**The seventh post is the November 24 version here. The next reading is 9.3-10.6, which is mostly about fibered products.** Section 11.1 is included in case you want to read ahead. As always: skip the starred bits (unless you are moved to read them), and skip the double-starred bits (unless you are really really moved to read them).

As promised, I will be slow a little longer: I expect the eighth posting to be in three weeks, on December 18. The ninth will likely be three weeks later (roughly January 8), and then I will begin catching up with many loose ends. Some brief thanks to people who have emailed me corrections and suggestions: I’ve incorporated significant chunks of ideas from the Duke group communicated by Ezra Miller, and Lindsay Erickson at Washington.

**For learners.**

*9.3* consists of constructions related to “smallest closed subschemes”. The hardest thing about this is the argument that the scheme-theoretic image can be computed affine-locally (on the target) in reasonable circumstances.

*10.1* shows that fibered products of schemes always exist. The proof is intended to set up how to think about similar arguments in the future. This argument is key, and quite subtle the first time you learn about it, so it is crucial that this section be readable. A double-starred section interprets this in a way that is actually quite pretty.

*10.2* is about how to compute fibered products in practice, and is useful and not too hard.

*10.3* is about pulling back families, and fibers of morphisms. It is mainly language.

*10.4* is about properties preserved by base change, and the non-starred portion is short.

*10.5* is on the Segre map.

*10.6*, on normalization, is here because the argument that it exists parallels the existence of the fibered product. It is short, and there are concrete problems to work on.

“Theory” exercises most worth doing: 10.1.B, 10.2.A, 10.2.B, 10.2.E, 10.4.B, 10.4.E, 10.6.A, 10.6.B, 10.6.I.

(It is also worth getting hands-on practice. Exercises of this sort include: 10.2.K, 10.3.C, 10.3.D, 10.6.F, 10.6.H, 10.6.J, 10.6.K.)

As usual, I have some questions for you:

(a) If you read the (double-starred) section 10.1.5 on describing the existence of fiber products in the language of representable functors, can you please tell me how it went, and what was most confusing (and what perhaps was enlightening)?

(b) If you try Exercise 10.2.L (that various definitions of the graph of a rational map are the same), could you please let me know how hard it is (e.g. in an email or a comment below)?

(c) Can anyone solve Exercise 10.4.C (the fact that quasifinite morphisms are preserved by base change) from the hints given, without referring to some other source?

(d) If anyone is reading the starred sections on “geometric fibers” etc., could they please try 10.4.J, and let me know how hard it is for them? I’m hoping it is easy.

(e) If anyone tries 10.6.K(b) (normalization of P^1 in a function field extension), please let me know how hard it is. I have reworded it so it is hopefully more gettable.

**For experts.**

I’ll begin with some language questions.

(L1) Dan Abramovich emailed me the following: *I noticed that you are following Hartshorne’s translation of the French “immersion ouverte” as “open immersion”. I believe the correct translation is “open embedding”.* A question to people whose french is better than mine: is he right? If so, a possible radical proposal is to be originalist, and use “open embedding” (and “closed embedding” and “locally closed embedding”). This would be making a case for a cultural change, so I presume I shouldn’t do this.

(L2) Is it reasonable to say “X is finite type” when one means to say “X is of finite type”? How about “X is a finite type A-scheme”?

(L3) Is a geometic point of X a map from Spec of an algebraically closed field into X (which is what I’d thought, and Mumford backs me up in lecture 3 of Curves of an Algebraic Surface), or the map from a separably closed field?

Next, some remarks.

(R1) Brian Osserman once told me an interesting cultural note (cf. 9.3): “EGA seems to have missed the fact that scheme-theoretic image always exists, although the proof isn’t too hard.”

(R2) The difficult point in section 9.3.1 on scheme-theoretic image is that in good circumstances, it can be computed affine-locally. Behind some of the discussion is the key fact that what matters is that (for f: X –> Y) f_* O_X is quasicoherent. But I don’t mention this — the reader doesn’t yet know about quasicoherent sheaves. I never come back to this point, and currently don’t feel guilty about it.

