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\begin{document}

\begin{center}
{\bf Math 601 -- Spring 2026 -- Harry Tamvakis}
\end{center}


\begin{center}
{\bf PROBLEM SET 5 -- Due March 12, 2026}
\end{center}


\skipline \noin {\bf A1)} Prove that if an abelian group $A$ is
injective as a $\Z$-module, then $A$ is divisible. [Hint: There is a
divisible group $D$ with a subgroup isomorphic to $A$.]


\skipline \noin {\bf A2)} Let $R$ be a commutative ring. Prove that a
direct product $\dis \prod_{i\in I} Q_i$ of any family of $R$-modules
$Q_i$ is injective if and only if each $Q_i$ is injective.


\skipline
\noin
{\bf A3)} Suppose that $R$ is a commutative ring and $M$ and $N$ are
flat $R$-modules. Prove that $M\otimes_R N$ is a flat $R$-module.



\skipline \noin {\bf A4)}  Problem 2 in Chapter 10.5 of Dummit and Foote 
(page 403).


\skipline \noin {\bf A5)}
Consider a commutative diagram
\[
\xymatrix{
0 \ar[r] & \A_1 \ar[r] \ar[d] &  \B_1 \ar[r] \ar[d] 
& \cC_1 \ar[r] \ar[d] & 0 \\
0 \ar[r] & \A_2 \ar[r]  &  \B_2 \ar[r] & \cC_2 \ar[r] & 0 }
\]
where each row is an exact sequence of complexes, and each square 
commutes. Prove the {\em naturality of the connecting homomorphisms}, 
i.e., show that for each $n \geq 0$ the following diagram commutes.
\[
\xymatrix{
\HH^n(\cC_1) \ar[r]^{\delta_1} \ar[d] &  \HH^{n+1}(\A_1)  \ar[d] \\
\HH^n(\cC_2) \ar[r]^{\delta_2} &  \HH^{n+1}(\A_2)  }
\]



\skipline \noin {\bf B1)} Let $R$ be a commutative ring. 
Suppose that $T:F\to F$ and $T':F'\to F'$ 
are endomorphisms of the free $R$-modules $F$ and $F'$ of ranks 
$m$ and $n$, respectively. Then 
\[
T\otimes T' : F \otimes F' \to F\otimes F'
\]
is also an endomorphism of a free $R$-module.

\medskip
\noin
(a) Show that the trace of $T\otimes T'$ is the product of the traces of 
$T$ and $T'$.

\medskip
\noin
(b) Express the determinant of $T\otimes T'$ in terms of the determinants 
of $T$ and $T'$. 

\medskip
\noin
(c) Do the characteristic polynomials of $T$ and $T'$ determine the 
characteristic polynomial of $T\otimes T'$?

\medskip
\noin
[NOTE: You may assume that $R=\C$ for this problem. We will discuss later in
class a technique which allows one to deduce the more general result above
from this special case.]


\skipline \noin {\bf B2)} Let $R$ be a commutative ring. 

\medskip
\noin
(a) Suppose $\{M_i\}_{i\in I}$ is a family of $R$-modules and
$f:R\to \bigoplus_{i\in I}M_i$ is a homomorphism of $R$-modules. Prove
that there is a finite subset $H$ of $I$ such that $f(R)$ is contained
in $\bigoplus_{i\in H}M_i$.
    
\medskip
\noin
(b) Suppose that $J_1\subset J_2\subset J_3\subset \cdots$ is an
ascending chain of ideals in $R$ and let $J:=\bigcup_{i=1}^\infty
J_i$.  For each $i\geq 1$, embed $J/J_i$ into an injective $R$-module
$M_i$. Assume that $\bigoplus_{i=1}^\infty M_i$ is injective. Prove
that there is an integer $n\geq 1$ such that the image of $J$ under
the natural map
\[
    J \to \bigoplus_{i=1}^\infty J/J_i \to \bigoplus_{i=1}^\infty M_i
\]
is contained in $\bigoplus_{i=1}^n M_i$.
  
\medskip
\noin
(c) Prove that if $R$ has the property that every countable direct sum of
injective $R$-modules is injective, then $R$ is Noetherian.




\skipline \noin {\bf B3)} Problems 25 and 26 in Chapter 10.5 of Dummit
and Foote (page 406).






\end{document}
