’t sure of it; I was hoping that it would be readable and direct. – R.]
5: 14.3.I includes ‘… is also a short exact sequence of finite rank locally free sheaves’. But the involved exact sequence isn’t short.
[Thanks, fixed. -R.]
6: 15.2.G these points should be k-valued points.
[Thanks, fixed. -R]
7: 15.4.16 includes ‘as a section of the new line bundle L(D)’. This needs the hypotheses that D is locally principal.
[I’ve reworded this slightly to fix it. -R]
8: 15.5.B includes O(-D). D is defined to be a closed subscheme, so -D makes no sense. Maybe it will make sense as a Cartier divisor, but it is not defined in the book.
[I’ve now formally defined it, and indeed O(nD) for any integer n, where D is an effective Cartier divisor. -R]
9: 15.5.C includes ‘this section s_D’, which is not mentioned previously in this subsection.
[I’ve now reworded this to make it clearer. -R]
10: 15.6.F.(b) as a S.-module, \Gamma.\tilde{S.} may have negative dimension components. Maybe \Gamma.\tilde{S.} should be limited to nonnegative dimensions?
[I’m not quite sure what you mean here. Here S. is a polynomial ring, so there shouldn’t be anything in negative dimension.
Oh, maybe you mean in the case where it is a polynomial ring in zero variables — I’ve now mentioned that it is in at least one variable. Or did you mean something else? -R.]
Also I met difficulties with 16.6.G: I could not prove S. -> \Gamma.\tilde{S.} defines a morphism on all of Proj \Gamma.\tilde{S.}, nor could I prove \Gamma.\tilde{S.} is finitely generated in degree 1. Could I get some help on these statements?
[By this I think you meant 15.6.G, but let me know if I am misunderstanding. I don’t think you actually need to prove that \Gamma. \tilde{S.} is finitely generated in degree 1! Part of the point here is that on the “geometry” side things are easier, while you have to work harder on the algebra side. The secret fact is that we’ll never need to understand the algebra side (at least for any of the fancy things we’ll do in algebraic geometry). So instead of trying to show an isomorphism of graded rings, try to show isomorphisms over “projective distinguished open sets”. After doing this problem, I recommend skipping all the later sections on the reverse map, and the saturation map. They feel like they should be useful, important, and not too hard. In fact they are none of these three. -R.]
Thanks again for the wonderful book!
Sincerely, AprilGrimoire.
[And thank you very much for these comments! Of course, I still owe you responses for your next batch, below. -R.]
November 4, 2023 at 2:58 pm
1. It seems simpler to me to define blow ups in 22.2 without mentioning the exceptional divisor. For a closed embedding X->Y, you can just say it is a Y-scheme, terminal with respect to the property that the inverse image of X is an ECD. This is subtle, but it does simplify some of the diagrams that follow, in that you do not need to always write out the thing lying above X.
[I like this point, and hadn’t considered it before! -R]
2. In 22.2.6, you say “Can you see why this is locally a nonzero divisor?” I found this more complicated than this would suggest. In order to apply the previous description for affine things, 22.2.D-F, you need to know that you can compute the closure affine locally. And we know that this can be done in general just for morphisms that are quasicompact. And this open embedding is quasicompact, because it is the complement of a locally principle closed subset.
[I agree it is tricky! -R]
3.Proving the blow up closure lemma did not feel tricky at all to me (though perhaps there is a flaw in my solution). Coming up with the answer to the question mentioned above was much harder for me. In proving the lemma, I essentially just used all of the universal properties of everything,
[Nicely done – that’s the way to do it! -R.]
4. In 22.3.6, an easier example of a normal cone not embedding into the scheme is of a smooth point on a non-rational curve.
[I literally groaned when I read this. How did I not see this? I’ve now added it. – R.]
5. In 22.3.11, I do not see a reason to use smooth instead of regular. The proof becomes simpler, not having to go back between smoothness and regularness.
[I agree. I’m hoping the reader will realize this, if they are not overwhelmed. I’m keeping the discussion in section centered on smooth things. -R.]
6. In 23.4.J, I think you want Z to be cut out by one equation. Also, it is not exactly clear what the multiplicity of the exceptional divisor should mean.
[Interestingly enough, the problem still works even if Z isn’t cut out by one equation. I’ve added a bit more text to make clear that part of the problem is to figure out how to make things precise, as you say. -R.]
