Let f : X —> S be a morphism of finite type. The relative assassin Ass(X/S) of X/S is the set of points x of X which are embedded points of their fibres. So if f has reduced fibres or if f has fibres which are S_1, then these are just the generic points of the fibres, but in general there may be more. If T —> S is a morphism of schemes then it isn’t quite true that Ass(X_T/T) is the inverse image of Ass(X/S), but it is almost true, see Remark Tag 05KL.

Definition: We say X is S-pure if for any x ∈ Ass(X/S) the image of the closure {x} is closed in S, and if the same thing remains true after any etale base change.

Clearly if f is proper then X is pure over S. If f is quasi-finite and separated then X is S-pure if and only if X is finite over S (see Lemma Tag 05K4). It turns out that if S is Noetherian, then purity is preserved under arbitrary base change (see Lemma Tag 05J8), but in general this is not true (see Lemma Tag 05JK). If f is flat with geometrically irreducible (nonempty) fibres, then X is S-pure (see Lemma Tag 05K5).

A key algebraic result is the following statement: Let A —> B be a flat ring map of finite presentation. Then B is projective as an A-module if and only if Spec(B) is pure over Spec(A), see Proposition Tag 05MD. The current proof involves several bootstraps and starts with proving the result in case A —> B is a smooth ring map with geometrically irreducible fibres.

I challenge any commutative algebraist to prove this statement without using the language of schemes. You will find another challenge in the next post.

This morning I finished incorporating the material from sections 1 through 4 of the paper by Raynaud and Gruson into the stacks project. Most of it is  in the chapter entitled More on Flatness. There is a lot of very interesting stuff contained in this chapter and I will discuss some of those results in the following blog posts. Note that I previously blogged about this paper here, here, here, here, here, here, and here.

It turns out that it was kind of a mistake to do this, as the payoff wasn’t as great as I had hoped for. Moreover, I don’t think you are going to find the chapter easy to read. So the benefit of having done this is mainly that I now understand this material very well, but I’m not sure if it is going to help any one else. Maybe the lesson is that I should stick to the strategy I have used in the past: only prove those statements that are actually needed to build foundations for algebraic stacks. This will sometimes require us to go back and generalize previous results but (1) we can do this as the stacks project is a “live” book, and (2) it is probably a good idea to rewrite earlier parts in order to improve them anyway.

The long(ish) term plan for what I want work on for the stacks project now is the following: I will first add a discussion of Hilbert schemes/spaces/stacks parameterizing finite closed subscheme/space/stacks. I will prove just enough so I can prove this theorem of Artin: A stack which has a flat and finitely presented cover by a scheme is an algebraic stack. A preview for the argument is a write-up of Bhargav Bhatt you can find here.

Curiously, Artin’s result for algebraic spaces is already in the stacks project: It is Theorem TAG 04S6. It was proved by a completely different method, namely using a Keel-Mori type argument whose punch line is explained on the blog here.

In the last two and and a half weeks I’ve updated the material on derived categories and derived functors. You can now find this material in a new chapter entitled Derived Categories.

The original exposition defined the bounded below derived category as the homotopy category of bounded below complexes of injectives. This is actually a very good way to think about derived categories if you are mainly interested in computing cohomology of sheaves on spaces and/or sites. On the other hand, it does not tell you which problem derived functors really solve. Let’s discuss this a bit more in the setting of sheaves of modules on a ringed space (X, O_X). I will assume you know how to define cohomology of sheaves by injective resolutions, left derived functors by projective resolutions, you have heard that D(A) is complexes up to quasi-isomorphism, but you don’t yet know exactly why one makes this choice.

Let F : Mod(O_X) —> A be a right exact functor from the abelian category of O_X-modules into an abelian category A. The category Mod(O_X) usually does not have enough projectives. Hence it wouldn’t work to define the bounded above derived category in terms of bounded above complexes of projectives. You could still make this definition but there wouldn’t be a functor from the category of modules into it and hence it wouldn’t suffice to compute left derived functors of F. In fact, what should be the “left derived functors” of F in this setting? Grothendieck, Verdier, and Deligne’s solution is the following: Let M be an O_X-module. Consider the category of all resolutions

… —> K^{-1} —> K^0 —> M —> 0

where K^i is an arbitrary O_X-module. For any such resolution we can consider the complex

F(K^*) = ( … —> F(K^{-1}) —> F(K^0) —> 0 )

in the abelian category A. We say that LF is defined at M if and only if the system of all F(K^*) is essentially constant up to quasi-isomorphism, i.e., essentially constant in the bounded above derived category D^-(A). If one can choose K^* so that F(K^*) is actually equal to this essentially constant value, then one says that K^* computes LF(M). These definitions are motivated by the case where there do exist enough projectives: in that case one shows that given a projective resolutions P^* there always exists a map P^* —> K^*, hence the system is essentially constant with value F(P^*). We say an object M is left acyclic for F if M computes LF. Note that this makes sense without knowing that LF is everywhere defined! It turns out that LF is defined for any M which has a resolution K^* where all K^n are left acyclic for F and that in this case F(K^*) is the value of RF(M) in D^-(A). For example, why is one allowed to use bounded above flat resolutions to compute tors? The reason is that flat modules are left acyclic for tensoring with a sheaf (this is not a triviality — it is something you have to prove; hint: use Lemma Tag 05T9).

