Derived category of sheaves
1. Generalities on categories
The categories we shall encounter are
- Top, Diff.
- Abelian Groups.
- Sh, sheaves on .
Examples of functors that we shall encounter are
- Forgetful functor to Sets.
- Sections of a sheaf : OpenSets Abelian Groups.
There is a notion of morphism of functors. A functor is representable if it is for some object .
1.1. Additive categories
An additive category is a category such that
- There is an object such that .
- All are abelian groups, and composition is bilinear.
- There is a direct sum, i.e. for all , , an object with arrows such that for .
- There is a direct product, i.e. for all , , an object with arrows such that for and all .
Ab, -Mod, Sh are additive.
A functor is additive if the map is a group homomorphism.
If is an additive category, the category of complexes is additive as well. We denote by the shift functor that re-numbers spaces in a complex. A homotopy between chain maps is a morphism such that . The intermediate category is the additive category with objects complexes and morphisms homotopy classes of chain maps.
This category is merely an intermediate step.
1.3. Abelian categories
Homology cannot be defined for general additive categories, since we do not have kernels and images. For , the kernel functor Sets, is defined as follows :
If this functor is representable, this provides us with an object, also denoted by , such that .
Similarly, given , there is a cokernel functor Sets. If this functor is representable, this provides us with an object, also denoted by .
Remark 1 is a functor, but it is not representable, usually. So this is not the right definition for cokernel.
Indeed, in Ab, let is multiplication by , and . Then whereas .
Definition 1 In an abelian category,
- For all , and exist.
- Let and be the natural maps. Then and are canonically isomorphic.
In fact, has a canonical factorization where .
If is an abelian category, given a complex , for all degrees , the differential factors as and factors as . One defines the homology of as .
A quasi-isomorphism between complexes is a morphism which induces and isomorphism on homology. We want to think of quasi-isomorphisms as as good as isomorphisms.
Example 1 Consider the following two complexes.
There is an obvious chain map between them. This is a quasiisomorphism, but there is no homotopy from it to 0.
1.5. Derived category
The derived category of an abelian category has for objects complexes and for morphisms a localization of chain maps mod homotopy, which we now define as the solution of a universal problem.
Definition 2 The derived category of an abelian category is the unique category such that there exists a functor such that
- For any category , any functor such that
factorizes through .
Concretely, is constructed as follows. Let be the collection of quasiisomorphisms in . Define
where the equivalence relation roughly means that , and precisely means that there exist quasiisomorphisms such that
The functor is defined as identity on objects (i.e. complexes) and on morphisms , by .
1.6. Localizing systems
The proof of the statement included in the definition of the derived category is not straightforward.
For the definition to work, one needs the following properties of (say is a localizing system) :
- Identity belongs to .
- is stable under composition (whenever defined).
- Given and a quasiisomorphism , there exists a complex , and a quasiisomorphism such that .
- For all , existence of such that implies existence of such that .
Proposition 3 The collection of quasiisomorphisms is a localizing system.
For the proof, we shall use the notion of mapping cone. Given , define the mapping cone as with differential . There is an exact sequence .
Now we can prove property 3 of localizing systems. Given and , let where arises in the mapping cone exact sequence for . Let and arise in the mapping cone exact sequence for . Then is a quasiisomorphism, there is a natural map making the diagram commute.
Property 4 is proven in a similar way.
1.7. Derived functors
Any functor from to itself induces a functor from to itself, but quasiisomorphismsquasiisomorphism in general (counterexample ?).
Say a functor if it maps short exact sequences to short exact sequences. Say is left exact if it maps short exact sequences to short sequences which are exact on the left and middle.
Proposition 4 An exact functor maps quasiisomorphisms to quasiisomorphisms, and so induces a functor on derived categories.
Indeed, is a quasiisomorphism iff is acyclic. For an exact functor, acyclic implies acyclic (split into short exact sequences using kernels and images).
So life with exact functors is easy. Unfortunately, the functors we are interested in are not exact. For instance, is only left exact. is right exact. The section functor for a sheaf is left exact.
Definition 5 The derived functor of an additive left exact functor from to is a pair
- of an exact functor from to ,
- and a morphism of functors from to ,
such that for any exact functor from to and morphism of functors from to , there exists a unique morphism of functors from to such that
Looks like a mess.
1.8. Strategy for computing derived functors.
Select a subclass of objects from (typically, resolutions). If functor is nice on subclass, work componentwise. If subclass is large enough, its localization is equivalent to the derived category .