**The fifteenth post is the May 13 version in the usual place.** This post covers until 25.3 (although 25.4 is also included). The next post should appear around June 11 (four weeks rather than three, as I’m away for a chunk of the next month.)

This one is a long one!

**Update.**

We’re nearly done the notes! As you will see from the table of contents, there is not much more to go (and only a couple of chapters are still missing from the table of contents). I’ve deliberately been moving at a too-fast clip, and I’m far ahead of a number of vigilant readers. I can now start to make smaller postings, and spend more time catching up on editing what is already out there. I’ve made reasonable progress on responding to earlier comments, and I’ve tagged those that I still have to respond to.

**For learners.**

Read the unstarred sections, and only look at the starred sections you feel compelled to glance through. I’ll discuss some of the starred sections below.

Recall that Chapter 23 is about differentials.

In 23.4, we define *k*-smoothness correctly, and discuss various birational invariants of smooth projective varieties. We prove “generic smoothness” and introduce unramified morphisms.

In 23.5, we prove Riemann-Hurwitz, and discuss some related issues.

Chapter 24 is on derived functors.

As a warm-up, we discuss Tor in 24.1. The reason for doing this is that it is direct and straightforward, but if you look back on it, you find that you have accidentally defined derived functors in complete generality!

In 24.2, we make this precise, and then define Ext as well (twice!).

In 24.3, we use spectral sequences to get some useful properties of (and useful perspective on) derived functors. In particular, we prove Grothendieck’s composition-of-functors spectral sequence.

Chapter 25 is on flatness. We’ll begin this topic in this post, and continue it next time.

We introduce flatness in 25.1.

In 25.2, we describe some “easy facts” that will give you some experience with flatness.

In 25.3, we use Tor to understand flatness better; we see flatness for the first time as a cohomological property.

I’ll list the starred sections for your convenience, so you can choose what to dip into.

24.4.9 Infinitesimal deformations and automorphisms

23.4.10 A first glimpse of Hodge theory (necessarily from a purely algebraic perspecive)

24.2.5 The category of A-modules has enough injectives.

24.4 Derived functor cohomology of O-modules

24.5 Cech cohomology and derived functor cohomology agree.

Part of this argument relies on a slick explanation by Martin Olsson.

*Problems to do*

I have picked a selection of various sorts of problems, so you can concentrate on the style of problem you would like.

*Twelve theory exercises*

23.4.A (verifying that the new definition of k-smoothness is the same as the old)

23.4.E (properties of unramified morphisms)

23.4.F (maps of smooth varieties)

24.1.A (homotopic maps of complexes give the same map on homology)

24.1.B and 24.1.C (technical exercises; hard to explain out of context)

24.3.A (symmetry of Tor — good practice of an important technique)

24.3.C (derived fucntors can be computed with acyclic resolutions — good practice of an important technique)

24.3.D (Grothendieck composition-of-functors spectral sequence)

25.2.D (transitivity of flatness)

25.2.J (cohomology comutes with flat base change)

25.2.O (flat morphisms are open in good situations)

About 24.3.D and 25.2.O: Are the hints enough for you to do it? Please let me know if you try it and (i) get stuck, or (ii) solve it. Or even if you are too frightened to try it. Feel free to let me know by email. I want these important exercises to be solvable.

*Seven “more applicable” exercises*

24.4.B (showing that the geometric genus and related invariants are in fact birational invariants)

23.5.A (Jacobian of map from one smooth n-fold to another)

23.5.C (ramification divisor for map of curves, in terms of number of preimages)

23.5.H (geometric genus = topological genus)

24.2.C (defining Ext)

24.4.D (arithmetic genus = geometric genus for smooth curves)

25.2.L (flat maps send associated points to associated points)

*Six “Example” Exercises*

23.5.E (no map from genus 2 curve to genus 3 curve)

23.5.F (no connected unbranched cover of A^1)

25.2.A, 25.2.G, 25.2.M (examples of (non)flatness)

25.3.A (examples of Tor)

*Question:*

If you feel like checking that A-modules have enough injectives (24.2.5), please let me know if you have any trouble with the series of exercises establishing this fact. Or if you are able to do it.

**For experts.**

I state that Tor_i(M, *) is an additive functor (24.1.F), but haven’t actually checked it! But we don’t seem to use this fact.

Update July 14+15 2011: I should have mentioned to experts that I don’t use delta functors (or universal delta functors, or effaceable functors, etc.). This will upset some people, but I don’t see what it adds to the current discussion (other than giving the reader even more heavy machinery to carry around until they day they actually use it), and I see a heavy cost. Feel free to complain in a comment!

