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.

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