### Big lists

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

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

Diagram of Chapter Dependencies

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

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

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

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

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

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

5.4.N   ${\mathbb{Z}}[\sqrt{-5}]$ is not a unique factorization domain, but its Spec can be covered with 2 (distinguished) affine subsets, each of which are Specs of UFD’s.  Is there some good reference for this?  (Presumably it becomes a UFD upon inverting either 2 or 3, but I can’t see why this is the case.   And of course I don’t just want to know what is true; I’d like a reference for why it is true.)  Added later:  I should also have added, is there a well-loved reference that shows that the class group of ${\mathbb{Z}}[\sqrt{-5}]$ is $\mathbb{Z}/2$(answer here

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

6.7  In this section, I mention the Schubert cell decomposition of the Grassmannian.  The key idea is that any $k$-dimensional subspace of $K^n$ (where $K$ is a field; and say $e_1$, …, $e_n$ is the standard basis of $K^n$) has a canonical basis, where the first $e_i$ to appear in each basis element appears with coefficient 1, and that $e_i$ appears in no other basis element, and that special $e_i$ for that basis element is “to the right” of the $e_i$ of the previous one.  Is there a standard name for this?  (Normal form?  Row-reduced echelon form?)  Is there a good (fairly standard) reference for it?  (Perhaps this gets too far into how linear algebra is taught in different countries, and I should just not give a reference, and instead give it as an exercise.)   (answer here, although I’m also happy to get more references)

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

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

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

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

11.3.13 Over an algebraically closed field, every smooth hypersurface of degree at least $n+1$ in $\mathbb{P}^n$ is not uniruled.  What is a good reference?  (I know why it is true!  As with many of these questions, I’d like to know where to point people to.) (answer here)

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

19.9.B  In (19.9.7.2), we have $j=2^8 (\lambda^2-\lambda+1)^3/(\lambda^2(\lambda-1)^2)$, and the discussion is away from characteristic $2$.  I want to say that the normalization factor $2^8$ is because of characteristic $2$, but I couldn’t convince myself that this was true.  Presumably it is.  Is there a good reference?  (Remark for comparison:  one can also write $j$ in terms of $\tau$$j = 1728 g_2^3 / \Delta$.  Here the prime factors of $1728$ are $2$ and $3$; but the reason for the $3$ is not characteristic $3$.)  (answer:  yes, see here)

20.2.H  Suppose $E$ is a complex elliptic curve.  Then  $\dim_{\mathbb{Q}} N^1_{\mathbb{Q}}(E \times E)$ is always $3$ or $4$.  It is $4$ if there is a nontrivial endomorphism from $E$ to itself (i.e. not just multiplication by $n$ followed by translation); the additional class comes from the graph of this endomorphism.  Is there a reference for this fact that I can/should direct learners to?   (answer:  yes, see here)

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

21.7.8  (not needed)  It is a nontrivial fact that irreducible smooth projective curves of
genus $g \geq 2$ have finite automorphism groups.  I know three arguments:  using the Neron-Severi theorem (and the Hodge index theorem) (see Hartshorne V.1.9, for example); the fact that the automorphisms are reduced and form a scheme (too hard); and by action on Weierstrass points.  I am surprised that this is so hard.  (Note:  I know that the idea can be quickly outlined to someone learning.  But I want an easy complete rigorous proof.  As long as I am asking, I also want someone to give me a Tesla Roadster.)

21.7.9  Smooth curves in positive characteristic  can have way more than $84(g-1)$ automorphisms.  Is there a “best” reference?

28.1.L   Is there a canonical reference for Tsen’s theorem, that any proper flat morphism  $X \rightarrow Y$ to a curve, whose geometric fibers are isomorphic to $\mathbb{P}^1$ is a Zariski $\mathbb{P}^1$-bundle?  Follow-up question (posted March 5), in response to David Speyer’s comment here:   Does anyone have a (loved) reference for the fact that the universal plane conic (over the space of smooth plane conics) is not a $\mathbb{P}^1$-bundle?  (See David Speyer’s comment for a little more detail.)  (possible answer to the first question here; answer to the second question here)

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

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

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

What are your favorite properties of varieties over a field k (or more generally finite type k-schemes) that can be checked after base change to the algebraic closure of k? This is of course secretly about faithfully flat descent, but my question is partially pedagogical.

Sometimes the implication goes only one way (e.g. if something is true over the algebraic closure, then it is true for k); please be clear on that.

I would like later to add references, so people can know not just what is true, but also why it is true. If you know references off the top of your head (even somewhat vague ones, such as “I remember a nice proof in [reference here]”, please post them).

For simplicity, please put one per comment, so people can respond and comment further.

What are your favorite properties of finite type k-schemes that can be checked at closed points (or possibly at “closed geometric points” — at closed points after you base change to the algebraic closure of k)? My reason for asking: this gives a connection to the classical theory of varieties.

Any open condition will work, but please list those here. I’m looking for some “variety-specific” facts. There seem to be remarkably few.

For simplicity, please put one per comment, so people can respond.

What are your favorite open and closed and locally closed conditions (under reasonable hypotheses). (My motivation for doing all three of these at once: to keep people saying “the reduced locus is open” and “the nonreduced locus is closed”.)

For simplicity, please put one per comment, so people can respond.

Related question: I was intending to have one list on properties that can be checked at closed points, but I realized that all examples I had in mind were of two sorts, which I am asking separately. (i) Some are true because they are open conditions, and thus for a quasicompact scheme it suffices to check them at closed points (see the comment after Important Exercise 6.1.E). (ii) Some are true because they a variety-specific result, usually involving the Nullstellensatz. Are there any things you like to check at closed points that aren’t of either of these two sorts?

Semicontinuous functions are a useful tool. Because I want to make sure I don’t miss any important ones, I took an inventory of the ones I use, and found surprisingly few — they just get used a lot.

What are your favorite semicontinuous functions of a scheme (with reasonable hypotheses)?

For simplicity, please put one per comment, so people can respond. I’ve posted my favorite five (ranked).

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.

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.