7. In 23.4.L, I believe the notation is supposed to mean that you are twisting B*L by O(-E), but is O(-E) defined? I don’t think you have defined what this should mean unless X is normal, because you are avoiding general Cartier divisors. But if you just say O(E)^{-1}, that should be fine, because you have defined O(D) when D is an ECD.
[I now more firmly define it earlier, including adding a new index entry, but don’t expect the reader to flip back and remember it. I’m hoping they will realize what it has to mean from the context, as you did.-R.]
8. In 17.2.B, I believe it is important that the maps \iota_U are U-isomorphisms, not just isomorphisms.
[You are right — I’ve changed the wording ever so slightly to fix this. -R.]
9. I believe 18.8.A and 18.8.B need the hypothesis that A is Noetherian, because otherwise, you cannot conclude there is a power killing the ideal sheaf or that there are finitely many components, respectively.
[Right! I’ve added the hypotheses to the statements. -R.]
10. In 18.6.Q-S, and maybe during other parts of 18.6, it feels like you are implicitly using the fact “complete intersections in P^n do not have embedded points” in order to repeatedly apply Bezout’s theorem.
[I think we’re okay there. In the definition of complete intersection, each “slice” is an effective Cartier divisor on the previous one. So even if you had embedded points at some stage (which secretly we know we can’t have), the results should still hold. Let me know if there is one of the problems where this is not the case though. – R.]
11. In 24.6.L, I believe the locally Noetherian hypothesis is unnecessary. It seems that what I presume to be the intended proof does not use the hypothesis, though maybe I am missing something.
[That is a good and important point. I think you are right (and the fact that you have a proof on your own makes me feel more secure about it). I’ve now removed that hypothesis. -R.]
February 4, 2024 at 11:23 am
Thanks for all of that! My responses are now embedded in what you wrote, in italics.
November 20, 2023 at 9:11 pm
8.4.1 – Is there any particular reason for the Noetherian hypothesis in the definition of a constructible subset? I think this should be explained in the text because it seems very mysterious otherwise.
January 22, 2024 at 5:39 pm
I deliberately avoided stating that things become more complicated in the non-Noetherian setting, but 10.3.I mentions this setting, and gives a reference (in EGA). Basically, the right definition of constructible subset is the one that makes Chevalley’s Theorem true!
November 20, 2023 at 10:02 pm
Also, it seems like Grothendieck’s generic freeness lemma has completely disappeared. There is an index entry that says “Grothendieck’s Generic Freeness Lemma, see Generic Freeness Lemma” , but “Generic Freeness Lemma” doesn’t exist in the index.
January 22, 2024 at 5:19 pm
You’re right, the generic freeness lemma is now history. I’ve now removed that orphaned index entry!
December 2, 2023 at 12:46 am
Dear Ravi,
in 21.2.E. the quotient should probably be
(\bigoplus A dx_i)/(A df_1 + \dots + A df_r)
instead of two direct sums. Take for example n=r=2, A = k[x,y], and f_1 = f_2 = x. Then I fail to make sense of the statement, if there is a direct sum.
Best,
Clemens
January 22, 2024 at 6:51 pm
Ah yes, I see your point. I know what I meant, but it isn’t clear from what I had written. I have replaced it with what you suggested. Thanks!
December 13, 2023 at 12:39 pm
In 15.4.A, I think “irreducible divisors correspond on” should be “irreducible divisors on”.
December 31, 2023 at 7:27 pm
Thanks, fixed!
January 1, 2024 at 7:49 am
P. 138, 4.3.7, “By Exercise 4.1.F, the stalk…”. The reference to Ex. 4.1.F is stale. There used to be an exercise 4.3.F that stated exactly this. I’m guessing that 4.3.F was replaced
by 4.1.E, but the focus of that exercise is slightly different (it centers on localization of modules) and the fact that A_p is a local ring isn’t mentioned there.
February 4, 2024 at 7:38 am
That is an important point. I’ve re-added the exercise to just before the start of the proof of Theorem 4.1.2.
January 10, 2024 at 12:09 pm
In Exercise 17.2.H. I believe that the relative Proj of S_\bullet should be replaced by that of its dual, so as to be able to apply 9.3.B. locally to the surjective morphism
Sym^\bullet (S_1^\vee) -> (S_\bullet)^\vee
– assuming of course that I do not miss something.
