\documentclass[12pt]{amsart}
\title{Lecture 13}
\author{}

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\newtheorem*{rem}{Remark}
\newtheorem*{rems}{Remarks}
\newtheorem*{ex}{Example}

\begin{document}
\maketitle

\PP{13.0}{Overview of these notes}
%--------------------------------
In this set of notes we will discuss {\em derived functors}. A brilliant
observation made in Grothendieck's T\^{o}hoku paper was to view numerous
cohomology (or homology) theories as sequence of functors {\em derived}
from a left (or right) exact functor.\\

There are {\bf a lot} of cohomology theories in mathematics. For example,
de Rham cohomology (for manifolds), sheaf cohomology and its special
cases: $\ell$--adic cohomology, \'{e}tale cohomology (for algebraic
schemes), intersection cohomology (for singular spaces), Lie algebra
cohomology (Chevalley--Eilenberg cohomology), group cohomology, Galois
cohomology, cyclic cohomology... the list goes on.\\

One systematic way to view them is via the machinery of derived functors:\\

\[
\xy
(0,0)*{\text{Left exact functor }F};
(50,0)*{\boxed{\text{Derived functors}}};
(100,3)*{\text{Sequence of functors}};
(100,-3)*{\{R^nF\}_{n\geq 0}};
{\ar@{~>} (20,0); (33,0)};
{\ar@{~>} (67,0); (80,0)};
\endxy
\]

satisfying some naturally imposed axioms (analogous to long exact sequence
in cohomology).
An introduction to this axiomatic is the purpose of these notes.
The material of these notes is for inspirational purposes only,
and shows how an attempt to construct derived functors leads to
the structures we will study in the next few weeks (see the ``To do list"
in the last paragraph).\\

\PP{13.1}{Right derived functors of a left exact covariant functor}
%------------------------------------------------------------------
Assume $F:\CalA\to \CalB$ is an additive, covariant, left exact functor.
{\em Right derived functors} of $F$ is a sequence of (additive) covariants functors
$\{R^nF : \CalA\to \CalB\}_{n\geq 0}$ together with the following data:\\

\begin{itemize}
\item A natural isomorphism $F\isom R^0F$. We often just identify
the two and write $F$ for $R^0F$.\\

\item For every short exact sequence $S: 0\to X\to Y\to Z\to 0$ in
$\CalA$, morphisms $\ds \left\{\delta^n(S) : R^nF(Z)\to R^{n+1}F(X)\right\}_{n\geq 0}$ in
$\CalB$ (called {\em connecting morphisms}).\\
\end{itemize}

This data is subject to two axioms:\\

\begin{enumerate}
\item For every short exact sequence $S: 0\to X\morph{f} Y\morph{g} Z\to 0$
in $\CalA$,
the following (long) sequence is exact:\\

\[
\xy
(0,40)*{0}="i";
(20,40)*{F(X)}="fx"; {\ar "i"; "fx"};
(50,40)*{F(Y)}="fy"; {\ar^{F(f)} "fx"; "fy"};
(80,40)*{F(Z)}="fz"; {\ar^{F(g)} "fy"; "fz"};
{\ar "fz"; (90,40)};
(90,40); (90,30) **\crv{(100,40)&(100,30)};
{\ar@{-} (90,30); (10,30)}; (50,32)*{\scriptstyle{\delta^0(S)}};
(10,30); (10,20) **\crv{(0,30)&(0,20)};
(20,20)*{R^1F(X)}="rfx"; {\ar (10,20); "rfx"};
(50,20)*{R^1F(Y)}="rfy"; {\ar^{R^1F(f)} "rfx"; "rfy"};
(80,20)*{R^1F(Z)}="rfz"; {\ar^{R^1F(g)} "rfy"; "rfz"};
{\ar "rfz"; (90,20)};
(90,20); (90,10) **\crv{(100,20)&(100,10)};
{\ar@{-} (90,10); (10,10)}; (50,12)*{\scriptstyle{\delta^1(S)}};
(10,10); (10,0) **\crv{(0,10)&(0,0)};
(20,0)*{R^2F(X)}="rrfx"; {\ar (10,0); "rrfx"};
(50,0)*{R^2F(Y)}="rrfy"; {\ar^{R^2F(f)} "rrfx"; "rrfy"};
(80,0)*{R^2F(Z)}="rrfz"; {\ar^{R^2F(g)} "rrfy"; "rrfz"};
(90,0)*{\cdots}
\endxy
\]
\ \\

