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.]

Max Lieblich asked if one could find an abelian sheaf G on the category of schemes in the étale topology such that

1. G(X) = G(X_{red}),
2. G(X) = 0 when X has only one point,
3. G is not zero, and
4. G is limit preserving.

I’ll tell you why he asked in a minute, but first let me tell you an example: Let A be an abelian group. Let F be the presheaf on the category of schemes which associates to a scheme X the group of constructible functions a : |X| —> A modulo locally constant functions. Let G be the sheafification of F in the étale topology. Then G works. (For more details, look at the the section entitled “Sheaves and constructible functions” in the chapter “Examples” of the stacks project.)

Why did this come up? Consider the stack [Spec(Z)/G] classifying étale G-torsors. Then the morphism f : Spec(Z) —> [Spec(Z)/G] is an equivalence of categories of sections over the spectrum of any field, f is formally étale, and the stack [Spec(Z)/G] is limit preserving, but f is not an equivalence (as G is not zero). This answers a question posed by Dan Abramovich.

There exists a flat proper morphism f : X —> S all of whose geometric fibres are connected nodal curves such that f is not of finite presentation. An explicit example can be found in the examples chapter of the stacks project. Once you’ve understood why the example works, you easily see that you can even make an example where all the fibres are stable curves, S is connected, and the genus of the fibres jumps!

But let me go out on a limb here and make a wild guess: If you assume that there exists an integer g > 1 such that f is flat, proper, and all fibres are stable curves of genus g, then f is of finite presentation.

Why do I think this is true? I think it is true by analogy with the following results: A finite flat module need not be projective. A finite flat module over a local ring is free. Thus given a finite flat module over a scheme S then you get a well defined rank function. Then the module is finite locally free if and only if the rank function is locally constant in the Zariski toplogy (yet another characterization of finite projective modules, see Bourbaki, Commutative Algebra, Chapter II, Theorem 1).

I also think the following may be true: Given an integer d >= 0. If R —> A is a finite type, flat ring map all of whose geometric fibres are smooth and irreducible of dimension d, then R —> A is of finite presentation. (Irreducible implies nonempty. For this one I actually have a pretty good idea for how to prove it.)

Don’t do this at home kids!

What I mean by the last sentence is that, if you are doing actual moduli, you should just assume that X —> S is of finite presentation. In regards to this, note that if my wild guess is correct, then Definition 1.1 of Deligne-Mumford is the correct one.  Thanks to Michael Thaddeus for pointing out that Deligne and Mumford only assume proper + flat + conditions on fibres.