(R3) No one complained (in response to my previous post) when I proposed the phrase “the reduced subscheme structure on a closed subset” rather than the “reduced induced subscheme structure”. I have also realized that this concept is never used (aside from the definition of the “reduction” of a scheme, where we can work around it)! (It will get used only in the proof of the valuative criteria, but you will see that that in turn never gets used, and is mentioned only for cultural reasons, see below.) This makes me want to remove this notion — we shouldn’t keep things only for the sake of nostalgia. But it is short enough that so far it has stayed. But feel free to vote it off the boat (or to keep it on…).

Next, some questions for you.

(Q1) Is the description of the existence of fibered products using functors (10.1.5) reasonable? (Of course, it could be done very quickly and concisely; but it would be great if mortals could learn this in a comfortable and natural way soon after first seeing fibered products.)

(Q2) Notice my question for learners: Can anyone solve Exercise 10.4.C (the fact that quasifinite morphisms are preserved by base change) from the hints given, without referring to some other source? I’d like to ask experts this too: Is there a slick proof I’m missing?

(Q3) I do normalization of reduced schemes, using a universal property. (a) What is the right universal property of normalization?

(b) (less important) I could imagine that one could define normalization more generally; is there any value in this (even in the sense of giving cleaner statements)? My current opinion is “no”.

Finally, I want to give an open challenge about valuative criteria. I’ll save this for a separate post.

**For everyone.**

I’m thinking of including arguments (in a starred or double-starred section) showing that to check geometric reducedness, it suffices to check after extending only to the perfect closure, and the analogous statements for geometric irreducibility and geometric integrality, based on **this note** (thanks to Greg Brumfiel and Brian Conrad — revised link Nov. 27 thanks to Brian’s suggestions). Votes or opinions for or against? I’m partially motivated by the fact that I realized after learning some algebraic geometry that I didn’t know why the product of two irreducible complex varieties was also irreducible.

If anyone reads the (sketched, double-starred) proof of finiteness of integral closure (10.6.4), please let me know if it is readable. (I hope someone reads it…)

**Chevalley’s Theorem**

The earlier Chevalley exposition of 8.4 was indeed botched, and I am grateful to many of you (notably Charles and D.H.) for helping me sort this out. It is rewritten in this version, but I realize I’ve tried everyone’s patience too much to ask you to look at it again. My rearrangement of the proof of closedness of finite morphisms (thanks to Matt) has led to a rearrangement here: I no longer use elimination theory to prove Chevalley, and just use closedness of finite morphisms. I want the proof of Chevalley to have all the steps handed to the reader on a silver platter. I hope this is now the case, and I want to tweak it until I get it right.

November 24, 2010 at 6:27 pm

In 9.3.2, make clearer that you are defining the entire phrase “image lies in” (and not just “lies in”, relative to a concept of “image” that may have been defined earlier but in fact was not).

Exercise 9.3.F is vague: what is the definition of this “locus” prior to saying it is open? (The “where” aspect is unclear.) A stalk-wise property? Clarify it.

Associativity of fiber products is a good simple example of the tremendous simplicity attained by thinking in terms of the functor and not in the more explicit language of “universal mapping properties”.

In 10.1.E and 10.1.F and elsewhere, use \rightrightarrows to make double right arrows. Somewhere near the end of 10.1 it may be nice to emphasize that the functor language has no content, but is an incredibly efficient way of working with many maps all at once without getting buried by commutative diagrams. That is why it is useful.

For the start of 10.4: a nice example is that injectivity of a scheme morphism is not a reasonable notion in general!

For 10.4.2, you are right to require k to be algebraically closed, since the theory of “geometric fibers” needs it. It is only in the context of etale sheaf theory (essentially, studying fibers of etale morphisms) that separably closed fields deserve to be regarded as “geometric points”. No need to get into that in these notes.

In 10.6.D, the phrase “has irreducible components” seems not what you mean to say: every non-empty scheme has generic points, the closures of which are the irreducible components (by definition). Do you mean to stress that the collection of irreducible components is locally finite (hence the need for a locally noetherian hypothesis)? As a weird example, let X = Spec(A) where A is an infinite product of copies of F_2, so all elements of A are idempotent and hence every local ring of A is F_2, every prime is minimal and maximal, and the quotient by every minimal prime is F_2. Let X’ be the disjoint union of the spectra of the quotients of A by its minimal primes (equivalently, local rings of A at its primes). Then X’ and X are both normal and X’ –> X is bijective between generic points inducing isom. between local rings, yet not an isomorphism. Since X is normal, it shouldn’t be called the normalization. And yet…somewhat of an enigma. (It makes me doubtful about 10.6.B.) The moral is that “normalization” seems to be a problematic concept away from reduced schemes with locally finite set of irreducible components. This is what seems to be the key feature, not the noetherian property. Among such schemes, normalization always exists, and such a scheme is normal (in sense of having local rings that are integrally closed domains) iff its normalization map is an isomorphism.

