derived functorDerivedFunctor2003-02-10 20:26:202006-05-15 16:05:49Definition% almost certainly you want these
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\newcommand{\Zp}{\Z/p}Let $\ES{A},\ES{B}$ be abelian categories, $\ES{A}$ having enough injectives, and $F:\ES{A}\to\ES{B}$
be a covariant \PMlinkname{left-exact functor}{ExactFunctor}.
Given an object $A\in \ES{A}$, we can construct an injective resolution
$$\xymatrix{A\to \mathbf{A}\colon 0\ar[r]& A\ar[r]& I^1\ar[r]& I^2\ar[r]&\cdots}$$
which is unique up to chain homotopy equivalence. Then we apply the functor
$F$ to the injectives in the resolution to to get a complex
$$\xymatrix{F(\mathbf{A})\colon 0\ar[r]&F(I^1)\ar[r]& F(I^2)\ar[r]&\cdots}$$
(notice that the term involving $A$ has been left out. This is not an accident,
in fact, it is crucial). This complex is also independent of the choice of $I$'s
(up to chain homotopy equivalence). Now, we define the \emph{classical right
derived functors} $R^iF(A)$ to be the cohomology groups $H^i(F(\mathbf{A}))$.
These only depend on $A$.
A completely analogous construction can be carried out for right-exact functors and for contravariant functors exact on either side, but it is traditional to only describe one case, as doing the others mostly consists of reversing arrows (and replacing ``injective'' with projective when appropriate), and the result is that of a left derived functor $L^iF$ for a given right-exact covariant functor $F$ via a projective resolution.
Important properties of the classical derived functors are these:
If the sequence $0\to A\to A'\to A''\to 0$ is exact, then there
is a long exact sequence
$$\xymatrix{0\ar[r]&F(A)\ar[r]& F(A')\ar[r]& F(A'')\ar[r]& R^1F(A)\ar[r]&\cdots}$$
which is natural (a morphism of short exact sequences induces a morphism
of long exact sequences). This, along with a couple of other properties
determine the derived functors completely, giving an axiomatic definition,
though the construction used above is usually necessary to show existence. One can often confirm that some other functors constructed in some different way are equal to the right derived functors by checking that they satisfy these properties; such a proof is called a ``universal $\delta$-functor argument''.
From the definition, one can see immediately that the following are equivalent:
\begin{enumerate}
\item $F$ is exact
\item $R^nF(A)=0$ for $n\geq 1$ and all $A\in \ES{A}$.
\item $R^1F(A)=0$ for all $A\in \ES{A}$.
\end{enumerate}
However, $R^1F(A)=0$ for a particular $A$ does not imply that $R^nF(A)=0$ for
all $n\geq 1$.
Important examples are:
\begin{itemize}
\item The Ext functors $\mathrm{Ext}^n$ are the right derived functors of $\mathrm{Hom}$.
\item The Tor functors $\mathrm{Tor}_n$ are the left derived functors of the tensor product.
\item Sheaf cohomology arises as the right derived functors of the global section functor on sheaves.
\item Group cohomology arises as the right derived functors of the ``fixed submodule'' functor on the category of $G$-modules for some group $G$.
\item \'Etale cohomology arises as the right derived functors of the global sections functor on the category of \'etale sheaves; this example includes as special cases the previous two.
\end{itemize}