February 6, 2024 at 11:10 am
I think what you means is that in 17.2.H, the upper right should be Proj Sym^\bullet S_1, and then I agree with you. The current revised version is here.
This is part of a larger issue, that I need to make consistent: on the choice of definition of “P F” where F is a coherent sheaf on X. Some would define it as Proj Sym^\bullet F, and others would define it as Proj Sym^\bullet F^\vee. It would be easiest and more natural to take the first definition, but there are geometric reasons (when F is locally free) where the second definition is the one that people are often used to taking. My personal conclusion is to try to be clear on what I mean whenever I use the notation, and I’ve put a paragraph to that effect in 17.2.3 (in the file linked here). One motivation for the seemingly weird definition as Proj Sym^\bullet F^\vee is in Exercise 17.2.I (also in this two-page snippet).
But I am still thinking about this, and there are still errors/contradictions in the notes, as pointed out by Swapnil Garg, whose email I am working through.
January 17, 2024 at 10:47 pm
Hello Professor Vakil, thanks for your great book! Again I’ve found some potential issues in the book, and I hope this could make the book better:
17.3.A.(a): The usage of P here is inconsistent with definition in 17.2.3. In the definition, the sheaf given is taken dual of. However, when I_1 is not locally free, taking its dual might make no sense. Also this exercise seems to use ideas from 15.6.J, which is behind something starred. So maybe some hint should be given.
[Yes, I had not correctly updated the use of P, and I still haven’t elsewhere (but will shortly). In this exercise, I’ve removed any use of that notation now. I hope it now works, and hopefully all remaining issues have disappeared. The revised version is here. Some changes here came from comments of David Speyer and Swapnil Garg, so I thank both of them as well. – R.]
17.3.B: This exercise seems broken. Someone has posted on MSE about this exercise: https://math.stackexchange.com/questions/1863379/how-to-complete-vakils-proof-that-the-composition-of-projective-morphisms-are-p
Some people have worked on this exercise, sometimes even using concepts not yet mentioned in the book. However, they still couldn’t get the result in this exercise. The exercise states something even stronger than what Stacks Project proved. Please double check if this exercise is possible.
[Aha, I now understand the issue people were concerned about; I had thought it was something else. The exercise indeed needs an additional hypothesis. More on this soon (certainly within the next month). – R.]
18.4.R: Before definition 18.4.7, C is required to be irreducible. However, a fiber K over r in P^5 might not be irreducible, as I think it could also be two lines.
[In the second paragraph of definition 18.4.7, it is pointed out that the irreducibility condition can be dropped. But then you would reasonably ask what might happen if C is nonreduced, i.e. a double line. I might then reply that definition 18.4.7 still makes sense even though I didn’t say that it did. But forget all that — for the purposes of this exercise and its applications, it suffices to consider the case where C is integral, so I will add that hypothesis. -R.]
-R]
18.6.A: There seem to be a grammer mistake in ‘and where D does not meet the associated points of F’.
[Right, now changed to “does not contain the associated points” – R.]
18.8.3: The usage of m_p is inconsistent with the definition in the index, 2.1.1, where m_p is defined as the ideal of germs vanishing at p, which in general is not coherent.
[I don’t follow you here – I don’t see how it isn’t coherent. (Don’t forget the Noetherian hypotheses!) Note to reader: follow-up is here. – R.]
18.8.5: Maybe I|q^bar should be used instead of I|q? Otherwise I -> I|q might not be a surjection of quasicoherent sheaves.
[Yes, that’s a good point. I think making this change once in the statement of 18.8.F resolves the issue. But if I’ve missed anything (e.g. another place where it needs to be fixed), please let me know. -R.]
18.8.C: The exact sequence in the hint seems invalid. I’ve checked an example and it seems no such exact sequence could exist in some cases. Is the direction of arrows wrong?
[Indeed the map coming from adjointness was wrong! I have now put the maps in the right way, and then the argument as I intended should work very cleanly and nicely. The revised version is here. I hope someone looks at this closely — this is (as are many things) my own argument (although of course I fully expect not to be the first person to have done this, and very likely this is how the argument was first shown). -R.]