\item If we have the following commutative diagram, with
exact rows in $\CalA$, denoted by $S_1$ and $S_2$ respectively,

\[
\xy
(0,20)*{0}="i1";
(20,20)*{X_1}="x1"; {\ar "i1"; "x1"};
(40,20)*{Y_1}="y1"; {\ar^{f_1} "x1"; "y1"};
(60,20)*{Z_1}="z1"; {\ar^{g_1} "y1"; "z1"};
(80,20)*{0}="t1"; {\ar "z1"; "t1"};
(0,0)*{0}="i2";
(20,0)*{X_2}="x2"; {\ar "i2"; "x2"};
(40,0)*{Y_2}="y2"; {\ar^{f_2} "x2"; "y2"};
(60,0)*{Z_2}="z2"; {\ar^{g_2} "y2"; "z2"};
(80,0)*{0}="t2"; {\ar "z2"; "t2"};
{\ar_x "x1"; "x2"};
{\ar_y "y1"; "y2"};
{\ar_z "z1"; "z2"};
\endxy
\]

then, the following diagram commutes, for every $n\geq 0$:

\[
\xy
(0,20)*{R^nF(Z_1)}="z1"; (40,20)*{R^{n+1}F(X_1)}="x1";
(0,0)*{R^nF(Z_2)}="z2"; (40,0)*{R^{n+1}F(X_2)}="x2";
{\ar^{\delta^n(S_1)} "z1"; "x1"};
{\ar_{\delta^n(S_2)} "z2"; "x2"};
{\ar_{R^nF(z)} "z1"; "z2"};
{\ar^{R^{n+1}F(x)} "x1"; "x2"};
\endxy
\]
\end{enumerate}
\ \\

\PP{13.2}{Contravariant, left exact functor}
%------------------------------------------
If $F:\CalA\to \CalB$ is {\em contravariant} additive, left exact functor,
its right derived functors is a sequence of contravariant functors
$\{R^nF:\CalA\to\CalB\}_{n\geq 0}$, together with\\

\begin{itemize}
\item An identification of $R^0F$ with $F$.\\

\item For every short exact sequence $S: 0\to X\to Y\to Z\to 0$ in
$\CalA$, connecting 
morphisms $\ds \left\{\delta^n(S) : R^nF(X)\to R^{n+1}F(Z)\right\}_{n\geq 0}$ in
$\CalB$.\\ 
\end{itemize}

Again we have two axioms analogous to the ones from the previous paragraphs.

\begin{enumerate}
\item For every short exact sequence $S: 0\to X\morph{f} Y\morph{g} Z\to 0$
in $\CalA$,
the following (long) sequence is exact:\\

\[
\xy
(0,40)*{0}="i";
(20,40)*{F(Z)}="fx"; {\ar "i"; "fx"};
(50,40)*{F(Y)}="fy"; {\ar^{F(g)} "fx"; "fy"};
(80,40)*{F(X)}="fz"; {\ar^{F(f)} "fy"; "fz"};
{\ar "fz"; (90,40)};
(90,40); (90,30) **\crv{(100,40)&(100,30)};
{\ar@{-} (90,30); (10,30)}; (50,32)*{\scriptstyle{\delta^0(S)}};
(10,30); (10,20) **\crv{(0,30)&(0,20)};
(20,20)*{R^1F(Z)}="rfx"; {\ar (10,20); "rfx"};
(50,20)*{R^1F(Y)}="rfy"; {\ar^{R^1F(g)} "rfx"; "rfy"};
(80,20)*{R^1F(X)}="rfz"; {\ar^{R^1F(f)} "rfy"; "rfz"};
{\ar "rfz"; (90,20)};
(90,20); (90,10) **\crv{(100,20)&(100,10)};
{\ar@{-} (90,10); (10,10)}; (50,12)*{\scriptstyle{\delta^1(S)}};
(10,10); (10,0) **\crv{(0,10)&(0,0)};
(20,0)*{R^2F(Z)}="rrfx"; {\ar (10,0); "rrfx"};
(50,0)*{R^2F(Y)}="rrfy"; {\ar^{R^2F(g)} "rrfx"; "rrfy"};
(80,0)*{R^2F(X)}="rrfz"; {\ar^{R^2F(f)} "rrfy"; "rrfz"};
(90,0)*{\cdots}
\endxy
\]
\ \\

