Challenge Accepted

Remember this challenge? Probably not. But wait, don’t click! Namely, I will do something more general in this post.

Suppose we have a ring A and a contravariant functor F on (Sch/A) with the following properties:

1. F satisfies the sheaf property for fpqc coverings
2. the value of F on a scheme is either a singleton or empty
3. for every quasi-compact scheme T/A such that F(T) is nonempty, there is an ideal I of A such that F(Spec(A/I)) is nonempty and such that T —> Spec(A) factors through Spec(A/I).

Example: A = k[x, y] for a field k and F(T) is nonempty if and only if the generic point of Spec(A) is not in the image of T —> Spec(A). Here F is not a representable functor.

I’d like to add some conditions that guarantee that F is representable by a closed subscheme of Spec(A). Here is what I just came up with; I think it is obviously correct and the right thing to do. If A is Noetherian we add the following two conditions

1. If s_1, s_2, s_3, … is an infinite sequence of points of Spec(A) such that F(s_i) is nonempty and s is a limit point of the sequence, then F(s) is nonempty.
2. If A —> …. —> A_n —> A_{n – 1} —> … —> A_1 are surjections such that the kernels A_n —> A_{n – 1} are locally nilpotent ideals and F(Spec(A_i)) is nonempty, then F(Spec(A_∞)) is nonempty where A_∞ = lim A_n.

I leave it as an exercise to show that 1 — 5 imply F is representable in the desired manner. If A is not Noetherian, then somehow these should still be enough although maybe you need to replace the natural numbers by a bigger directed set.

Why am I excited by this observation? It is because I want to apply this to the situation of the other blog post mentioned above: X is an algebraic space of finite presentation over A, u : H —> G is a map between quasi-coherent O_X-modules. We assume G is flat over A, of finite presentation, and universally pure relative to A (this is a technical condition which is satisfied if the support of G is proper over A). The functor F is defined by F(T) is nonempty if and only if the base change u_T of u is zero.

Properties 1, 2, 3 hold for F and are easy to prove. The proof of property 4 still doesn’t use purity of G relative to A (I think because we already have 3 it follows from an argument using generic freeness, but I also have an argument using \’etale localization). The key is to prove property 5.

To see 5 is true, I argue as follows. Suppose that the base change u_∞ to A_∞ is nonzero. Choose a weakly associated point ξ of the image of u_∞. This is also a weakly associated point of G_∞. The image t’ of ξ in Spec(A_∞) specializes to a point t in V(I_1) = Spec(A_1) because I_1 is contained in the radical of A_∞. Because G is universally pure relative to A, there is a specialization θ of ξ which lies over t (indeed this is the definition of being pure relative to the base). Then since u_∞ is zero at θ (in a suitable \’etale neighbourhood Edit: Argh… I just discovered this doesn’t work!) it is zero at ξ, a contradiction.

Enjoy!

PS: A finitely presented module G on X flat and pure over A is universally pure relative to A. However, this is harder to prove than the above and it is easy to see that support proper over A implies universal purity.

Limit preserving and finite presentation

Let A —> B be a ring map. Let R = colim R_i be a filtered colimit of A-algebras. Then there is a canonical map

colim Hom_A(B, R_i) ——> Hom_A(B, R)

By Tag 00QO the following are equivalent

1. A —> B is of finite presentation,
2. the map above is bijective for all R = colim R_i
3. the map above is surjective for all R = colim R_i

Let S be a scheme. Let X be a scheme over S. Let T = lim T_i be a directed limit of affine schemes over S. Then there is a canonical map

colim Mor_S(T_i, X) ——> Mor_S(T, X)

By Tag 01ZC and Tag 0CM0 the following are equivalent

1. X —> S is locally of finite presentation,
2. the map above is bijective for all T = lim T_i
3. the map above is surjective for all T = lim T_i

The same thing is true if X and S are algebraic spaces (Tag 04AK and Tag 0CM6).

I didn’t know you could replace bijectivity by surjectivity in the criterion. But somewhere in the Stacks project we used this fact without proof, so it had better be true, right?

A related result is that to check a morphism f of algebraic stacks is locally of finite presentation, you need only check f is limit preserving on objects (this is the analogue of the above and it says that certain functors are essentially surjective). You can find this in Tag 0CMQ.

Caveat: as this only applies to situations where you already know your functors (or stacks in groupoids) are algebraic spaces (or stacks), it probably won’t be that useful. Often when we try to show a stack is limit preserving, it is part of applying Artin’s criteria and then we don’t yet know our stack is algebraic of course.

[Edit on 6/30/2016: Matthew Emerton just pointed out that this observation was already in Lemma 2.3.15 of his paper with Toby Gee. I must have read it and then forgotten that I had. Apologies to everybody.]

Up to date again

Worked through your comments online as well as those emailed to me. Please let me know if I forgot to do something. Thanks for all your help!

Up to date

OK, I worked through your comments online as well as emailed ones.

Thanks for the help!

Completeness

For everybody reading and using the Stacks project here is something to keep in mind:

Of course the results in a particular section or chapter do not cover all possible results about the topic discussed in that section or chapter.

First of all, taken literally, this is simply not possible. But even trying to do justice to a topic and mention all the wonderful things one can say, would slow down progress to a halt. Adding all possible deductions and recombinations of lemmas in the section and earlier ones also often takes too much work.

Instead what we try to do is as follows. Each time we broach a new topic we try to have a skeleton outline of the basic material. Often we do this when there is just one tiny result we want to use. Then over time, we come back to the section/chapter with more material as needed. Also, sometimes a result cannot be formulated or proved immediately because it needs more terminology or results proven later in the Stacks project. When this happens we try to put in a pointer to this material in the earlier section.

