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\begin{document}

\begin{center}
{\Large Advanced Algebra I}\\
 Homework 15\\
 due on Jan. 19, 2007
\end{center}


\begin{enumerate}
\item Complete the proof of Theorem 4.5.6 and 4.5.7

\item Let $F: \mathcal{A} \to \mathcal{B}$ be a left exact functor and $\mathcal{A}$
has enough injectives. Show that $R^iF(A):=H^i(F(I^\bullet))$ is
well-defined. That is, independent of choice of injective resolution
$I^\bullet$.

\item Given a complex $K^\bullet$, construct an injective resolution
$I^\bullet$ of $K^\bullet$. That is, a quasi-isomorphism $f:
K^\bullet \to I^\bullet$.

\item In the category of abelian groups, show that an injective
object is a divisible group. 

\item We can define projective in a similar way (with arrow
reversing ). That is for any exact sequence $B
\stackrel{\alpha}{\to} C \to 0$ and $f: P \to C$, there exist $g: P
\to B$ such that $g \alpha = f$.

Show that for any exact sequence $ 0 \to A \to B \to P \to 0$ with
$P$ being projective, the sequence splits.

We now consider the category of abelian groups $Ab$. Determine the
projective objects. (Hint: every abelian group is a quotient of free
abelian group.)


\end{enumerate}

\end{document}
\end