\item If we have the following commutative diagram, with
exact rows in $\CalA$, denoted by $S_1$ and $S_2$ respectively,

\[
\xy
(0,20)*{0}="i1";
(20,20)*{X_1}="x1"; {\ar "i1"; "x1"};
(40,20)*{Y_1}="y1"; {\ar^{f_1} "x1"; "y1"};
(60,20)*{Z_1}="z1"; {\ar^{g_1} "y1"; "z1"};
(80,20)*{0}="t1"; {\ar "z1"; "t1"};
(0,0)*{0}="i2";
(20,0)*{X_2}="x2"; {\ar "i2"; "x2"};
(40,0)*{Y_2}="y2"; {\ar^{f_2} "x2"; "y2"};
(60,0)*{Z_2}="z2"; {\ar^{g_2} "y2"; "z2"};
(80,0)*{0}="t2"; {\ar "z2"; "t2"};
{\ar_x "x1"; "x2"};
{\ar_y "y1"; "y2"};
{\ar_z "z1"; "z2"};
\endxy
\]

then, the following diagram commutes, for every $n\geq 0$:

\[
\xy
(0,20)*{R^nF(X_1)}="z1"; (40,20)*{R^{n+1}F(Z_1)}="x1";
(0,0)*{R^nF(X_2)}="z2"; (40,0)*{R^{n+1}F(Z_2)}="x2";
{\ar^{\delta^n(S_1)} "z1"; "x1"};
{\ar_{\delta^n(S_2)} "z2"; "x2"};
{\ar^{R^nF(x)} "z2"; "z1"};
{\ar_{R^{n+1}F(z)} "x2"; "x1"};
\endxy
\]
\end{enumerate}
\ \\

\PP{13.3}{Covariant, right exact case}
%-----------------------------------
Let $G:\CalA\to\CalB$ be a covariant, additive, right exact functor.
Its {\em left derived functors} consist of a sequence of (additive)
covariant functors $\{L_nG:\CalA\to\CalB\}_{n\geq 0}$, together with
the data of:
\begin{itemize}
\item An identification of $G$ with $L_0G$.\\

\item For every short exact sequence $\ds
S: 0\to X\to Y\to Z\to 0$ in $\CalA$, connecting morphisms
$\{\delta_n(S):L_{n+1}G(Z)\to L_nG(X)\}_{n\geq 0}$.\\
\end{itemize}

Again there are two axioms. In the case of right exact functors,
the long exact sequence continues indefinitely to the right side.

\begin{enumerate}
\item For every short exact sequence $S: 0\to X\morph{f} Y\morph{g} Z\to 0$
in $\CalA$,
the following (long) sequence is exact:\\

\[
\xy
(0,40)*{\cdots}="i";
(20,40)*{L_2G(X)}="fx"; {\ar "i"; "fx"};
(50,40)*{L_2G(Y)}="fy"; {\ar^{L_2G(f)} "fx"; "fy"};
(80,40)*{L_2G(Z)}="fz"; {\ar^{L_2G(g)} "fy"; "fz"};
{\ar "fz"; (90,40)};
(90,40); (90,30) **\crv{(100,40)&(100,30)};
{\ar@{-} (90,30); (10,30)}; (50,32)*{\scriptstyle{\delta_1(S)}};
(10,30); (10,20) **\crv{(0,30)&(0,20)};
(20,20)*{L_1G(X)}="rfx"; {\ar (10,20); "rfx"};
(50,20)*{L_1G(Y)}="rfy"; {\ar^{L_1G(f)} "rfx"; "rfy"};
(80,20)*{L_1G(Z)}="rfz"; {\ar^{L_1G(g)} "rfy"; "rfz"};
{\ar "rfz"; (90,20)};
(90,20); (90,10) **\crv{(100,20)&(100,10)};
{\ar@{-} (90,10); (10,10)}; (50,12)*{\scriptstyle{\delta_0(S)}};
(10,10); (10,0) **\crv{(0,10)&(0,0)};
(20,0)*{G(X)}="rrfx"; {\ar (10,0); "rrfx"};
(50,0)*{G(Y)}="rrfy"; {\ar^{G(f)} "rrfx"; "rrfy"};
(80,0)*{G(Z)}="rrfz"; {\ar^{G(g)} "rrfy"; "rrfz"};
(100,0)*{0}="t"; {\ar "rrfz"; "t"};
\endxy
\]
\ \\