I started rewriting the material on derived categories because I gave 2 lectures about derived categories and derived functors in my graduate student seminar, and I wanted to understand the details. Let me know if you find any typos, errors, or lack of clarity. Also, there is still quite a bit missing, for example a discussion of derived categories of dg-modules would be cool.

Let X be a smooth variety over a field k. The index of X over k is the gcd of the degrees [κ(x) : k] over all closed points x of X. The index is 1 if and only if X has a zero cycle of degree 1. If k is perfect, then the index of X is a birational invariant on smooth varieties over k: The reason is that given a nonempty open U of X and a closed point x in X you can find a curve C ⊂ X with x ∈ C, and it is easy to move zero cycles on curves. (I think the birational invariance also holds over nonperfect fields, but I haven’t checked this.)

Another birational invariant of a d-dimensional variety X over k is the gcd of the degrees of rational maps X —> P^d_k. This is the same as the gcd of closed subvarieties of P^n (any n) birational to X. Let’s temporarily call this the b-index. Note that by taking inverse images of k-rational points on P^d_k we see that index | b-index for smooth X (if k is finite you have to look at points over finite extensions). I claim that in fact index = b-index at least over a perfect field. After shrinking X we may assume that X is affine, hence quasi-projective, so X ⊂ P^N_k for some N >> 0 having some (super large and super divisible) degree D. On the other hand, consider the blow up b : X’ —> X of X in x. Then the invertible sheaf b^*O_X(N)(-Exceptional) will be very ample and will embed X’ into a large projective space where it has degree N^dD – [κ(x) : k]. This implies that b-index divides [κ(x) : k] and we win.

Let A be an abelian category. In the stacks project this means that A has a set of objects, and that

• A is a pre-additive category with a zero object and direct sums, i.e., an additive category,
• A has all kernels and cokernels (and hence all finite limits and all finite colimits), and
• Coim(f) = Im(f) for all morphisms f in A

Martin Olsson pointed out that there is a simple direct argument which proves that in such a category any epimorphism (called a surjection in the following) is a universal epimorphism, see Lemma Tag 05PK. Using this fact we obtain a site C whose underlying category is simply A and where a covering is the same thing as a single surjective morphism. Then the Yoneda functor gives a fully faithful, exact functor

A —> Ab(C), X —> h_X

into the category of abelian sheaves, see Lemma Tag 05PN. Combining this with results on abelian sheaves one obtains a proof of Mitchell’s embedding theorem for abelian categories, see Remark Tag 05PR.

I like the argument phrased in this way because I already know about sites, sheaves, etc. It in some sense explains to me (and hopefully an additional handful of readers here) why the embedding theorem should be true. Moreover, I want to make the point that for all applications I can imagine the embedding into the category of abelian sheaves on a site is sufficient.

Why is a product of varieties over an algebraically closed field k a variety?

After some preliminary reductions this reduces to the question: Why is A ⊗ B a domain if A, B are domains over k (tensor product over k). To prove this suppose that (∑ a_i ⊗ b_i) (∑ c_j ⊗ d_j) = 0 in A ⊗ B with a_i, c_j ∈ A and b_i, d_j ∈ B. After recombining terms we may assume that b_1, …, b_n are k-linearly independent in B and also that d_1, …, d_m are k-linearly independent in B. Let A’ be the k-subalgebra of A generated by a_i, c_j. Unless all of the a_i and c_j are zero, we can find (after rearranging indices) a maximal ideal m ⊂ A which does not contain a_1 and c_1 (use that A’ is a domain). Denote a_i(m) and c_j(m) the congruence classes in A/m. By the Hilbert Nullstellensatz A/m = k and we can specialize the relation to get

(∑ a_i(m) b_i) (∑ c_j(m) d_j) = 0

in B! This is a contradiction with the assumption that B is a domain and we win.

This blog post is my atonement for having “forgotten” this argument. What are some standard texts which have this argument? (Ravi will add it to his notes soon he just told me…)

Today I encountered the following lemma:

Let A be a ring. Let u : M —> N be an A-module map. Let R = colim_i R_i be a directed colimit of A-algebras. Assume that M is a finite A-module, N is a finitely presented A-module, and u ⊗ 1 : M ⊗ R —> N ⊗ R is an isomorphism. Then there exists an i ∈ I such that u ⊗ 1 : M ⊗ R_i —> N ⊗ R_i is an isomorphism. (All tensor products over A.)

