The String Coffee Table

Last time I reviewed the definition of the derived category $\mathbf D (C)$ of any abelian category $C$. $\mathbf D (C)$ is like the category $\mathbf K (C)$ of (cochain) complexes in $C$ modulo chain homotopy but modded out by

(cf. Weibel 10.3.1)

which sends quasi-isomrphisms to true isomorphisms.

Now, given any additve functor

it lifts in the obvious way to a functor

The problem is to turn this into a functor

in a way that respects $q$, i.e. which preserves quasi-isomorphisms.

So what one wants is to have an $\mathbf R F$ which makes it true that there is a natural transformation from

to

If such an $\mathbf R F$ exists it is called the total derived functor (10.5.1).

The idea here is simple enough. Things become more involved when this implicit definition of $\mathbf R F$ is to be made more explicit. In order to do that the rest of the theory revolves around determining the cohomology of $\mathbf R F$.

So if $\mathbf R F$ is applied to some cochain complex $X$, what is the $n$-th cohomology of the complex $\mathbf R F (X)$, i.e. what is the right hand side of

?

Here the $\mathbb{R}^n F$ are called the hyper-derived functors, by definition (5.7.4).

In order to understand hyper-derived functors it is finally necessary to first understand ordinary derived functors (to be distinguished from the ‘total’ derived functors that I started with).

So the idea is this:

Given any object $A$ in an abelian catgeory we can find resolutions of this object. A resolution is an exact cochain complex

which hence has $H^0 = A$.

(This is familiar in physics from BRST quantization, where $A$ would be the space of physical states at ghost number 0.)

There are certain such resolutions which enjoy a property called injectivity. This is the case when each of the $I^n$ is injective which in turn means that $Hom(\cdot,I)$ is an exact functor.

So here is what an ordinary right derived functor $R^n F$ does to an object $A$:

it assigns to it the $n$th-cohomology of the result of applying $F$ to any (fixed but arbitrary) injective resolution $I_A$ of $A$.

Fine, but that’s not enough. One needs to boost this construction by one dimension to get hyper-derived functors.

Given any chain complex $A$, we can find its right Cartan-Eilenberg resolution, which is a (cochain) complex of cochain complexes, i.e. a double complex, with injective objects everywhere, being a resolution of the complex $A$ in analogy to the above construction in an appropriate sense. The $n$th hypercohomology of this (the cohomology of the total complex) is the $n$th hyper-derived functor of $F$ applied to $\mathbf R^n F (A)$.

Using all this the implicitly defined total right derived functor $\mathbf R F$ can be described finally as indicated at the beginning:

Its $n$th cohomology is given by

Furthermore, if $X$ is the cochain representing an object $x$ then hypercohomology on $X$ coincides with cohomology

and at 0th position it reproduces $F$ applied to the original object

This gives the relation of the total derived functor $\mathbf R F$ to the original functor $F$.