\documentclass[12pt]{amsart}
\title{Lecture 16}
\author{}

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\begin{document}
\maketitle

\PP{16.0}{Road to derived functors}
%---------------------------------
Recall from Lecture 13, our objective is to follow the path sketched
below to get to derived functors. For definiteness, $F:\CalC\to\CalD$
is a left exact, covariant functor between two abelian categories
$\CalC$ and $\CalD$.\\

\[
\xy
{\ar@{~>}^{\boxed{\ast}} (0,20)*{\CalC}="a"; (30,20)*{\CoComp{\CalC}}="k"};
{\ar^F "k"; (30,0)*{\CoComp{\CalD}}="kb"};
{\ar^{H^n} "kb"; (60,0)*{\CalD}="b"};
"a"; "b" **\crv{(0,-10)&(30,-20)}?(0.9)*\dir{>};
(30,-15)*{R^nF};
\endxy
\]

\begin{itemize}
\item Here $F:\CoComp{\CalC} \to \CoComp{\CalD}$ is applied to
a complex, term-by-term, and also to the differentials.
\[
\lp\cdots X^n\morph{d^n} X^{n+1}\morph{d^{n+1}}\cdots\rp\qquad
\mapsto \qquad
\lp\cdots F(X^n)\morph{F(d^n)} F(X^{n+1})\morph{F(d^{n+1})}\cdots\rp
\]

Similarly, on morphisms $F$ is applied term-by-term.\\

\item We studied $\{H^n\}_{n\in\Z}$ functors in the last two lectures.
They give rise to long exact sequences, and do not distinguish
between morphisms homotopic to each other.\\

\item $\ds \boxed{\ast}$ is not going to be a functor. This is the
step where we will have to construct {\em resolutions} by special
objects - injective, projective, free or flat (the meaning of these
words will be explained, don't worry). We will also have to prove
that the choice of a resolution in this construction becomes
immaterial after $H^n$ is applied - namely, {\em resolutions are
unique up to homotopy}. Proofs of these assertions are the focus
of this and the next lecture.\\
\end{itemize}

As before, $\CalA = \Rmod{R}$ is the category of left $R$--modules.
The definitions and results of this lecture work for any abelian
category $\CalA$, and will be stated and proved in that generality.
However, for the purposes of this course, you may as well assume
that we are talking about modules over $R$.\\

\PP{16.1}{Injective and projective objects}
%-----------------------------------------

\begin{defn}
An object $Q\in\CalA$ is said to be {\em injective} if, for every
injective morphism $f:Q\to X$, there exists $Q'$, and an isomorphism
$g:Q\oplus Q'\isom X$ such that $g\circ \iota_Q = f$, where
$\iota_Q : Q\to Q\oplus Q'$ is the canonical inclusion.\\

Similarly, an object $P\in\CalA$ is said to be {\em projective} if, for every
surjective morphism $f:Y\to P$, there exists $P'$, and an isomorphism
$g:Y\isom P\times P'$ such that $\pi_P\circ g = f$, where
$\pi_P : P\times P' \to P$ is the canonical surjection.\\
\end{defn}

Recall (Lecture 8, Lemma 8.3) that finite direct sums and direct
products are naturally isomorphic in any additive category.
The following characterization of direct sum/product of two objects
was used in Lecture 10 (see Claim on page 2). We record here
as a lemma for future use, its proof is given in Lecture 10 (page 2).\\

\begin{lem}
Let $X,X_1,X_2$ be three objects in $\CalA$. Then $X\cong X_1\oplus X_2$
if, and only if, we have morphisms $f_\ell:X_\ell\to X$ and $g_\ell:X\to
X_\ell$ ($\ell=1,2$) such that:
\[
g_k\circ f_\ell = \delta_{k,\ell},\ k,\ell\in\{1,2\}, \qquad
\Id_X = f_1\circ g_1 + f_2\circ g_2.
\]
\end{lem}
\ \\