What I like about this statement is that M only needs to be a finite A-module. This is similar to what happened in this post.

Just a brief note about the results produced by looking up a tag on the query page. If the tag you’re looking up points to a lemma, proposition, theorem, remark, example, exercise, situation, or definition, then the search will also return the latex code of the corresponding environment. Here is are two examples to see what it looks like. Enjoy!

Of course eventually we’ll want to render the latex directly (I prefer not do this until there is a good way to render diagrams coded with the xypic package). But in some sense seeing the latex source is better: it tells you the latex label for the result, and it allows you to copy and edit the latex source directly and email me your improvement. It would be trivial to have the page also print the proof (in case the tag points to a lemma, proposition, or theorem). Let me know if you think this is a good idea.

Denote Sch the site of schemes over Q endowed with the fppf topology. Let F = Q(x_1, x_2, x_3, …) be the purely transcendental extension of Q generated by countably many elements. Let X = Spec(F). Let G = Z[X] be the free abelian sheaf on (the sheaf represented by) X. This sheaf has the following amusing property: If k is a field then

• G(Spec(k)) = 0 if trdeg(k/Q) finite, and
• G(Spec(k)) is not zero else.

The reason is that Mor_{Sch}(Spec(k), X) = ∅ if the transcendece degree of k is finite.

Here is another amusing abelian sheaf H. For any scheme S in Sch let I_S be the category of arrows f : T —> S where T is a nonempty connected scheme which is locally of finite type over some field of finite transcendence degree over Q. (Yes, this is a bit contrived.) A morphism (f : T —> S) —> (f’ : T’ —> S) in I_S is a morphism a : T —> T’ such that f = f’ o a. Define H(S) to be the set of maps σ: Ob(I_S) —> Z such that σ(f : T —> S) = σ(f’ : T’ —> S) if there is a morphism between f and f’ in I_S. In other words, σ is constant on “connected components” of Ob(I_S). In the case that Ob(I_S) = ∅ we set H(S) = 0. I claim that H is a sheaf (see remark below). Then H has the following property: if k is a field then

• H(Spec(k)) = Z if trdeg(k/Q) is finite, and
• H(Spec(k)) = 0 else.

The reason is that if there exists a morphism T —> Spec(k) with T nonempty and locally of finite type over a field of finite transcendence degree over Q, then k has finite transcendence degree over Q.

Remark: Suppose Sch’ ⊂ Sch is a full subcategory consisting of locally Noetherian schemes such that if T is in Sch’ and T’ —> T is locally of finite type, then T’ is in Sch’. Then Sch’ is also a site (with fppf topology) and the inclusion functor u : Sch’ —> Sch is cocontinuous. This gives rise to a morphism of topoi g : Sh(Sch’) —> Sh(Sch), see the chapter on Sites and Sheaves in the stacks project. Warning: this morphism of topoi is in the “wrong” direction. The sheaf H above is the sheaf g_*Z when we take Sch’ the category of schemes which are locally of finite type over a field of finite transcendence degree over Q. (Note that in our example Sch’ does not have all fibre products, but that doesn’t matter.)

Conclusion: The category of all schemes (over a given base) is too large to expect (fppf) sheaves to exhibit any kind of “coherent” behaviour as the input ranges over spectra of fields.

Let S be a scheme. Suppose that G is a sheaf of groups on (Sch/S)_{fppf}. What kind of conditions guarantee that any fppf torsor is actually an étale torsor? I know that if G is representable by an affine smooth group scheme this is OK. So this suggests looking at formally smooth sheaves G. Is there a counter example?

PS: What about the sheaf of the preceding blog post? Here G is formally étale.

[Edit Tuesday January 04, 2011. Bhargav just send me the following example of a formally smooth G with an fppf torsor which is not an etale torsor. Let S = Spec(k) where k is separably closed but not algebraically closed of characteristic p > 0. Let F be the sheaf which for a k-algebra R gives

F(Spec(R)) = {r in R | r^{p^n} = 0 for some n > 0}.

In other words F is the colimit of the sheaves alpha_{p^n} of p^n roots of zero. Allowing arbitrary p-power roots of 0 gives formal smoothness. The injective map alpha_p -> F gives fppf F-torsors that are non-trivial because H^1_{fppf}(Spec(k), alpha_p) -> H^1_{fppf}(Spec(k), F) is injective. And H^1_{fppf}(k, alpha_p) is non-zero by non-perfectness of k. But H^1_{etale}(Spec(k), F) = 0 since S has no connected etale covers.]