This has turned out to work fairly well. But if you find cases where it didn’t then please let us know. For example, if you find a case where some elementary results are being used which have no formulation in earlier chapters, then please let us know so we can fix that. On the other hand, if you have a result you would like to see mentioned, then send us a latex file with the actual math and we will consider it for inclusion. Thanks!

Blowing down exceptional curves

Let X be a Noetherian separated scheme. Let E ⊂ X be an effective Cartier divisor such that there is an isomorphism E → P1k where k is a field. Then we say E is an exceptional curve of the first kind if the normal sheaf of E in X has degree -1 on E over k.

You can get an example of the situation above by starting with a Noetherian separated scheme Y and a closed point y such that the local ring of Y at y is a regular local ring of dimension 2 and taking the blowup b : X → Y of y and taking E to be the exceptional divisor.

Conversely, if E ⊂ X is gotten in this manner we say that E can be contracted.

The following questions have been bugging me for a while now.

Question 1: Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E?

Question 2: Given an exceptional curve E of the first kind on a separated Noetherian scheme X is there a contraction of E but where we allow Y to be an algebraic space?

Question 3: Suppose that Y is a separated Noetherian algebraic space and that y is a closed point of Y such that the henselian local ring of Y at y is regular of dimension 2. Is there an open neighbourhood of y which is a scheme?

Question 4: With assumptions as in Question 3 assume moreover that the blow up of Y in y is a scheme. Then is Y a scheme?

In these questions the answer is positive if we assume that X or Y is of finite type over an excellent affine Noetherian scheme (and I think in the literature somewhere; I’d be thankful for references).

But… it might be interesting and fun to try and find counter examples for the general statements. Let me know if you have one!

Canonical divisor

Let k be a field and let X be a finite type scheme over k. Let F be a coherent O_X-module which is generically invertible. This means there exists a an open dense subscheme such that F is an invertible module when restricted to that open.

Lemma: There exists an open subscheme U containing all codimension 1 points, an invertible O_U-module L, and a map a : L → F|_U which is generically an isomorphism, i.e., there exists an open dense subscheme of U such that a restricted to that open is an isomorphism.

Proof. We already have a triple (U, L, a) for some dense open U in X. To prove the lemma we can proceed by adding 1 codimension 1 point ξ at a time. To do this we may work over the 1-dimensional local ring at ξ, where the existence of the extension is more or less clear.

Now assume that X is equidimensional of dimension d. Then we have a Chow group A_{d-1}(X) of codimension 1 cycles. If X is integral this is called the Weil divisor class group. For F as above we pick (U, L, a) as in the lemma. Observe that A_{d-1}(U) = A_{d-1}(X).

Def: The divisor associated to F is c_1(L) ∩ [U]_d + [Coker(a)]_{d-1} – [Ker(a)]_{d-1}

The notation here is as in the chapter Chow Homology of the Stacks project. The first term c_1(L) ∩ [U]_d is the first chern class of L on U and the other two terms involve taking lengths at codimension 1 points. Using the lemma to compare different triples for F it is easy to verify this is well defined as an element of A_{d-1}(X).

Def: Assume in addition X is generically Gorenstein, i.e., there exists a dense open which is Gorenstein. Let ω and ω’ be the cohomology sheaves of the dualizing complex of X in degrees -d and -d+1. The canonical divisor K_X is the divisor associated to ω minus [ω’]_{d-1}.

There you go; you’re welcome!

Rmks:
1. Fulton’s “Intersection Theory” defines the todd class of X in complete generality.
2. If X is generically reduced, then X is generically regular, hence generically Gorenstein and our definition applies.
3. The term [ω’]_{d-1} is zero if X is Cohen-Macaulay in codim 1.
4. If X is Gorenstein in codimension 1, then our canonical divisor agrees with the canonical divisor you find in many papers.
5. A canonical divisor of an equidimensional X can always be defined: either by Fulton or by generalizing the definition of the divisor associated to F to the case where F and O_X define the same class in K_0(Coh(U)) for some dense open U. This will always be true for ω. Just takes a bit more work.
6. If X is proper and equidimensional of dimension 1, then χ(F) = deg(divisor asssociated to F) + χ(O_X) whenever F is generically invertible.
7. If X is proper and equidimensional of dimension 1, then deg(K_X) = – 2χ(O_X).
8. If X is a curve and f : Y → X is the normalization, then K_X = f_*(K_Y) + 2 ∑ δ_P P where δ_P is the delta invariant at the point P (Fulton, Example 18.3.4).
9. If X is equidimensional of dimension 1 and Z ⊂ X is the largest CM subscheme agreeing with X generically, then K_X = K_Z – 2 ∑ t_P P where t_P is the length of the torsion submodule in O_{X,P}.

Edit 3/1/2016: Jason Starr commented below that there is a refinement which is sometimes useful, namely, one can ask for a Todd class and Riemann-Roch in K-theory and he just added by email: “In our joint work on rational simple connectedness of low degree complete intersections, we need to know that certain (integral) Cartier divisor classes on moduli spaces are Q-linearly equivalent. It is not enough to know that the pushforward cycles classes to the (induced reduced) coarse moduli scheme are rationally equivalent. So we need the Riemann-Roch that works on K-theory. In fact, the relevant computations are in our earlier manuscript about “Virtual canonical bundle …”, and we slightly circumvent Riemann-Roch in the computation. But, morally, we are using a Todd class that lives in K-theory, not just in CH_*.”