19.1.1: There seems to be a typo: ‘Those allergic to algebraically closed fields’ seems should have been ‘allergic to not algebraically closed fields’.
[I think that’s as it was intended — even if people are mainly interested in fields that are not algebraically closed should still pay attention. – R.]
19.2.2: Here w_C is stated to have g sections. I think this should be g linear independent sections, or is this an accustomed shorthand?
[It is a common shorthand, but I’d rather say it correctly, so I’ve changed the wording. – R.]
19.5.2: Here it is stated ‘Thus in K(C), we must have z^2=u^r f(1/u)’. However, I think there might be a constant multiplier between each side of the equation.
[Right; that multiplier needs to be pulled in to the variable z. I’ve made this more explicit — and actually correct — here. -R.]
19.8.4: There seem to be a missing word: ‘there is only _one_ genus 0 curve’, ‘one’ is missing.
[That’s right, now fixed. – R.]
21.2.12: Here after ‘or more formally’, the map delta is given by 1 tensor d: B/I tensor_B I -> B/I tensor_B Omega_B/C. However, when writing functions with domain A tensor_C B as f tensor g, f and g need to be C-linear. Here d is not B-linear.
[Indeed – and yet it still works! If you see what you need for this to be well-defined, you will see that a miracle occurs, and just the right miracle. – R.]
21.2.22: Here in the formula Sum x_i(1 tensor y_i – y_i tensor 1), I think it would be better to state explicitly that A acts on A tensor_B A by multiplying on the left component.
[I think I see what you mean – here in the equation display, instead of x_i, it would be better to say (x_i \otimes 1). That makes sense. If you disagree, just let me know. – R.]
21.3.C.(c): How can (a) be used to show that any differential preserved by the involution must be pulled back from P^1? I’ve paid a strong peer student to help me solve it, but he failed too. Someone posted this question on MSE, and there’s only a comment suggesting Lefschetz fixed point formula could be used here, which is not covered in the book. May I get more hints on this?
[Part (a) was not the right hint to give! It is much easier than that (I think). A revised version of the exercise is here. I want this exercise to be a pleasant and straightforward one, and so far it hasn’t been (and I suspect that roughly the same exercise isn’t so available elsewhere, or else someone would have pointed to it). But I hope this current version helps set up how things work, and points the reader in the right direction. To everyone (not just Grimoire April), please try it out and let me know how it goes. -R]
21.3.D.(a): What does ‘Interpret a generator as x^-1 dx’ mean? Am I supposed to calculate manually using Čech cohomology, or am I supposed to show it has dimension 1 and interpret it arbitrarily?
[Try calculating it manually using Čech cohomology. Then you’ll solve the problem, and also be led to wonder about other things (including – doesn’t this depend on choice of cover?) -R.]
21.4 p610: Here it is stated ‘Then as pi is finite’. However, when X is an open subscheme of Y, pi might not be finite.
[Good point. I have no idea what I was thinking there, and it is irrelevant to our local study. Instead, I have replaced it with: “By restricting to an affine
open neighborhood of $p$ in $X$, we assume that $X$ is affine
too.” – R.]
21.4.A: Here it is stated ‘assuming the image of the ramification divisior is indeed a divisor’. What is the image of a divisor? It is not defined previously. Is the divisor implicitly identified with a closed subscheme?
[That’s right – the only reasonable meaning is the associated closed subset, which can be called the support of the divisor. – R.]
21.4.5: Here the definition of simply connected is incomplete. The covers also need to be connected.
[You are right! The word “connected” is now added. – R.]
There are two definitions of degree of a coherent sheaf on a projective curve: as in Riemann-Roch in one definition, and by Hilbert polynomial in another. Hopefully there would be some disclaimer about this.
[Can you tell me which definitions you mean? Here are the four possibilities. (1) There is the initial definition, for a line bundle on a regular curve (take a non zero rational section, count zeros minus poles). (2) 18.4.2 has the degree of a line bundle on a curve, and is cohomological in nature. This generalizes the older definitions of degrees of line bundles on projective regular curves, via Riemann-Roch. (3) In 18.4.7, there is the degree of a coherent sheaf, but this time on an integral projective curve. But it clearly agrees with the definition of 18.4.2 where both definitions apply, since they are the same cohomological definition. (4) Finally, there is Exercise 20.1.C, but it also agrees with 18.4.2 (indeed that is what the exercise is about). I’m going to have to have proper references to these three in the index, and will get to that soon. The index is still very rough though, and is a big challenge.