\PP{16.2}{Idempotents}
%--------------------
The following result is going to be crucial in obtaining different
characterizations of injective and projective objects.\\

\begin{lem}
Let $X\in\CalA$ and let $p\in\End_{\CalA}(X)$ be such that $p\circ p = p$
(such a morphism is called an {\em idempotent} or {\em projection}).
Then $\Ker(p)\oplus\Im(p)\isom X$.\\
\end{lem}

\begin{pf}
Let $q\in\End_{\CalA}(X)$ be defined as $\Id_X-p$. Then:
$\ds q^2 = (1-p)(1-p) = 1-2p+p^2 = 1-p = q$. Moreover,
$pq = p(1-p) = p-p^2 = 0$, and $qp=(1-p)p = p-p^2=0$. That is,
we have:

\[
q\circ q = q,\qquad\text{and}\qquad p\circ q = 0 = q\circ p.
\]

\noindent {\bf Claim.} $\Im(q) = \Ker(p)$ and $\Ker(q)=\Im(p)$.\\

Let us assume this for now, and see how the lemma follows. Let us
denote by $X_1 = \Ker(p) \morph{i} X$ the canonical inclusion.
Moreover, $X\morph{\ol{q}} X_1=\Im(q)$ is the natural surjection
which factors $q = i\circ \ol{q}$. Similarly,
\[
X_2=\Ker(q)\morph{j} X \morph{\ol{p}} X_2 = \Im(p).
\]
From the properties of $p,q$ listed above, it is easy to see
that this $4$-tuple of morphisms satisfies the equations listed
in Lemma 16.1 above, hence identifies $X$ as a direct sum
(or product) of $X_1$ and $X_2$.\\

{\em Proof of the claim.} Let us prove this in a more categorical
framework. Consider the following diagram:
\[
\xy
{\ar^{\ol{q}} (0,20)*{X}="x"; (30,20)*{\Im(q)}="i"};
{\ar^i "i"; (60,20)*{X}="x2"};
{\ar^p "x2"; (90,20)*{X}="x3"};
"x"; "x2" **\crv{(30,40)}?(0.96)*\dir{>};
(30,32)*{\scriptstyle{q}};
\endxy
\]
The composition $p\circ i\circ \ol{q} = p\circ q = 0$. As $\ol{q}$
is surjective, we conclude that $p\circ i=0$. Now
to prove that $\Im(q)\morph{i}X$ is the kernel of $p$, we have
to show that for every morphism $z:Z\to X$ such that $p\circ z=0$,
$z=i\circ \ol{z}$ for a unique $\ol{z}$ (uniqueness is clear since
$i$ is injective, we only have to check it exists).
\[
\xy
{\ar^{\ol{q}} (0,20)*{X}="x"; (30,20)*{\Im(q)}="i"};
{\ar^i "i"; (60,20)*{X}="x2"};
{\ar^p "x2"; (90,20)*{X}="x3"};
"x"; "x2" **\crv{(30,40)}?(0.96)*\dir{>};
(30,32)*{\scriptstyle{q}};
{\ar^z (60,0)*{Z}="z"; "x2"};
{\ar_0 "z"; "x3"};
{\ar@{-->}_{\exists \ol{z}} "z"; "i"};
\endxy
\]

Since $p\circ z=0$ we get $z = z-p\circ z = (1-p)\circ z = q\circ z$.
Hence $z=i\circ (\ol{q}\circ z) = i\circ \ol{z}$ as we wanted.
The proof of $\Im(p)=\Ker(q)$ is analogous and therefore omitted.
\end{pf}
\ \\

\PP{16.3}{Pull--backs and push--forwards}
%---------------------------------------
(See Homework 5, Problems 3 and 4). Recall that pull--back diagram
is summarized as:

\[
\xy
{\ar_{u_1} (0,0)*{M_1}="a"; (30,0)*{N}="z"};
{\ar^{u_2} (30,20)*{M_2}="b"; "z"};
{\ar@{~>} (40,10); (60,10)};
(15,-10)*{\text{Given}};
{\ar_{u_1} (70,0)*{M_1}="ap"; (100,0)*{N}="zp"};
{\ar^{u_2} (100,20)*{M_2}="bp"; "zp"};
(70,20)*{\boxed{M_1\times_N M_2}}="pb";
{\ar^{\pi_1} "pb"; "ap"};
{\ar^{\pi_2} "pb"; "bp"};
(85,-10)*{\text{Definition}};
\endxy
\]
\ \\