\item If we have the following commutative diagram, with
exact rows in $\CalA$, denoted by $S_1$ and $S_2$ respectively,

\[
\xy
(0,20)*{0}="i1";
(20,20)*{X_1}="x1"; {\ar "i1"; "x1"};
(40,20)*{Y_1}="y1"; {\ar^{f_1} "x1"; "y1"};
(60,20)*{Z_1}="z1"; {\ar^{g_1} "y1"; "z1"};
(80,20)*{0}="t1"; {\ar "z1"; "t1"};
(0,0)*{0}="i2";
(20,0)*{X_2}="x2"; {\ar "i2"; "x2"};
(40,0)*{Y_2}="y2"; {\ar^{f_2} "x2"; "y2"};
(60,0)*{Z_2}="z2"; {\ar^{g_2} "y2"; "z2"};
(80,0)*{0}="t2"; {\ar "z2"; "t2"};
{\ar_x "x1"; "x2"};
{\ar_y "y1"; "y2"};
{\ar_z "z1"; "z2"};
\endxy
\]

then, the following diagram commutes, for every $n\geq 0$:

\[
\xy
(0,20)*{L_{n+1}G(Z_1)}="z1"; (40,20)*{L_nG(X_1)}="x1";
(0,0)*{L_{n+1}G(Z_2)}="z2"; (40,0)*{L_nF(X_2)}="x2";
{\ar^{\delta_n(S_1)} "z1"; "x1"};
{\ar_{\delta_n(S_2)} "z2"; "x2"};
{\ar_{L_{n+1}G(z)} "z1"; "z2"};
{\ar^{L_nG(x)} "x1"; "x2"};
\endxy
\]
\end{enumerate}
\ \\

\PP{13.4}{Some heuristic discussions}
%--------------------------------------
To fix ideas, let us assume $F:\CalA\to\CalB$ is covariant and left
exact. Let us try to see if our definition constraints what $R^1F:\CalA\to\CalB$
could possibly be. That is, if we can define $R^1F(X)$ for $X\in\CalA$.\\

By requirement of long exact sequence,
we know that if $0\to X\to Y\to Z\to 0$ is exact in $\CalA$,
then the following sequence is supposed to be exact:
\[
0\to F(X)\to F(Y)\to F(Z)\to R^1F(X)\to R^1F(Y)\to\cdots
\]
So, if we picked $Y$ ``cleverly" so that $R^1F(Y)$ has to be $0$
(we will be more precise later), then $R^1F(X) = \Coker(F(Y)\to F(Z))$
by the required exactness.\\

What does it mean for an object $Y$, to vanish under $R^1F$?
It means that no matter what short exact sequence starting from $Y$
we pick: $0\to Y\to \wt{Y}\to U\to 0$, the following sequence
is exact:
\[
0\to F(Y)\to F(\wt{Y})\to F(U)\to 0=R^1F(Y).
\]

Now we straighten our logic.\\

\begin{enumerate}
\item {\em Definition.} An object $Y\in\CalA$ is called {\em $F$--acyclic} (or, $F$--exact)
if for every short exact sequence $0\to Y\to \wt{Y}\to U\to 0$, the 
following sequence
is exact:
\[
0\to F(Y)\to F(\wt{Y})\to F(U)\to 0.
\]

\item {\em Assumption.} $\CalA$ has enough $F$--acyclic objects.
Meaning, for every $X\in \CalA$, there exists some $F$--acyclic $Y$
and an injective morphism $f:X\to Y$.\\

\item {\em Construction.} For $X\in\CalA$, choose an injective
morphism $f:X\to Y$, where $Y$ is $F$--acyclic. Let $\pi:Y\to Y/X$ denote
the cokernel of $f$, and define:
\[
\boxed{
R^1F(X) := \Coker(F(\pi):F(Y)\to F(Y/X)).
}
\]

(Hope for the best: that this definition does not depend on the choice
of $Y$).\\
\end{enumerate}