Perhaps you mean: if you embed a curve with a degree d line bundle in projective space, why does the image have degree d? – R.]
21.6.1: Here X also needs to be reduced. An unreduced point is never smooth.
[You’re right, and I’ve fixed this. (Swapnil Garg simultaneously noticed this.) – R.]
Also it is stated by an exercise in a starred section ‘every extension of a perfect field is separably generated’, but this is only true for finite extensions.
[If I understand you correctly: I should have (and now have) added “finitely-generated” (not finite) in this proof, but it indeed applies happily without change. -R.]
21.6.5: Here what does K(X’) mean? We have not shown X’ is integral so far. Also in the definition of X’ only the minimal polynomials for x_i is used, without using relations between different x_i. How is this enough?
[Note to reader: follow-up including complete response is here. -R]
22.3.D.(a): Should ‘normal sheaf’ be replaced with ‘conormal sheaf’? This naming convention appears a few times also in this section, inconsistent to the naming conventions discussing differentials.
[The normal sheaf is the dual of the conormal sheaf, see 22.2.15. -R.]
Also there’s a small issue with the index: ‘Bibliography’ and ‘Index’ is placed under ‘Part VI. More cohomological tools’, contrary to ‘Preface’ placed on the first layer. I think they should be moved to the first layer too.
[This is something I have no idea how to fix, as it is a latex issue! Hopefully someone can tell me what to do… – R.]
February 10, 2024 at 2:33 pm
I have now responded fully to your comments, inline in italics! (I’m mostly done responding to your later comment here, but at the time I type this, I have one thing to think through, and so I’ve not officially replied. But almost all responses are there too.
February 11, 2024 at 2:19 am
21.6.1: Here X also needs to be reduced. An unreduced point is never smooth.
[You’re right, and I’ve fixed this. (Swapnil Garg simultaneously noticed this.) – R.]
Also it is stated by an exercise in a starred section ‘every extension of a perfect field is separably generated’, but this is only true for finite extensions.
[If I understand you correctly: I should have (and now have) added “finitely-generated” (not finite) in this proof, but it indeed applies happily without change. -R.]
I think the ‘applies happily without change’ part is not true: for example, consider an algebraically closed characteristic p field k. Then the direct limit of k(x^(1/p^n)) seems not separably generated. The issue is that the proof in the book uses descent of inseparable degree, but in non-finitely-generated cases it could be infinite.
February 17, 2024 at 6:03 pm
Aha, I think I understand the point you are making. I think your comment is about the last two sentences of 10.5.18: “A field extension is said to be separable if all finitely generated subextensions are separably generated. Thus Theorem 10.5.17 implies that all extensions of perfect fields are separable.” The last sentence is indeed nonsense, and I’ve just removed these last two sentences because even the second-last sentence adds nothing and is never used. I think these two sentences don’t affect anything else in the notes (I tried to be careful to see if I ever speak of separable extensions that are not algebraic and I don’t think I do, and I tried to be careful to say “finite separable” and “algebraic separable”). If there is some mathematical error somewhere, of course please let me know.
January 18, 2024 at 10:51 am
There were some issues with Grimoire April’s comments being posted in the past, so there is some chance that something slipped through the cracks. Rather than carefully checking, I will simply post this (even though it maybe partly or completely a duplicate). (Also, although there are a number of comments not replied to because I’ve not had a chance to read them, but I certainly will before long!)
Update February 4, 2024: Grimoire April’s comments indeed successfully posted, so for simplicity (and so I don’t get confused as I reply to them), I have deleted my copied versions (originally in this comment), and will instead give you the links to the comments:
first batch
second batch
January 18, 2024 at 9:46 pm
I think the way to approach 2.7.F is to generalize. The diagram of topological spaces induces a corresponding diagram of categories of sheaves on each space, and the α’s and β’s give you adjunctions between those categories. You can just generalize now to the case of a diagram of categories and adjunctions that commutes in the same way. Those adjunctions give *bijections* between sets of arrows in those categories. So now you have a diagram of isomorphisms that commutes in one direction, but because all the arrows are isomorphisms, it also commutes in the “other” direction, and that gives you the equality of the two push-pull maps.