Concretely $M_1\times_N M_2$ is defined as the kernel of $M_1\times M_2\to N$,
with $\pi_1$ and $\pi_1$ being the following two compositions:
\[
M_1\times_N M_2 \hookrightarrow M_1\times M_2 \to M_1, \qquad\text{and}\qquad
M_1\times_N M_2 \hookrightarrow M_1\times M_2 \to M_2.
\]

Similarly, the following diagram defines push--forward:\\

\[
\xy
{\ar^{v_2} (0,20)*{L}="y"; (30,20)*{M_2}="b"};
{\ar_{v_1} "y"; (0,0)*{M_1}="a"};
{\ar@{~>} (40,10); (60,10)};
(15,-10)*{\text{Given}};
{\ar^{v_2} (70,20)*{L}="yp"; (100,20)*{M_2}="bp"};
{\ar_{v_1} "yp"; (70,0)*{M_1}="ap"};
(100,0)*{\boxed{M_1\oplus^LM_2}}="pf";
{\ar^{\iota_1} "ap"; "pf"};
{\ar^{\iota_2} "bp"; "pf"};
(85,-10)*{\text{Definition}};
\endxy
\]
\ \\

Again, concretely $M_1\oplus^LM_2$ is the cokernel of $L\to M_1\oplus M_2$,
with morphisms $\iota_1$ and $\iota_2$ given as compositions of natural
inclusions and surjections:
\[
M_1\to M_1\oplus M_2 \twoheadrightarrow M_1\oplus^LM_2,\qquad\text{and}\qquad
M_2\to M_1\oplus M_2 \twoheadrightarrow M_1\oplus^LM_2.
\]

The proof of the following proposition is left as part of the homework.\\

\begin{prop}\hfill
\begin{enumerate}
\item If $u_1$ is surjective, then so is $\pi_2$.\\

\item If $v_2$ is injective, then so is $\iota_1$.\\
\end{enumerate}
\end{prop}


\PP{16.4}{Characterizations of injective and projective objects}
%---------------------------------------------------------------
Now we are ready to state and prove various different ways injective
and projective objects can be defined.\\

\begin{thm}\hfill
\begin{itemize}
\item Let $Q\in\CalA$. Then the following conditions on $Q$ are equivalent.
\begin{enumerate}
\item $Q$ is injective.
%\item Every short exact sequence $0\to Q\to Z\to Y\to 0$ splits
%{\footnote{See Homework 5, Problem 8.}}.
\item For any injective morphism $f:A\hookrightarrow B$, and arbitrary $g:A\to Q$,
there exists a lift $\wt{g}:B\to Q$ such that $\wt{g}\circ f = g$.
\[
\xy
{\ar^{f} (0,0)*{A}="a"; (30,0)*{B}="b"};
{\ar_{g} "a"; (0,-20)*{Q}="q"};
{\ar (-20,0)*{0}; "a"};
{\ar@{-->}^{\exists\wt{g}} "b"; "q"};
\endxy
\]
\item $\Hom_{\CalA}(-,Q)$ is exact.
\end{enumerate}

\item Let $P\in\CalA$. Then the following conditions on $P$ are
equivalent.
\begin{enumerate}
\item $P$ is projective.
%\item Every short exact sequence $0\to X\to Z\to P\to 0$ splits.
\item For any surjective morphism $f:B\twoheadrightarrow C$, and arbitrary $g:P\to C$,
there exists a lift $\wt{g}:P\to B$ such that $f\circ \wt{g} = g$.
\[
\xy
{\ar^{f} (0,0)*{B}="b"; (30,0)*{C}="c"};
{\ar^{g} (30,20)*{P}="p"; "c"};
{\ar "c"; (50,0)*{0}};
{\ar@{-->}_{\exists\wt{g}} "p"; "b"};
\endxy
\]

\item $\Hom_{\CalA}(P,-)$ is exact.
\end{enumerate}

\end{itemize}
\end{thm}

We will prove this theorem for injective objects only. The proof for
projective objects is absolutely analogous and is left to the reader.\\

\PP{16.5}{Proof of Theorem 16.4: injective case}
%----------------------------------------------
Let $Q\in\CalA$.\\