\PP{13.5}{Higher derived functors}
%--------------------------------
If all the wishes from the previous paragraph come true, we will be able 
to define higher derived functors inductively
as follows. Given $X\in\CalA$, find an injective morphism $X\to Y$, where
$Y$ is $F$--acyclic. Set $\ds R^{k+1}F(X) = R^kF(Y/X)$, for every $k\geq 1$.\\

For instance, in order to get $R^2F(X)$, we will have to compute
$R^1F(Y/X)$. Denoting $Y/X$ by $\ol{Y}$, this was done by choosing 
another $F$--acyclic $Y'$, an injective
map $\ol{Y} \to Y'$, and taking the cokernel of $F(Y')\to F(Y'/\ol{Y})$.

\[
\xy
(0,20)*{X\ }="x"; (30,20)*{Y}="y"; (60,20)*{Y'}="yp";
(45,0)*{Y/X}="yb";
{\ar@{^(->} "x"; "y"}; {\ar@{->>} "y"; "yb"};
{\ar@{^(->} "yb"; "yp"};
{\ar "y"; "yp"};
\endxy
\]

This line of reasoning leads us to the notion of an {\em
F--acyclic resolution}.

\begin{defn}
Let $F:\CalA\to\CalB$ be a left exact, covariant functor. Given
$X\in\CalA$, a {\em resolution} of $X$ by $F$--acyclic objects
is a sequence of morphisms:
\[
Y^{\bullet}:\qquad
Y^0\morph{f^0} Y^1\morph{f^1} \cdots 
\]
such that:
\begin{itemize}
\item $\Ker(f^0) = X$.
\item The sequence is exact at $Y^n$, for every $n\geq 1$.
\end{itemize}
\end{defn}

The logic worked out preceeding this definition yields the following
recipe for defining $\{R^nF(X)\}_{n\geq 0}$. Choose an $F$--acyclic
resolution $Y^{\bullet}$ of $X$ as above. Apply $F$ to it (as $F$
is additive, we are only promised to get a complex, not necessary
exact sequence):

\[
0\to F(Y^0)\morph{F(f^0)} F(Y^1)\morph{F(f^1)} F(Y^2)\morph{F(f^2)} \cdots
\]

and define (with the convention $F(f^{-1})=0$):

\[
\boxed{
R^nF(X) := \Ker(F(f^n))/\Im(F(f^{n-1})) = H^n(F(Y^\bullet))
}
\]

Here, we are borrowing notation from algebraic topology: for any
complex $Z^\bullet: 0\to Z^0 \to Z^1\to \cdots $ in an abelian
category, its $n^{\scriptstyle{\text{th}}}$ cohomology is defined
as:

\[
\boxed{
H^n(Z^{\bullet}) := \frac{\Ker(Z^n\to Z^{n+1})}
{\Im(Z^{n-1}\to Z^n)}
}
\]

\begin{rem}
At this point, nothing is obviously true. We don't know that
the thing defined above on objects is a functor. We don't know
how the connecting morphisms will be defined.
\end{rem}

\PP{13.6}{To do list}
%-------------------
Let us pause and reflect on the amount of work laid out before
us. This ``to do list" contains precisely the topics we will
cover in this part of the course (i.e, Homological Algebra).\\

\begin{enumerate}
\item Study category of complexes and functorial properties of
$H^n : \text{Complexes over } \CalA \to \CalA$. In this part,
we will encounter:
\begin{itemize}
\item The famous ``snake lemma" which gives rise
to the connecting morphisms.\\

\item The notion of {\em homotopy} between two morphisms of
complexes, and how homotopic morphisms induce
the same morphism under $H^n$. This will be needed to prove
that our construction is {\em independent of the chosen
resolution.}\\
\end{itemize}

\item Study $F$--acyclic objects. As we will see, when we get
to this part, that:
\begin{itemize}
\item {\em Injective objects} are $F$--acyclic for any
(i) left exact, covariant or (ii) right exact contravariant $F$.\\

\item {\em Projective objects} are $F$--acylic for any
(i) left exact contravariant, or (ii) right exact covariant $F$.\\
\end{itemize}

\item Prove uniqueness (up to homotopy) of injective and projective
resolutions.\\
\end{enumerate}

Once we reach the end of this long road, we will be able to define
$\Ext$ (and $\Tor$) as right (and left) derived functors of
$\Hom$ (and $\otimes$). At that point, we will let go of the
abstraction and see how to compute these things explicitly.

\end{document}