January 18, 2024 at 10:38 pm
(Full disclosure: it took me a while to come up with that solution. I would never describe this as an easy exercise.)
February 4, 2024 at 11:57 am
Very nicely done Mark! This is indeed hard, and I would even say that it is surprisingly hard. I think I will add a star and a warning to it. It seems like a necessary exercise to state. Otherwise one glibly just uses the push-pull in life without realizing that there is something very subtle in knowing what it actually means, and that there are potentially (at least) two definitions which are not obviously the same.
January 21, 2024 at 3:27 am
I gather that this book will be formally published at some point? Are you going to keep the PDFs available?
January 29, 2024 at 6:54 pm
Yes, I think it makes sense to have an official published version, and I also very much prefer to keep these pdfs available. These goals are definitely in tension but in the end not in contradiction.
I might post something about these issues at some point, because there are some complicated and nonobvious issues involved, and things require compromise and sacrifices of various forms (not all visible). I want to say that the all of the potential publishers have been very good about it, and I want to recognize and thank them for their willingness to be flexible on this issue. I have some serious hopes of having a version ready for publication quite soon, as I’m now going to be very busy with trying to be a particularly good citizen in the american mathematical society for next four years. But even after a version is ready for a “snapshot”, I intend to keep fixing errors and infelicities as people point them out.
I want to say that the support and comments of the wider community (on this site and elsewhere) has made the essential difference in helping publishers realize the importance of keeping these notes available. But also the comments and support are what made the notes happen in the first place — in giving me the energy and motivation to progressively draft them, and then over the years in relentlessly explaining to me how to refine and improve them.
While I hope that many people will want to have a physical copy that hold in their hands, and write in, and put by their bed, and throw at a wall, and use as a paperweight, it won’t be for the benefit of my own pocket. (This is not the way mathematicians make money…) But I also want people who want to glance through a few pages, or browse through it, or read a few key chapters to be able to do so freely. If they are just curious they should take a look. And I like the fact that people who can’t so easily afford books (or have to make hard decisions on what to buy) can still learn essentially just as easily, no matter where they are in the world. And this particular subject is very hard, so if clearing a path to the front door tips someone into taking the risk and starting to read, because what the heck, it’s free, then that’s a good thing too.
January 26, 2024 at 6:44 am
Dear Ravi,
I finished reading your notes some years ago. I left a comment with some math errors under your announcement of the August 2022 version of the notes.
The new version seems to have fixed over half of the mistakes I found! However, some remain. My goal is to write here the mistakes which are still in the notes, by order of appearance. Thank you very much for writing these notes, they were very helpful and enjoyable to read.
Here is the list:
1) The proof of Theorem 18.1.3 is flawed, as one cannot conclude G is coherent just from its definition as the kernel of O(m)^j->F. This is because O(m) might not be coherent in the non-Noetherian case.
2) The proof of Theorem 18.2.2 is incomplete, as it is not clear that the definition of Cech cohomology of a cover is independent of the order of the sets U_i in the cover, so you cannot assume without loss of generality that the added open set is the 0-th one.
3) Related to the previous point, the proof of Theorem 18.1.2 is flawed because the Cech complex is not symmetric with respect to permutations of the variables, so you lose generality when assuming which variables in the monomial have negative exponents.
4) Theorem 19.1.3 is stated in a not well-defined manner. First, it’s not trivial that C^{reg} is indeed an open set (it’s true, using tools from Chapters 21 and 24 one can show C has a nonempty smooth locus). Second, it’s not necessarily true that the pullback of C^{reg} to the algebraic closure C_{\bar{k}} is the regular locus of C_{\bar{k}}. This is true if one replaces “regular” with “smooth”, however again I cannot prove this without tools from Chapter 21. It’s also possible to show that if p, q are in the smooth locus than so is t(p,q), so that indeed replacing “regular” with “smooth” fixes everything.
5) The proof of Proposition 20.1.6 is flawed, as one cannot apply Exercise 18.6.A without the base field being infinite. This could be fixed by first base changing to a field extension, however it is not clear that this base change preserves numerical equivalence. One can show that any algebraic extension preserves numerical equivalence, but this is nontrivial.
6) In the hint for Exercise 20.2.I, the notion of normal bundles is used, which is only defined in Chapter 21.