November 27, 2010 at 1:41 pm

Thanks Brian! I’ve made most of the changes (except for 10.6.D, which I haven’t gotten around to), but have 2 questions.

In 10.1.E and 10.1.F, I don’t see what you mean. I suspect you mistyped the sections, and somewhere I used \xymatrix to make rightrightarrows and it doesn’t look good, but I’m not sure.

In 10.4, I’m almost certain you mean injectivity as sets, and find it a good thing to add, but I want to make sure — is that what you meant?

December 1, 2010 at 9:25 am

Yes, for 10.4 I meant on underlying sets. For Exercises 10.1.E and 10.1.F, I double checked that what I wrote was as intended (and the comment applies to Exercise 10.1.D too). For instance, you write “$X, Y \rightarrow Z$”, which is bad since there are really *two* maps here but you have just a single arrow. Better to write “$X, Y \rightrightarrows Z$”. Ditto for the pair of open subfunctors, etc.

December 8, 2010 at 2:34 pm

Thanks! All done (including the addition of radicial = universally injective morphisms) except for 10.6.D, which I will do as I gradually catch up.

September 2, 2011 at 11:08 am

About 10.6.D: good point. The question just asks for a statement of the meaning, but there is no point in asking it in a situation where the answer won’t be nice. So I’ve changed it to the locally finite case. (Incidentally, thanks to Jakub Byszewski convinced me to fix my discussion of irreducible components, and not just keep retreating to the Noetherian or irreducible case.) I’m still not doubtful about 10.6.B, as that’s still in the very nice “reduced and locally finite number of irreducible component” case. But if you (or anyone) remains suspicious, please let me know!

November 27, 2010 at 1:43 pm

Interesting comments from Brian Conrad (from an email).

(i) regarding things not immediately obvious to the pioneers: Grothendieck “missed quasi-separatedness until the writing of EGA IV_1 forced him to go back and fit it into the theory.”

(ii) “Later in EGA IV_3, section 11.9 or so there is the very useful notion of a collection of maps {Z_i —> X} being “schematically dense” in X (as well as an interesting story for its relative version). My experience is that this concept is mainly motivated by two examples: {G[p^i] —> G} for a semi-abelian scheme G over a base S (with p a fixed prime), and the collection of etale quasi-sections of a smooth scheme over a base S (or quasi-finite flat quasi-sections of an lfp flat scheme over a base S). It shows up a fair amount in SGA3, for instance. Your example of the collection of infinitesimal nbhds of 0 in A^1 naturally fits into this setting too. It seems sort of weird (to me) to take the viewpoint of “schematic image” for that example.”

(Brief response to this: it seems reasonable to want the notion of scheme-theoretic image in the two cases we discuss; and I guess we could just say “give it this name in these cases” or else “give it this name in general, and note that it only clearly behaves well in certain cases”. I’ve gone with the latter, for ease of exposition. I think of this of a case of a universal property, where existence always happens, but isn’t necessarily nice except in special cases.)

December 1, 2010 at 9:30 am

Ravi, for (i) I should have said IV rather than IV_1, since 95% of IV_1 is Chapter 0 commutative algebra stuff (with quasi-separatedness tucked into the tiny bit of non-Ch.0 stuff at the end), and the real applications of quasi-separatedness don’t come until later in IV_2 (and really IV_3). In keeping with his style, A.G. laid out in IV_1 what he’d need later, even though it would look pretty unmotivated to a reader at the time who wasn’t in a time machine reading backwards from IV_3.

[Thanks! — R.]November 27, 2010 at 2:07 pm

Update on why the “quasifinite preserved by base change” is hard (for experts): at this point, the reader isn’t assumed to know any dimension theory (of finitely generated algebras over fields), or anything about Artinian rings. They have seen the statement of the Nullstellensatz, and are free to use it, as it will be proved shortly (in a non-circular fashion!).