\noindent (1) $\Rightarrow$ (2). Assume that $Q$ is injective,
and we have the diagram:
\[
\xy
{\ar^{f} (0,0)*{A}="a"; (30,0)*{B}="b"};
{\ar_{g} "a"; (0,-20)*{Q}="q"};
{\ar (-20,0)*{0}; "a"};
%{\ar@{-->}^{\exists\wt{g}} "b"; "q"};
\endxy
\]

Let $\wt{Q} = Q\oplus^A B$ be the push--forward (see \S 16.3 above). 
As claimed in Proposition 16.3 (2), the natural map $Q\to \wt{Q}$ is injective.
By definition, we have an object $Q'$ and an isomorphism
$Q\oplus Q' \isom \wt{Q}$.

\[
\xy
{\ar^{f} (0,0)*{A}="a"; (30,0)*{B}="b"};
{\ar_{g} "a"; (0,-20)*{Q}="q"};
{\ar (-20,0)*{0}; "a"};
{\ar (-20,-20)*{0}; "q"};
{\ar_i "q"; (30,-20)*{Q\oplus Q'=\wt{Q}}="qp"};
{\ar^{\ol{g}} "b"; "qp"};
%{\ar@{-->}^{\exists\wt{g}} "b"; "q"};
\endxy
\]

Define $\wt{g}=p \circ \ol{g}$. Here $p:Q\oplus Q' = Q\times Q' \to Q$
is the natural surjection. Thus,
\[
g = p\circ i\circ g = p\circ \ol{g}\circ f = \wt{g}\circ f,
\]
as claimed.\\

\noindent (2) $\Rightarrow$ (3). Now we assume (2), and prove that
$\Hom(-,Q)$ is exact. Since $\Hom$ functors are always left exact, we only
need to show that it is right exact. That is, given a short exact
sequence:

\[
0\to A \morph{f} B \to C\to 0\ ,
\]

the natural map $\Hom(B,Q)\to \Hom(A,Q)$ is surjective. That is,
any morphism $g:A\to Q$ lifts to a morphism $\wt{g}:B\to Q$. This is
exact what condition (2) says, and we are done. Note that this
argument in fact shows that (2) and (3) are equivalent.\\

\noindent (3) $\Rightarrow$ (1). Assume that $Q\in\CalA$ is such
that $\Hom(-,Q)$ is exact. Let $f:Q\to X$ be an injective morphism.
We need to prove that there exists $Q'$ and an isomorphism
$h:Q\oplus Q'\isom X$ such that $h\circ i = f$, where $i:Q\to Q\oplus Q'$
is the natural inclusion.

We use (3) to conclude that $-\circ f: \Hom(X,Q)\to \Hom(Q,Q)$ is a surjective
map. Let $g:X\to Q$ be a morphism so that $g\circ f = \Id_Q$. Let $p:X\to X$
be the composition $p=f\circ g$. Then $p\circ p = f\circ g\circ f\circ g
= f\circ g = p$. Using Lemma 16.2 we conclude that $X\cong \Ker(p)\oplus\Im(p)$.
Note that $\Im(p) = \Im(f\circ g) \cong Q$, since $g$ is surjective and $f$
is injective. Hence $Q$ is injective.\\

\PP{16.6}{Remarks}
%--------------

\noindent {\bf I.} The following is a defining property whose proof follows
along the lines of Theorem 16.4 above. It is left as an exercise.
\begin{prop}
$M\in\CalA$ is injective (resp. projective) if, and only if every short
exact sequence $\ds 0\to M\to Y\to Z\to 0$ (resp.
$\ds 0\to X\to Y\to M\to 0$) splits{\footnote{See Homework 5, Problem 8 for
what it means.}}.
\end{prop}

\noindent {\bf II.} We have not yet proved whether injective/projective
objects exist. The standard examples of these, which can be easily verified
using Theorem 16.4 (we will go over these in more detail next week), 
are following. (i) $\Q\in\Rmod{\Z}=\Ab$ is an injective $\Z$--module. 
(ii) For any indexing set
$I$, let $R^{(I)}$ be direct sum of $I$--many copies of $R$ (viewed
as a left module over itself). Then $R^{(I)}\in\Rmod{R}$ is projective
(it is in fact {\em free}). (iii) Over a field, every module is both
injective and projective.\\

\end{document}