*Questions for experts*

(a) I call elements of N^1_\Q(X) “\Q-line bundles” (20.4.11). Is there some official name?

(b) In 24.5.D, I mention that under reasonable hypotheses on a topological space, the simplicial homology is computed by taking the derived functor cohomology (in the category of sheaves of abelian groups) of the constant sheaf Z. Does anyone know offhand what the right reasonable hypotheses are?

(c) I haven’t thought about the following type of fact, beyond vaguely thinking that it is useful: do derived functors (of left-exact functors, say, for concreteness) automatically commute with some other type of functors (say with exactness properties, or adjoints)? Or even are there maps in one direction under weaker hypotheses? An such statement would be some variant of the FernbaHnHopF (FHHF) Theorem (Exercise 2.6.H). Follow-up if the answer is “yes”: does this come up often enough that it is worth mentioning? And do you have a favorite example?

(d) I want to prove (in 25.2.9): Suppose f: X –> Y is a flat morphism of schemes. I want that the dimension of X_y at x plus the dimension of Y at y is the dimension at x. Unfortunately, I seem to need that codimension “behaves well” (precisely, is the difference of dimensions), so I have this awkward hypothesis that all the stalks are localizations of finite type k-algebras. Is there a better hypothesis? (Or a reference to a better proof? Or a better proof?) *(Update June 23: now done “correctly”, following Georges’ advice.)*

(e) Theorem 25.4.2 says that flat = free = projective for coherent modules over local rings. More precisely it says that if A is a local ring, and M is a flat coherent A-module, then M is free. I don’t seem to use Noetherian hypotheses here, so I feel nervous. Can someone confirm or deny that Noetherian hypotheses aren’t necessary? *(Update June 23: now confirmed. Update June 27: in fact, only finitely presented is necessary; will be fixed in next posting.)*

(f) Coming up at the start of the next post (25.5) is the central fact that euler characteristic is locally constant in flat families. I seem to recall that this, and its converse (assuming the target is reduced) are due to Serre, but I don’t know a reference. Does anyone know offhand (because they have seen it before)?

**Random list of things I’ve belatedly added.**

Spectral sequences (2.7 and later): I’ve just changed the indexing to be correct (i.e. (x,y)). It was absolutely no fun doing it, and I’m sure I’ve screwed something up. If anyone tries to read the spectral sequence section after this, please let me know. (Darij Greenberg’s comments were very handy in an earlier iteration.)

Critch’s additions to discussion of gluing along closed subschemes now added (17.4.8). (Question: does anyone have an example of when you can’t glue a scheme to itself along two disjoint isomorphic closed subschemes? Ideally with proof, but not necessarily. The argument may need to be a bit vague.) Neat fact: the construction is by gluing in locally ringed spaces. However, the fact that the resulting diagram is not just a cofibered diagram but also a fibered diagram uses that you are gluing together schemes, not just locally ringed spaces! Maybe it is true of locally ringed spaces as well, but it’s not clear to me; I need the local model to make it work.

The fact that a blow-up of a smooth variety along a smooth variety is smooth (19.4.12).

All proper curves over a field are projective (20.6.E).

**Coming soon**

I want to collect people’s favorite examples of the following criteria, to make sure I don’t miss anything important. This will be a (short) new category of posts: “big lists” (of which this is the first). The examples I have in mind are the following.

1. Favorite open and closed and locally closed conditions (under reasonable hypotheses). (Reason for doing both at once: to keep people saying both “the reduced locus is open” and “the nonreduced locus is closed”.)

2. Favorite things that can (in reasonable cases) be checked at closed points. (Exception: any open condition needn’t be repeated, as for any quasicompact scheme, any open condition can be checked at closed points.)

3. Favorite semicontinuous functions.

4. Favorite things about finite type k-schemes that you can check at closed points, perhaps after base change to the algebraic closure. (Reason: this gives a connection to the classical theory of varieties.)

I’ll put up pages for each of these soon, along with many of my favorite examples. *(Update June 23, 2011: now done!)*

I want to point out that comprehensive and very useful lists are given in Appendices C, D, and E of Gortz and Wedhorn’s wonderful book *Algebraic Geometry I*. Appendix C tracks which propertoes of schemes satisfy which “permanence properties” (e.g. stable under base change; local on the source; etc.). Appendix D gives relations between properties of morphisms of schemes, including a great flowchart. Appendix E lists constructible and open properties.

May 19, 2011 at 3:13 pm

Lalit Jain and I think there may be an error in problem 17.3.B from the May 13th notes.