December 2, 2010 at 6:43 am

About quasi-projective schemes over a base: Don’t you want to be consistent with EGA and assume they are quasi-compact relative to the base? Right now in your definition 5.5.5 over a ring this is not assumed.

December 6, 2010 at 5:30 am

Whoops, and thanks! That is now fixed. I need to also decide how much of EGA’s definition of “quasiprojective morphism” to include (when introducing projective morphisms later). I just chatted with Brian Conrad to help me think this through. I’m now leaning toward saying that the reader can flip to EGA II.5.3 for a general definition, and that in reasonable situations, it is just a (quasicompact) open immersion followed by a projective morphism. (Later comment: now done.)

December 3, 2010 at 12:02 pm

Dear Ravi,

I hope you will forgive me the following nit-picking remark. Bézout, whom you mention on page 188, is spelled with an acute accent on the “e”. Here is a moving photograph of the title page of a book he wrote in 1779, which displays his name.

http://imgbase-scd-ulp.u-strasbg.fr/displayimage.php?album=1022&pos=0

Hoping that your book too will be scrutinized in 2010-1779=231 years from now, I send you my warmest regards.

Georges.

December 4, 2010 at 6:00 am

Thanks Georges! I was at some point intending to fix accents, so I’ve also done Cech (sic) as well. The changes will be in the next post. If there are more I’m missing, just let me know. Also, do you have any opinions on question (L1) above?

December 5, 2010 at 9:27 am

Dear Ravi,

I have always been astonished by Dieudonné and Grothendieck’s decision to use the terminology “open immersion” for what I would have thought should obviously be called “open embedding”. This all the more strange as

a) Dieudonné in volume 3 of his impressive treatise Eléments d’Analyse introduces the concepts “immersion”, “submersion “, “étale”

and even “subimmersion” (= locally of constant rank)in their accepted meaning in differential calculus.

b) I would have thought that Grothendieck’s philosophy in introducing the concept “étale” was to allow algebraic geometers to stay as close as possible, at least formally, to their differential topology colleagues.

Be that as it may, the issue is clearly not one of translation. Viscerally I am all for your radical proposal to write “open embedding” in your course henceforth, which indeed would be quite a cultural change.

I am sure you will take the right decision and am looking forward to your next instalment.

Very friendly,

Georges.

December 5, 2010 at 9:45 am

PS Let me emphasize that *I* don’t presume to know what the right decision is! (My formulation was a bit ambiguous)

December 12, 2010 at 12:43 pm

I’m convinced that I should leave it as it is. Thank you Georges!

January 26, 2011 at 10:15 am

Rob Lazarsfeld in his (fantastically well thought out) “Positivity in Algebraic Geometry” uses “closed embedding”, which may give me some cover. (He also uses “amplitude” instead of “ampleness”, and “free” instead of “base point free”.) I’m now leaning toward using “open embedding” and “closed embedding” and “locally closed embedding”, but I’m far from making a decision (because I’m far from having a chance to implement it).

Update March 17, 2011: I’m moving all future discussion related to this here, in the “Notation” section.

December 6, 2010 at 5:42 pm

I have generally found that most proofs of the existence of fiber products without functors are far more painful than they need to be. I left a note at http://mathoverflow.net/questions/33149 sketching my thoughts on this, although apparently after you had stopped responding to new answers.

Also, I know you imply that you could write the proof much more succinctly, but you feel it would be intimidating. I would like to suggest that the many-stepped proof that is commonly used is not an improvement; the length of the proof is intimidating by itself.

March 17, 2011 at 10:40 am

I’m still going back to the mathoverflow question periodically, but I think I’ll respond in bulk. Toward the end of the academic year, I’ll take a long look there and see what things should be added to the notes.

December 6, 2010 at 6:00 pm

Concerning the “reduced induced structure”: In my (quite limited) experience, the primary importance is the fact that any closed subset can be given the structure of a closed subscheme, and not so much the fact that we can make this structure reduced. In particular, I have seen the former used to prove that every ample line bundle has a very ample power. (See the proof in Hartshorne.)

