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\begin{center}
  {\large Mathematics 111}\\Spring 2007\\Homework 3
\end{center}

\begin{enumerate}
\item (Pullbacks) Given a ring $A$ with identity and
  $A$-modules $M, M', M''$, consider the following diagram with
  $A$-linear maps $f, g$:
\[
\xymatrix{& M'' \ar[d]_g \\
 M' \ar[r]^f & M }
\]
A \textit{pullback} for this diagram (also called a fiber product of
$f$ and $g$) consists of the following data:

\begin{enumerate}
\item An object
$X$ and $A$-linear maps $p: X \to M'$, $q: X \to M''$ making the
following diagram commute.
\[
\xymatrix{
X \ar[d]_p \ar[r]^q & M'' \ar[d]_g \\
 M' \ar[r]^f & M }
\]
\item For every commutative diagram of linear maps (same $f, g$)
\[
\xymatrix{
X' \ar[d]_{p'} \ar[r]^{q'} & M'' \ar[d]_g \\
 M' \ar[r]^f & M }
\]
there is a unique $A$-linear map $h : X' \to X$ such that the
following diagram commutes:
\[
\xymatrix{ X' \ar@/_/ @{->} [ddr]_{p'} \ar@{->}[dr]^h
\ar@/^/[drr]^{q'} \\
&X \ar[d]^p \ar[r]_q & M'' \ar[d]_g \\
& M' \ar[r]^f & M }
\]

Let $X = \{ (m', m'') \in M' \times M'' \mid f(m') = g(m'')\}$, $p$
and $q$ the standard projections to the factors $M'$ and $M''$.  Show
that $X$ together with the associated data form a pullback, i.e.,
verify that $X$ is an $A$-module and that the universal mapping
property (b) holds for this choice of $X$ and maps $p,q$.
\end{enumerate}

\item Let $A$ be a ring, and consider two exact sequences of
  $A$-modules
\[\xymatrix{%
  0 \ar@{>}[r]& K \ar@{>}[r] &P \ar@{>}[r]^\varphi&M\ar@{>}[r] &0&
  0 \ar@{>}[r]& K' \ar@{>}[r] &P' \ar@{>}[r]^{\varphi'}&M\ar@{>}[r] &0
}
\]
where $P$ and $P'$ are projective.  Show that as $A$-modules $P\oplus
K' \cong P' \oplus K$.  \textit{Hint:} Show there is an exact sequence 
\[\xymatrix{%
  0 \ar@{>}[r]& \textrm{ker}\; \pi \ar@{>}[r] &X \ar@{>}[r]^\pi&P\ar@{>}[r] &0
}
\]
with $\textrm{ker}\; \pi \cong K'$ and where $X$ is the fiber product
of $\varphi$ and $\varphi'$ as in the first problem.  From this deduce
that $X \cong P \oplus K'$.  Similarly, show $X \cong P' \oplus K$.
\end{enumerate}






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