\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 5}
\author{}

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\newcommand{\problem}[1]{\noindent {\bf Problem #1.--}}

\begin{document}
\maketitle

\begin{center}
In the problems below, $\CalA=\Rmod{R}$ is the abelian category
of left $R$--modules, over a unital ring $R$. $\CoComp{\CalA}$
is the category of complexes of $R$--modules.\\
\end{center}

\hrule

\vspace*{0.2in}

\problem{1} Consider the following commutative diagram with exact rows.

\[
\xy
{\ar (0,20)*{M_1}="m1"; (30,20)*{M_2}="m2"};
{\ar^a "m2"; (60,20)*{M_3}="m3"};
{\ar "m3"; (90,20)*{M_4}="m4"};
{\ar (0,0)*{N_1}="n1"; (30,0)*{N_2}="n2"};
{\ar^b "n2"; (60,0)*{N_3}="n3"};
{\ar "n3"; (90,0)*{N_4}="n4"};
{\ar_u "m1"; "n1"};
{\ar_v "m2"; "n2"};
{\ar_w "m3"; "n3"};
{\ar_s "m4"; "n4"};
\endxy
\]

Assume that $u$ is surjective and $s$ is injective. Prove that
$\Ker(w) = a(\Ker(v))$ and $\Im(v) = b^{-1}(\Im(w))$.\\

\problem{2} Assume we have two morphisms of $R$--modules
$\ds M\morph{u} N\morph{v} P$. Prove that we get the following
{\em exact sequence}:
\[
0\to \Ker(u)\to\Ker(v\circ u)\to \Ker(v)\to \Coker(u)
\to \Coker(v\circ u) \to \Coker(v) \to 0.
\]
\ \\

\problem{3} (Pull--backs). Given two morphisms of $R$--modules
\[
\xy {\ar^f (0,0)*{A}="a"; (20,0)*{S}="s"};
{\ar^g (20,20)*{B}="b"; "s"};
\endxy
\] 
$A\times_S B$ is defined as $\{(a,b):f(a)=g(b)\}\subset A\times B$,
with canonical maps $p_1:A\times_S B \to A$ and $p_2:A\times_S B\to B$,
given by $p_1(a,b)=a$ and $p_2(a,b)=b$
(see Problem 3 of Mid Term 1, for a more categorical definition).

Let $M_1\to M_2\to M_3\to 0$ be an exact sequence of $R$--modules, and assume 
that we
have a morphism $X\to M_3$. Define $\ds \wt{X_2} = M_2\times_{M_3} X$
and $\wt{X_1} = M_1\times_{M_2} \wt{X_2}$. Prove that the first
row of the following diagram is exact:
\[
\xy
{\ar (0,20)*{\wt{X_1}}="x1"; (20,20)*{\wt{X_2}}="x2"};
{\ar "x2"; (40,20)*{X}="x3"}; {\ar "x3"; (50,20)*{0}};
{\ar (0,0)*{M_1}="m1"; (20,0)*{M_2}="m2"};
{\ar "m2"; (40,0)*{M_3}="m3"}; {\ar "m3"; (50,0)*{0}};
{\ar "x1"; "m1"};
{\ar "x2"; "m2"};
{\ar "x3"; "m3"};
\endxy
\]
\ \\

\problem{4} State and prove the result analogous to that of Problem 3, for
``push--forwards".\\

\problem{5} Consider the following diagram in $\CalA$ with exact rows and commuting squares.

\[
\xy
(0,20)*{M}="m"; (20,20)*{N}="n"; (40,20)*{P}="p";
(0,0)*{M'}="mp"; (20,0)*{N'}="np"; (40,0)*{P'}="pp";
{\ar^{u} "m"; "n"};
{\ar^{v} "n"; "p"};
{\ar_{u'} "mp"; "np"};
{\ar_{v'} "np"; "pp"};
{\ar_{f} "m"; "mp"};
{\ar_{g} "n"; "np"};
{\ar_{h} "p"; "pp"};
\endxy
\]

\begin{enumerate}
\item Prove that, if $u',f,h$ are injective, then $g$ is injective.\\

\item Prove that, if $v,f,h$ are surjective, then $g$ is surjective.\\

\item Prove that, if $g$ is injective and $f,v$ are surjective, then
$h$ is injective.\\

\item Prove that, if $g$ is surjective and $h,u'$ are injective,
then $f$ is surjective.\\
\end{enumerate}

\problem{6} Consider the subcategory of $\CoComp{\CalA}$ consisting of
exact complexes (i.e, $C^{\bullet}$ such that $H^n(C^{\bullet})=0$
for every $n\in \Z$). Give an example to illustrate that this subcategory
is not closed under taking kernels.\\

