The Derived Category 3: Derived Functors


Let’s get back to some math. Today I’ll restart a series that I started forever ago on the derived category of a variety. I briefly described what the derived category of a variety is in order to give a very sketchy outline of what Homological Mirror Symmetry is. If you’ve forgotten the construction you can go read about it. I’ll do a one paragraph recap here, so if you don’t care about the details then that should suffice.

For us, in this series we will assume our varieties are all smooth, projective (and irreducible) over a field {k}. The notation {D(X)} will mean the bounded derived category of coherent sheaves on {X}. It is no longer abelian, but it is a {k}-linear, triangulated category. One way to construct {D(X)} is to form the category of complexes of coherent sheaves, then consider morphisms of complexes up to homotopy equivalence, then invert all quasi-isomorphisms. Thus a morphism {\mathcal{A}^\bullet \rightarrow \mathcal{B}^\bullet} is a “hat.” There is a complex {\mathcal{C}^\bullet} quasi-isomorphic to {\mathcal{A}^\bullet} and a morphism in the homotopy category {\mathcal{C}^\bullet\rightarrow \mathcal{B}^\bullet}.

Recall how we form derived functors in classical-land. The most beloved derived functor in algebraic geometry is probably the global section functor, because the {i}-th right derived functor is just sheaf cohomology: {R^i\Gamma (X, \mathcal{F})=H^i(X, \mathcal{F})}. What do we do? We take our sheaf and replace it with an injective resolution (an exact sequence) {0\rightarrow \mathcal{F}\rightarrow \mathcal{I}^\bullet}. Then we take global sections of each term (and chop off the first guy) to get a complex which is possibly no longer exact. The {i}-th derived functor is now just cohomology at the {i}-th spot.

One of the beautiful things about the derived category is that we can keep track of all of this information all at once using a “total” derived functor. Secretly what is going on is that we started with an additive functor {\Gamma: Coh(X)\rightarrow Vec(k)} to vector spaces over {k} (in general, between any two abelian categories). When we first started talking about this we noted that the homotopy category of the full subcategory of injectives is isomorphic to the derived category: {\mathcal{K}^+(\mathcal{I})\simeq D^+(X)}. So step one of finding an injective resolution just amounts to replacing the complex {\cdots \rightarrow 0 \rightarrow \mathcal{F}\rightarrow 0 \rightarrow \cdots} with a quasi-isomorphic complex of injectives (i.e. use this equivalence!).

In the homotopy category it is perfectly fine to take global sections of everything and get another complex. Then we just go back to the derived category by the universal quotient functor (of inverting quasi-isomorphisms). What did this do? Well, we may as well generalize. If I have a left exact functor between abelian categories (and I have enough injectives) {F: \mathcal{A}\rightarrow \mathcal{B}}, then I can make a total derived functor {\mathbf{R}F: D^+(\mathcal{A})\rightarrow D^+(\mathcal{B})} by replacing a complex by a quasi-isomorphic complex of injectives and applying {F} to everything (and strictly speaking passing back to the derived category).

It takes a complex to a complex, but it is keeping track of all the information of a classical derived functor, because it is literally the exact same process but we just omitted that last step of taking cohomology. So what we end up with is a complex with the property that {\mathcal{H}^i(\mathbf{R}\Gamma(\mathcal{F}))=R^i\Gamma(\mathcal{F})=H^i(X, \mathcal{F})}. Since we’re in our nice variety situation all higher cohomology vanishes so we can actually stay in the bounded derived categories.

Note that this isn’t merely a way to keep track of all the {R^iF} at once. It wouldn’t be that useful if this was the only thing it was doing. If we have any of our common left exact functors {F:Coh(X)\rightarrow Coh(Y)}, we get a functor {\mathbf{R}F: D(X)\rightarrow D(Y)}. So we apply the derived functor to complexes and not just objects of {Coh(X)}! This is a vast generalization. A word of warning here. Strictly speaking we have to keep making use of two facts (presented previously).

First, the natural functor {D(X)\rightarrow D^b(QCoh(X))} induces an equivalence between {D(X)} and the full subcategory of complexes of quasi-coherent sheaves with coherent cohomology. This allows us to actually have enough injectives to form the resolutions. Second, we have enough conditions on {X} so that we can do things in either the bounded below or bounded above versions of the derived category, but keep landing inside the bounded derived category since our resolutions may not necessarily be finite a priori.

The common functors to which I referred above are the pushforward of a map {f: X\rightarrow Y}. If I input a sheaf and take cohomology, then we recover the higher direct images {\mathcal{H}^i(\mathbf{R}f_*(\mathcal{F}))=R^if_*(\mathcal{F})}. We have {\mathcal{H}om(\mathcal{F}, -): QCoh(X)\rightarrow QCoh(X)}. Again, smoothness saves us and we get a functor on the bounded derived category with the property that {\mathcal{H}^i(\mathbf{R}\mathcal{H}om(\mathcal{F}, \mathcal{G}))=\mathcal{E}xt^i(\mathcal{F}, \mathcal{G})}. We could keep going, but the only other major one I foresee coming up in the near future is the derived tensor product. Of course this will be left derived and so we have to do the whole process above but for right exact functors.

The subtlety about reversing everything is that when you unravel the definitions you’ll find that {\mathcal{H}^{-i}(\mathcal{F}\otimes^{\mathbf{L}}\mathcal{G})=\mathcal{T}or_i(\mathcal{F}, \mathcal{G})}. A note to the detail oriented reader. I may forget to put bullets in the superscripts, but everything from here on out should be read as a complex of sheaves and not just a sheaf. I also may get lazy and leave off the R and L for right and left derived, but functors between derived categories will ALWAYS be derived.

Overall, this was just an annoying technical post I had to do. Next time I want to get to some actual geometry of the derived category!

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