7) After Exercise 24.4.Q, the written formula for the flat limit is wrong, one needs to quotient by the ideal (m^2, mx, my, y^2-x^2-x^3). In fact, the written formula is identical to the formula before 24.4.P, which is stated to be wrong.
8) The proof of Theorem 28.7.1 is incomplete in the sense that Y is not shown to be smooth, but regular (which is a problem over non-perfect fields). There is a workaround though.
This concludes the list of mathematical errors remaining in the book that I know of. Most of the mathematical errors 1)-8) can be worked around with some extra work. I’d be happy to share the exact way this is done for each case if requested.
In any case I want to express deep appreciation for you leaving these wonderful notes for the public to enjoy.
February 10, 2024 at 5:41 pm
Thanks for this! I realized that I didn’t get around to replying to your earlier comments before the posts moved on, so I’m glad you reposted.
Responses:
1) There are (at least now) Noetherian hypotheses, so this is okay.
2) You’re right; the reader has to realize how and why the order in Cech cohomology doesn’t matter.
3) I think this one is okay, in that here (more so than in 2)) the reader can really follow through what is going on.
4) I see your point. What is implicit here is that the regular points are the same as the normal points (for curves), and the normal points form an open set (here using that normalization is a finite morphism). I should figure out where to say this once and for all.
5) You are right — and there is a different patch, and I’ve edited it for the next version. We are using the fact that in projective space over a field, given any finite set of points, there is a hyperplane missing them all *so long as the field is infinite*. But even if the field is finite, there is a *hypersurface* missing them all.
6) Actually, normal bundles to effective Cartier divisors are defined in 15.5.2 (much earlier). I’ve now added a reference at 20.2.I pointing back to it, because certainly the reader shouldn’t be expected to remember.
7) You are absolutely right. If I am not mistaken, the only needed patch is just the change you say (the addition of my to the list of generators).
8) Good point; I now alert the reader explicitly to this fact. (Behind it: a k-valued point that is regular is also smooth.)
Thanks again!
February 4, 2024 at 12:21 pm
Exercise 12.1.D : I think this is false if the dimension is infinite. For example, $k[x_1, x_2, … ] \rightarrow k[x_2, … ]$ given by $x_1 = 0$. This is surjective, and the rings are isomorphic, but the given function is not an isomorphism.
Exercise 12.1.H : Would this approach work? It seems to be much easier than using the hint.
Assume $A$ is an integral domain. Then, since $k \rightarrow K$ is integral, and integrality is preserved by base change, $A \rightarrow A \otimes_k K$ is also integral. By flatness of $A$ over $k$, it’s injective. Conclude by exercise 12.1.F.
February 4, 2024 at 12:30 pm
On second thought, I think it’s the “pure dimension” part that might be an issue, as irreducibility isn’t preserved by base change.
February 4, 2024 at 12:44 pm
In 12.1.D, I certainly wanted the dimension to be finite, so thanks for catching that; I’ve added “finite” to the statement. I don’t fully follow what you are proposing for 12.1.H, but it sounds like it didn’t work out. If you end up with a different attack that is better (at least in some regard) please do pass it on.
February 5, 2024 at 7:29 pm
Warning 10.7.4 tells me to see Eisenbud, p 299 for a discussion of Noetherian rings with non-Noetherian integral closure. In the edition of Eisenbud which I just downloaded from Springer’s website, the only relevant thing I can find on p 299 is Exercise 13.10, which I don’t think is what you meant to point to.
My guess is that you wanted the portion of Eisenbud after the proof of Prop 13.14, which is pages 294-295 in the edition on Springer’s website.
February 6, 2024 at 6:04 am
Oh, interesting. I just checked the copy on my office book shelf, and it is page 299 there. So I guess you don’t need to fix anything.
February 6, 2024 at 9:22 am
The Springer e-book I have (which agrees with your original comment) says it’s a “softcover reprint of the hardcover 1st edition 1995”. Perhaps that is relevant? I find it hard to imagine that a reprint would change page numbers though. Bigger question: which of the two should we think of as “canonical”, in this digital age?
