
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 2}


\usepackage{tikz-cd}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 5 February. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

\begin{enumerate}


\item  Let $R$ be a ring with $1$, and consider a sequence of
  $R$-modules $(M_i)_{i \in I}$ for some interval $I$ in $\mathbb Z$, together
  with morphisms $\phi_i: M_i \rightarrow M_{i+1}$ defined when $i, i+1 \in I$. 
  Such a sequence is said to be {\em exact at $i$} if $i-1, i, i+1 \in I$ and
  the image of $\phi_{i-1}$ is equal to the kernel of $\phi_i$, and to be {\em exact}
  if it is exact at each relevant $i$.

  Note: usually we describe exact sequences informally, by writing things like
  
\begin{tikzcd}
  A \rar{\alpha} & B \rar{\beta} & C \\
\end{tikzcd}

Also we typically don't name the arrows when it is clear what they must be.
In particular the zero module $0$ is both initial and terminal, so
we just write $0 \rightarrow M$ or $N \rightarrow 0$.

  Prove that:

  \begin{enumerate}

  \item The sequence
    
\begin{tikzcd}
  0 \rar & B \rar{\alpha} & C \\
\end{tikzcd}

 is exact iff $\alpha$ is injective. 


  \item The sequence   

    \begin{tikzcd}
  A \rar{\alpha} & B \rar & 0 \\
    \end{tikzcd}
    
 is exact iff $\alpha$ is surjective. 

 
\item The sequence
  
\begin{tikzcd}
 0 \rar &  A \rar{\alpha} & B \rar & 0 \\
\end{tikzcd}

 is exact iff $\alpha$ is an isomorphism. 

\item   If the sequence
  
\begin{tikzcd}
 0 \rar &  A \rar{\alpha} & B \rar{\beta} & C \rar & 0 \\
\end{tikzcd}

is exact, then $C \simeq B/\alpha[A]$.

\end{enumerate} 

  
\item Let $H$ be a non-trivial torsion-free abelian group, and assume that $H$ has the following
  property: for all nonzero elements $a, b \in H$ there exist nonzero integers $m, n$ such that
  $m a = n b$ (groups with this property are sometimes said to have ``rank one'').

  For any nonzero $a \in H$ and any prime number $p$, define $n_p(a)$ as follows:
  $n_p(a)$ is the largest integer $k \ge 0$ such that there exists $b$ with $p^k b = a$
  ($b$ will necessarily be unique as $H$ is torsion-free), or $n_p(a) = \infty$ if
  such $b$ exists for all $k$. 

  Let $a, b \in H$ be nonzero. Prove that $n_p(a) = \infty$ if and only if $n_p(b) = \infty$,
    and also that $\{ p : n_p(a) \neq n_p(b) \}$ is finite.

    Hint: facts about gcd's may be helpful.

\item  Recall that a subset $A \subseteq {\mathbb R}$ is {\em open} if for all $a \in A$ there is
    $\epsilon > 0$ such that $(a - \epsilon, a + \epsilon) \subseteq A$, and {\em closed} if its
    complement is open. It is a standard fact that a  subset $C$ of $\mathbb R$ is closed
    if and only if every convergent sequence $(x_n)$ with $x_n \in C$ converges to a point of $C$.
    It is also standard that if $B$ is a subset of $\mathbb R$ and $C$ is the set of limits of convergent sequences
    of elements of $B$, then $C$ is closed and is the least closed set containing $B$: in this case
    we write $C = \bar B$ and call $C$ the {\em closure} of $B$. 
    
    
    \begin{enumerate}

    \item  Give an example of a non-closed subgroup of $(\mathbb R, +)$. 

    \item   Prove that the closure of a subgroup of $(\mathbb R, +)$ is also a subgroup.
      
    \item  Prove that a non-trivial closed subgroup of $({\mathbb R}, +)$ must either be an infinite cyclic
     group or be $\mathbb R$ itself.

    \item (Challenging, not for credit) Describe with proof the closed subgroups of $({\mathbb R}^2, +)$
  
\end{enumerate}  

  \item Let $R$ be a commutative ring with $1$, let $M$ be an $R$-module, and recall from class the definition of the 
  functor $Hom(M, -)$. 

  Give a proof or a counterexample for each of the following statements:

\begin{enumerate}


\item   If $\alpha: N_1 \rightarrow N_2$ is injective, then
     $Hom(M, \alpha): Hom(M, N_1) \rightarrow Hom(M, N_2)$ is injective.


\item   If $\alpha: N_1 \rightarrow N_2$ is surjective, then
     $Hom(M, \alpha): Hom(M, N_1) \rightarrow Hom(M, N_2)$ is surjective.


\item   If $\alpha: N_1 \rightarrow N_2$ is an isomorphism, then
     $Hom(M, \alpha): Hom(M, N_1) \rightarrow Hom(M, N_2)$ is an isomorphism.

\end{enumerate}


\item  A function from $\mathbb R$ to the set of real $N \times N$ matrices is differentiable if
     each of the $N^2$ functions corresponding to the entries is differentiable, and the 
     derivative is the matrix made up of the derivatives of the entries. Let $A$ be a real $N \times N$ matrix.
     Prove that the matrix exponential function $\exp(A t)$ is a differentiable function of $t$,
     and find its derivative.

     Hint 1: since $At Ah = Ah At$ you can simplify the expression $\frac{\exp(At + Ah) - \exp(At)}{h}$
     in a helpful way.

     Hint 2: You may find it useful to bound the entries in $\sum_{i=2}^\infty \frac{(A h)^i}{i!}$.

  

    

   \end{enumerate}  



\end{document} 