The exercise is asking us to show that is right adjoint to restriction, but we couldn’t get the proof to go through that way. It seems that should actually be left adjoint to restriction (and we can prove it this way).

If is left adjoint, then we’re also a little confused on how 17.3.C should go through.

Thank you!

David

September 2, 2011 at 10:39 am

I recall that I had a mistake to do with , and this presumably is it. I still haven’t had a chance to think about this, and I’m flagging this so I don’t forget it. Sorry for the delay!

March 25, 2012 at 5:52 pm

Very delayed response: you are completely right. That was garbage. It is now fixed, in the version that will be posted within the next few days.

May 19, 2011 at 9:00 pm

I’m not sure where to put this…

In exercise 2.4.C on page 33, a line runs off the page. Also, it should presumably read “(a_i,i)~(a_j,j)” instead of “(a_i,i)~(a_j,a_j)”.

The whitespace on page 36 is messed up.

In 3.2.11 (top of page 65) “neighborhood U of X” should be “neighborhood U of \pi(y)”.

I may be missing something, but I don’t see how you’re supposed to use exercise 3.4.D in 3.4.E. 3.4.A seems more relevant.

September 2, 2011 at 10:47 am

Hi Ted,

Thanks! I’ve fixed 2.4.C and 3.2.11. I think p. 36 is patched, but I’m not worrying too much about whitespace at this stage. And if you happily solved 3.4.E, then I’m already happy, even if you didn’t follow the hint. (The danger in giving hints is that people may assume that any solution must use the hint…)

May 20, 2011 at 12:06 pm

Dear Ravi,

concerning your Proposition 25.2.9, here is a statement that might be relevant, since it is fairly general and involves no field.

Let f:A–>B be a flat morphism of noetherian rings, P a prime ideal in B and p its trace in A. Then

height(P)= height(p)+dim(B_P/p.B_P)

This is Theorem 15.1 in Matsummura’s Commutative Ring Theory .

Friendly greetings,

Georges.

June 17, 2011 at 10:14 am

Dear Georges,

Thanks for reminding me of this! It has led me to restate some things. The inequality version of what you mention is now done quickly and easily back in the dimension chapter (using the “hard” fact of the strong form of Krull). Then I do this equality in 25.2, which allows me to (a) use faithful flatness for something, (b) state the going-down theorem for flat morphisms, and (c) state the “right” form of Proposition 25.2.9, and (d) give what is probably the right hypotheses for “flatness of relative dimension n”.

I’d like to say what is now written, in the hopes that someone will stop me if I say something silly.

(i) We say is flat of relative dimension if it is flat and all fibers have pure dimension . (Should I play it safe and stick in some locally Noetherian conditions, at least on the fibers?)

(ii) Exercise: if and are both locally Noetherian, and is equidimensional, then is flat of relative dimension $n$ if and only if has pure dimension . (I hope my hypotheses are neither too general to be true, nor too needlessly restrictive.)

(iii) Exercise: the composition of two morphisms that are flat of relate dimension and respectively is flat of relative dimension . I’ve put in locally Noetherian hypotheses here to use (ii), although it may be true more generally.

best,

Ravi

Update May 4 2012: More recent thoughts on “flatness of relative dimension n” are given

here.May 23, 2011 at 5:03 am

At the end of section 23.4, you write “Notice that in both cases is the Picard number $\rho$ (defined in section 20.4.11). This is always true.”

I’m not sure what you mean by “always”, but the most obvious interpretation this is false. Consider a product of two elliptic curves with no isogenies between them. Then is four dimensional, but Picard only has rank .

The complex analytic statement is that

. I’m not sure what the corresponding purely algebraic statement is, but it has the consequence that , and inequality can be strict.

June 9, 2011 at 10:44 am

Thanks David! It was the complex analytic statement that was in my head, and I have no idea why I wrote what I wrote. Now fixed!

July 8, 2011 at 9:31 am

I’ve heard it said often that the category of sheaves of abelian groups on a topological space X does not have enough projectives. Does anyone have a reference? (Or better yet, an example?)

July 22, 2011 at 10:51 am

See http://mathoverflow.net/questions/5378/when-are-there-enough-projective-sheaves-on-a-space-x

August 2, 2011 at 10:12 am

Thanks! That is so short that I added it as a double-starred remark at the end of 24.4, when we saw that the category of O-modules always has enough injectives.

