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\centerline{\Kopfont Commutative Algebra}

\smallskip
\centerline{Excercises 2}

\bigskip\noindent
{\bf 1.} Let $(I,\geq)$ be a partially ordered set which is directed.
Let $A$ be a ring and let $(N_i,\varphi_{i,i'})$ be a directed system of
$A$-modules indexed by $I$. Suppose that $M$ is another $A$-module. Prove
that 
$$\lim_{i\in I} M\otimes_A N_i\cong M\otimes_A \Big( \lim_{i\in I}N_i\Big).$$

\medskip\noindent
{\bf Remark.} A module $M$ over $R$ is said to be of finite presentation over
$R$ if it  is isomorphic to the cokernel of a map of finite free modules
$ R^{\oplus n} \to R^{\oplus m}$.

\medskip\noindent
{\bf 2.} Prove that any module over any ring is 
\item{a.} the limit of its finitely generated
submodules, and
\item{b.} in some way a limit of finitely presented modules.

\medskip\noindent
{\bf 3.} Let $S$ be a multiplicative subset of the ring $A$.
\item{a.} For an $A$-module $M$ show that $S^{-1}M = S^{-1}A\otimes_A M$.
\item{b.} Show that $S^{-1}A$ is flat over $A$.

\medskip\noindent
{\bf 4.} Find an injection $M_1 \to M_2$ of $A$-modules such that
$M_1\otimes N \to M_2 \otimes N$ is not injective in the following
cases:
\item{a.} $A=k[x,y]$ and $N=(x,y) \subset A$. (Here and below $k$ is a field.)
\item{b.} $A=k[x,y]$ and $N=A/(x,y)$.

\medskip\noindent
{\bf 5.} Give an example of a ring $A$ and a finite $A$-module $M$
which is a flat but not a projective $A$-module.

\medskip\noindent
{\bf Remark.} If $M$ is of finite presentation and flat over $A$,
then $M$ is projective over $A$. Thus your example will have to
involve a ring $A$ which is not Noetherian. I know of an example
where $A$ is the ring of ${\cal C}^\infty$-functions on $\mR$.

\medskip\noindent
{\bf 6.} Let $A=k[x,y]_{(x,y)}$ be the local ring of the affine plane at
the origin. Make any assumption you like about the field $k$. Suppose
that $f = x^3 + x^2y^2 + y^{100}$ and $g = y^3 - x^{999}$. What is the length
of $A/(f,g)$ as an $A$-module? (Possible way to proceed: think about the
ideal that $f$ and $g$ generate in quotients of the form $A/\gm_A^n=
k[x,y]/(x,y)^n$ for varying $n$. Try to find $n$ such that 
$A/(f,g)+\gm_A^n \cong A/(f,g)+\gm_A^{n+1}$ and use NAK.)

\end