February 6, 2024 at 9:33 am
I just looked at the version on my shelf at home (I also have a copy at work, which may not be the same edition). Here it is p. 294-5! I had tried to be careful with stating which edition of which book I refer to. But I obviously wasn’t careful enough. I prefer to refer to section numbers rather than page numbers (because page numbers migrate in books over editions more often than section numbers), but this comment in Eisenbud doesn’t have an obvious “anchor” to reference. (This is one reason I chose to have this system of “10.7.4” etc. in order to have as much as possible “anchored”, incidentally.) I think I will just change this reference to section 13.3 of Eisenbud, and let the reader figure out where in this short section the discussion is. It’s not a perfect solution.
February 6, 2024 at 12:18 pm
Thanks!
May I suggest adding a reference to Chapter 9 of Reid’s “Undergraduate Commutative Algebra”? What I was looking for last night was a clear write up of the infamous “counter-examples of Zariski and Akizuki” which are cited in so many places without detail, and this is the best I’ve found so far.
But I haven’t found a good write up yet of a Noetherian ring with non-Noetherian normalization.
February 6, 2024 at 12:49 pm
Ah, what a great dude Miles Reid is! I’ll take a look when I’m in my office tomorrow (and then replace this comment).
February 6, 2024 at 5:07 pm
At this point I’m trying hard not to add any more books to the bibliography, but in this case it is worth it, and I’ve indeed added a reference to section 9.4 of Reid, because the reader may be led to so much good stuff there!
February 5, 2024 at 8:37 pm
Hi professor Ravi. I think the sequence of inclusions (1.7.2.2) should be in the reverse order. Specifically, I think it should read
E^{0,k} –E^{1,k-1}–> ? –E^{2,k-2}–> … –E^{k,0}–> H^k(E^·)
(let’s call this (1.7.2.2′) to distinguish it from the current version (1.7.2.2))
Here are some reasons.
Following the tip in the next paragraph (“Here’s a tip…”) one would conclude that the “deeper” term (i.e. the smallest term in the filtration) should be the one close to the heads of the arrows, i.e. E^{0,k} which is on the vertical axis.
Also, following the proof that “spectral sequences work” from the “puzzling” picture book, I arrived at (1.7.2.2′) instead of (1.7.2.2).
Finally, IMPORTANT EXERCISE 1.7.E (the mapping cone) doesn’t work out for me using that filtration, while it works just perfect using the filtration in the other order.
Thanks for all your work.
February 6, 2024 at 10:30 am
I think you are right.
What makes me worried: (i) I always get confused about these things. So I’m easily swayed by suggestions such as yours. (ii) Furthermore, at some point I changed my convention for indexing spectral sequences (discussed here), and at that point changed *from* what you wrote to what is currently in the notes. (iii) No one mentioned this earlier, although it is an extremely well-traveled territory (both intellectually, and in the notes).
What makes me think you are right: (i) you thought through why spectral sequences are true, and it led you to this. (Plus, it makes me happy that you came to it through the picturebook, which means that the visual representation helps at least one person think clearly. I get much more confused when just shuffling indices.) (ii) You used my own words against me (the tip for remembering that later differentials point deeper and deeper in the filtration – I can’t remember now who I got that insight from). (iii) you actually used it in practice, for the mapping cone.
Right now my conclusion is that most people nod their heads (or avert their eyes) far too quickly when dealing with fundamental technical tools such as spectral sequences, and your willingness to fully engage with it means that you have particular mathematical strength and ability that others should aspire to. (This is true even if we end up being wrong on this…)
So I think the needed corrections are the following. (1.7.2.2) should be changed as you say. And also the “information from the second page” (Exercise 1.7.A) also needs the superscripts reordered, four times. If there are any other places (anywhere in the book) that need editing as a consequence (or that perhaps should be looked at with care), please let me know!
And I hope that someone jumps in to agree or disagree with this. It will make me feel much more secure about it.
February 6, 2024 at 11:01 am
Without actually thinking carefully about this, I can provide some further evidence in Jackozee’s favor. You reversed the index convention in (1.7.2.2) between the 2022-08-29 draft and the 2023-06-29 draft, but you did not change the indices in the “anticommutes” diagram a couple of pages earlier that more or less defines that convention. I read through Section 1.7 pretty carefully about two years ago using the 2017-11-18 draft, definitely before the switch, and I didn’t notice any issues with the indexing in (1.7.2.2) then.
February 6, 2024 at 11:21 am
Excellent. I’ve made the changes described, and so far it is looking like no disaster has happened…