July 21, 2011 at 10:31 am

[…] I have heard that they don’t have enough projectives. (I asked a variant of this question here.) Does anyone have a reference (ideally with a proof)? I’ve heard that locally free sheaves […]

November 1, 2011 at 5:42 pm

About the Grothendieck composition-of-functors problem: I’d asked if I’d given enough hints, or possibly messed up the problem. Likely the latter, as Yifei Zhu has tactfully pointed out to me. I won’t have a chance to deal with it really soon, so he’s allowed me to post his email, as a flag for myself, and for use for others. The rest is from him. — Ravi

Dear Ravi,

On the fifteenth post, you asked if anyone got stuck when doing 24.3.D (the Grothendieck composition-of-functors spectral sequence). Unfortunately I did, even with your encouraging hint. I was wondering if this exercise would be solvable without using a Cartan-Eilenberg resolution. The following is what I did.

Take an injective resolution 0 -> A -> I^* and apply F to it, and I get a cochain complex in the category B. I then put this as (the augmentation of) the bottom row of a first-quadrant double complex, and inductively take injective resolutions (going upward) of each term of this cochain complex, row by row, so that except for the bottom row, all rows and columns are exact. I think this is analogous to the construction of hint 24.3.3. Now applying G, I get a double complex as the E_0 page.

Running the spectral sequence first vertically, I get it collapsed on the E_1 page with the leftmost column R^iG(F(A)), as the F(I^i)’s are G-acyclic. Running it first horizontally, I get it collapsed again on the E_1 page with the bottom row R^i(GF)(A), as injectives in the category B are of course G-acyclic. So this shows that R^iG(F(A)) is isomorphic to R^i(GF)(A). And I got stuck.

I then looked it up in Weibel’s book “An introduction to homological algebra”. Grothendieck spectral sequences are discussed in section 5.8, and I was trying to go through the proof of theorem 5.8.3 on pp150-1 (there they swamp the notations F and G, and there are probably some typos on p151: in the statement of the theorem it should be the E_2 page of II (first vertically), rather than I; toward the end of the proof the last index p should be q, and the second but last index p should be p+q; unfortunately, I also feel there’s some inconsistency of horizontal-vertical within 5.7.9 on pp 149-50). In the proof they need a Cartan-Eilenberg resolution (cf 5.7.9) in order to have the stated E_2 page. I think the key property of this resolution is that after taking cohomology it is “still an injective resolution”. Also, if I understand correctly, for version II, we need the G-acyclicity of the F(I)’s so that taking cohomology and applying G commute, a variant of the Fernbahnhof theorem.

The above is all I have so far, and indeed it follows from discussion with a fellow graduate student Denis Bashkirov yesterday. Could you please give some further hint?

Thank you!

Yifei

April 8, 2012 at 5:01 pm

This is now, finally, patched (see

here). Thank you Yifei!May 4, 2012 at 11:22 am

On flatness of relative dimension n: I now define it as flat, locally finitely presented, and all fibers have pure dimension n. I believe this is the right definition (with the l.f.pr. hypothesis) for the following reason: it is closed under composition, and it is closed under base change. An instructive example is the fact that has dimension 1. (I had earlier mistyped this as “2”, as Peter Johnson pointed out in

the comment below; now fixed.) If anyone thinks I am wrong (or right) please let me know!May 5, 2012 at 4:58 am

Hi Ravi! Yes, that seems fine. Love the example! Maybe it is instructive to give an example of a flat morphism with special fibre of pure dimension d whose general fibre has components of dimension k[x, y, z, t]/(z(z – t), zy, zx). Note that the special fibre is a plane with an embedded point. And in fact, this is `required’: if the fibres are Cohen-Macaulay and the map is flat, then this kind of thing doesn’t happen.

May 16, 2012 at 1:20 pm

Hi Johan,

Thanks for writing! (I realize I should make it “flat + locally of finite type”; there is no need for local finite presentation.) Could you please restate your example? I think there is something missing after “components of dimension”, so it’s not clear to me what this is an example *of*.

May 20, 2012 at 5:30 am

Sorry! What I meant is that you can find a proper flat morphism towards the spectrum of a dvr whose special fibre is equidimensional of dimension 2 but whose generic fibre has 2 irreducible components: one of dimension 1 and one of dimension 2. If the special fibre is CM this can’t happen.

May 18, 2012 at 9:38 am

I am surprised that nobody commented on the dimension of k(x) \otimes_k k(y). It is 1, not 2: the points, except for the generic one, are closed. They correspond to monic irreducible polynomials over k(x), not over k; i.e. (up to scalars) to the irreducible polynomials in k[x,y] that use both variables.

May 20, 2012 at 5:28 am

Oops, yes, I should have seen that… I think I didn’t even notice the number.

May 18, 2012 at 10:46 am

Thanks Peter! I’d mistyped it, and had intended to write “1”; now fixed. (It’s right in the notes.)