December 12, 2010 at 12:48 pm

Thanks Charles — I agree. I’d also forgotten about the “very ample” argument, as that’s one of the four chapters still to be appropriately digested. I also mention when teaching that I think the main point is that you get a scheme structure and don’t really care that it is reduced; but in the notes, my very first example is to define the reduction of a scheme. (But one may certainly easily define the reduction directly, so this is misleading in a way I find unimportant…)

December 14, 2010 at 5:54 pm

Hmm. It is very useful that if $G$ is a group scheme (locally) of finite type over a perfect field $k$ then $G_{\rm{red}}$ is a smooth $k$-subgroup of $G$. (The point is that for perfect $k$ and a $k$-scheme $X$, the formation of $X_{\rm{red}}$ commutes with scalar extension to $\overline{k}$ because the algebraic closure is a limit of finite etale extensions of $k$. This implies that $X_{\rm{red}}$ is geometrically reduced, so its formation therefore commutes with direct products over $k$ via comparison with the case of algebraically closed ground fields. And a geometrically reduced group scheme locally of finite type is smooth.) However, over imperfect fields $G_{\rm{red}}$ can fail to smooth, nor even a $k$-subgroup scheme.

February 26, 2011 at 5:54 am

Updated response to Charles’ original comment: ampleness and (implicitly) very ampleness is now in the notes, and the reduced subscheme structure is not really used in this argument.

December 14, 2010 at 12:25 pm

I noticed one little thing out of order today: 10.2.K uses the notion of the graph of a rational map, but that notion isn’t introduced until 10.2.4 on the next page. I also thought it would be helpful to have the diagram at the very end of section 10.2 (middle of page 206) appear before 10.2.K.

And one typo: in 10.3.3(i), there’s an extra Spec after the tensor product in the first line.

By the way, we really enjoyed 10.2.E about Gal(Qbar/Q) being an affine scheme!

February 22, 2011 at 2:02 pm

Thanks Silas!

About 10.2.K and related issues: thanks for catching that. I decided to separate the rational map interpretation of the blow-up, which is a touch harder, and might mislead people. So I’ve moved that out of 10.2.K, and made it a new exercise after 10.2.L, which also follows your suggestion with the diagram. The new version is here.

10.3.3(i) is now fixed.

And I’m glad you enjoyed 10.2.E. I did too — I think if I saw it early in grad school, I may have ended up as a number theorist (or arithmetic geometer)! Kirsten Wickelgren and I thought about it further for our own entertainment — see this article if you are curious.

December 17, 2010 at 3:57 pm

[…] promised last time, I expect the next post to be on or about January 8, and I then hope to return to a fortnightly […]

February 21, 2011 at 4:13 am

Hi,

For excercise 10.4.f (January draft), can we assume that $Y\ra S$ (or more generally, the projection to S of any S-scheme) has P? Another thing I have thought of is that in the tower of fibre diagram in 2.3.P, being a fibre diagram the outside rectangle and the bottom rectangle forces the upper to be a fibre diagram.

I believe the first assertion is true (it is true for any property I can think of), but I find it really hard to prove it for an abstract ‘P’. However I believe the second one is false in general. I need any of the two to do the exercise.

February 22, 2011 at 1:46 pm

Hi Jesus,

Here’s a hint:

You want to show that X x X’ –> Y x Y’ has P. Show that X x X’ –> X x Y’ and X x Y’ –> Y x Y’ both have P. Please let me know if that doesn’t help!

March 3, 2011 at 9:23 am

Thanks for your answer.

I imagined so but at some point I needed some isomorphism between X x X’ and some fibred product with base S fibring something else fibred over Y’ (I do not have my notes with me so I cannot remember the details). I thought that was impossible but after trying it I proved it, so I guess it was OK.

I’m preparing a list of typos I’ve found, when I have them ready I’ll send you an e-mail.

March 3, 2011 at 9:40 am

Thanks — and I’m looking forward to the list of typos!

September 13, 2011 at 10:07 am

Dear professor Vakil!

In a hint to 9.3.F you write that the complement to the reduced locus of an affine noetherian scheme Y is the closure of the embedded associated points of Y. I think it’s wrong (for example for Y=Spec(k[x]/(x^2)) ). Still, it’s true that the complement to the reduced locus is the closure of the non-reduced associated points.

September 29, 2011 at 7:43 am

Good point! By chance, someone else (I’ve now forgotten who) recently caught this too, and it is fixed as of the Sept. 6 draft. Well, almost: I then forgot to say “closure of”. I’ll patch this now, and also include your better wording.