\problem{7} {\em Five lemma.} Consider the following commutative diagram
with exact rows.\\

\[
\xy
(0,20)*{M_1}="m1";
(20,20)*{M_2}="m2";
(40,20)*{M_3}="m3";
(60,20)*{M_4}="m4";
(80,20)*{M_5}="m5";
{\ar^{a_1} "m1"; "m2"};
{\ar^{a_2} "m2"; "m3"};
{\ar^{a_3} "m3"; "m4"};
{\ar^{a_4} "m4"; "m5"};
(0,0)*{N_1}="n1";
(20,0)*{N_2}="n2";
(40,0)*{N_3}="n3";
(60,0)*{N_4}="n4";
(80,0)*{N_5}="n5";
{\ar_{b_1} "n1"; "n2"};
{\ar_{b_2} "n2"; "n3"};
{\ar_{b_3} "n3"; "n4"};
{\ar_{b_4} "n4"; "n5"};
{\ar_{f_1} "m1"; "n1"};
{\ar_{f_2} "m2"; "n2"};
{\ar_{f_3} "m3"; "n3"};
{\ar_{f_4} "m4"; "n4"};
{\ar_{f_5} "m5"; "n5"};
\endxy
\]

\begin{enumerate}
\item Prove that if $f_2, f_4$ are injective and $f_1$ is surjective, then
$f_3$ is injective.\\

\item Prove that if $f_2, f_4$ are surjective and $f_5$ is injective, then
$f_3$ is surjective.\\
\end{enumerate}
{\em Hence, if $f_1,f_2,f_4,f_5$ are isomorphisms, then so is $f_3$.}\\

\problem{8} Let $\CalC$ be an arbitrary abelian categories. A short
exact sequence in $\CalC$, $\ds 0\to X\to Y\to Z\to 0$ is said to 
be {\em split} if there exists an isomorphism $Y\isom X\oplus Z$ making
the following diagram commute:
\[
\xy
{\ar (0,20)*{0}; (20,20)*{X}="x"};
{\ar "x"; (40,20)*{Y}="y"};
{\ar "y"; (60,20)*{Z}="z"};
{\ar "z"; (80,20)*{0}};
{\ar (0,0)*{0}; (20,0)*{X}="xp"};
{\ar "xp"; (40,0)*{X\oplus Z}="yp"};
{\ar "yp"; (60,0)*{Z}="zp"};
{\ar "zp"; (80,0)*{0}};
{\ar@{=} "x"; "xp"};
{\ar@{=} "z"; "zp"};
{\ar^{\cong} "y"; "yp"};
\endxy
\]
Prove that a sequence $\ds 0\to X\morph{i} Y\morph{\pi} Z\to 0$
is split if, and only if there exists $s:Z\to Y$ such that $\pi\circ s=\Id_Z$.\\

\problem{9} Let $C^{\bullet}$ and $D^{\bullet}$ be two cochain complexes
of $R$--modules. Let $\alpha^{\bullet} : C^{\bullet}\to D^{\bullet}$
be a morphism. Define:

\begin{itemize}
\item $C[1]^{\bullet} \in \CoComp{\CalA}$ is a complex with $C[1]^n = C^{n+1}$
and $d_{C[1]}^n = -d^{n+1}_C$, for every $n\in \Z$.\\

\item $\operatorname{Cone}(\alpha)$ is the following complex:\\

\begin{itemize}
\item For every $n\in \Z$, $\operatorname{Cone}(\alpha)^n = C^{n+1}\oplus D^n$.\\
\item For every $n\in \Z$, the differential, denoted by $\mathsf{d}^n$ below, is given by the following
expression, for every $x\in C^{n+1}$ and $y\in D^n$,
\[
\mathsf{d}^n(x,y) = (-d^{n+1}_C(x), d^n_D(y)-\alpha^{n+1}(x))
\]
\end{itemize}
\end{itemize}

\begin{enumerate}
\item Verify that $\operatorname{Cone}(\alpha)$ is a complex.\\

\item Prove that we have the following short exact sequence of cochain complexes.

\[
0\to D^{\bullet} \to \operatorname{Cone}(\alpha)
\to C^{\bullet}[1] \to 0
\]

\item Prove that $\alpha^{\bullet}$ is null homotopic if, and only if,
the sequence above is split (see the previous problem for its definition).\\

\item Prove that $H^n(\alpha)$ is an isomorphism, for every $n\in \Z$ if,
and only if $H^n(\operatorname{Cone}(\alpha))=0$ for every $n\in \Z$.\\

\end{enumerate}

\problem{10} {\em Koszul complex}. Let $K$ be a field, $A=K[x,y,z]$ be
the (commutative) ring of polynomials in three variables, with coefficients
from $K$. Let $A^{\oplus 3}$ be the $A$--module which is direct sum of
three copies of $A$, with $A$--basis denoted by $e_1,e_2,e_3$. 
Consider the following sequence of morphisms:
\[
0\to A \morph{d_2}A^{\oplus 3} \morph{d_1}  A^{\oplus 3} \morph{d_0} A\to 0
\]

\begin{itemize}
\item $d_0(e_1)=x$, $d_0(e_2)=y$, and $d_0(e_3)=z$.\\

\item $d_1(e_1) = ze_2 - ye_3$, $d_1(e_2) = xe_3-ze_1$, and
$d_1(e_3) = ye_1 - xe_2$.\\

\item $d_2(1) = xe_1+ye_2+ze_3$.\\
\end{itemize}

Verify that this is a complex of $A$--modules. Prove that it is
exact at each spot, except $\Coker(d_0) = K \cong K[x,y,z]/(x,y,z)$.\\

\noindent {\bf Bonus.} Generalize this to the polynomial ring of
$n$--variables.
\end{document}
