1/x_0}=[(M_{x_1})_0]_{x_0/x_1}$.

\begin{prop}[Serre]\label{pr:Serre}
The functor \eqref{eq:serre} induces an equivalence
\begin{equation*}
\frac{\mod^\mathbb Z k[x_0,x_1]}{\mod_0^\mathbb Z
k[x_0,x_1]}\lto[\sim]\coh\mathbb P^1_k.
\end{equation*}
\end{prop}
\begin{proof}
We refer to \cite{S} for the proof. It is clear from the definition
that the functor \eqref{eq:serre} is exact having kernel ${\mod_0^\mathbb Z
  k[x_0,x_1]}$.  This fact yields the induced functor which is
faithful by construction.
\end{proof}

For any $n\in\mathbb Z$ and ${\mathcal F}=(M',M'',\mu)$ in $\coh\mathbb P^1_k$, denote by
${\mathcal F}(n)$ the \index{sheaf!twisted} \emph{twisted sheaf}
$(M',M'',\mu^{(n)})$, where $\mu^{(n)}$ is the map $\mu$ followed by
multiplication with $y^{-n}$.  Given a module $M$ in $\mod^\mathbb Z
k[x_0,x_1]$, the \index{module!twisted} \emph{twisted module} $M(n)$
is obtained by shifting the grading, that is, $M(n)_i=M_{i+n}$ for
$i\in\mathbb Z$. Note that the functor \eqref{eq:serre} is compatible with
the twist functors defined on $\mod^\mathbb Z k[x_0,x_1]$ and
$\coh\mathbb P^1_k$.

\begin{prop}[Serre]
The category $\coh\mathbb P^1_k$ is a Hom-finite $k$-linear abelian
category satisfying Serre duality. More precisely, there is a
functorial $k$-linear isomorphism
$$D\Ext^1({\mathcal F},{\mathcal G})\cong\Hom({\mathcal G},{\mathcal F}(-2))\quad\text{for all}\quad {\mathcal F},{\mathcal G}\in\coh\mathbb P^1_k.$$
\end{prop}
\begin{proof}
See \cite[III.7]{H}.
\end{proof}

\subsection{Locally free and torsion sheaves}

A sheaf $(M',M'',\mu)$ in $\coh\mathbb P^1_k$ is called
\index{sheaf!locally free} \emph{locally free} or \index{vector
bundle} \emph{vector bundle} if $M'$ and $M''$ are free modules over
$k[y]$ and $k[y^{-1}]$ respectively. We denote the full subcategory of
vector bundles in $\coh \mathbb P^1_k$ by $\vect \mathbb P^1_k$.


The \index{structure sheaf} \emph{structure sheaf} is the sheaf ${\mathcal O}=(k[y],k[y^{-1}],
\id_{k[y,y^{-1}]})$. For any pair $m,n\in\mathbb Z$, we have a natural
bijection
\begin{equation}\label{eq:bij_pol}
k[x_0,x_1]_{n-m}\lto[\sim]\Hom({\mathcal O}(m),{\mathcal O}(n)).
\end{equation}
The map sends a homogeneous polynomial $P$ of degree $n-m$ to the
morphism $(\phi',\phi'')$, where $\phi'\colon k[y]\to k[y]$ is
multiplication by $P(1,y)$ and $\phi''\colon k[y^{-1}]\to k[y^{-1}]$ is
multiplication by $P(y^{-1},1)$. This bijection is a special case of
the next result.

Let $R=k[x_0,x_1]$ and denote by $\proj^\mathbb Z R$ the category of
finitely generated projective $\mathbb Z$-graded $R$-modules. Note that $R$
is a graded local ring since the homogeneous elements of positive
degree form the unique maximal homogeneous ideal. Thus finitely
generated projective $R$-modules are up to isomorphism of the form
$R(n_1)\oplus\dots\oplus R(n_r)$.


\begin{prop}[Grothendieck]\label{pr:Grothendieck}
The functor \eqref{eq:serre} induces an equivalence
$$\proj^\mathbb Z k[x_0,x_1]\lto[\sim]\vect \mathbb P^1_k.$$ In particular,
each locally free coherent sheaf on $\mathbb P^1_k$ is isomorphic to a sheaf of the form
${\mathcal O}(n_1)\oplus \dots\oplus{\mathcal O}(n_r)$.
\end{prop}
\begin{proof}
We need to show that the functor \eqref{eq:serre} is fully faithful
when it is restricted to $\proj^\mathbb Z R$, where $R= k[x_0,x_1]$. Every
finitely generated projective $R$-module is up to isomorphism of the
form $R(n_1)\oplus\dots\oplus R(n_r)$. Thus it suffices to show that
\eqref{eq:serre} induces a bijection
$\Hom_R(R(m),R(n))\to\Hom({\mathcal O}(m),{\mathcal O}(n))$ for each pair $m,n\in\mathbb Z$.
But this is clear since the map coincides with the bijection \eqref{eq:bij_pol}.

It remains to show that the functor is dense, that is, each locally
free coherent sheaf is isomorphic to one of the form ${\mathcal O}(n_1)\oplus
\dots\oplus{\mathcal O}(n_r)$. For this we refer to \cite{Gr}. An elementary
proof is based on an argument due to Birkhoff \cite{Bi,EET}. A vector bundle
${\mathcal F}=(M',M'',\mu)$ is basically determined by an invertible matrix over
$k[y,y^{-1}]$, which represents the isomorphism $\mu$. Now one uses
the fact that such a matrix can be transformed into a diagonal matrix
with entries $(y^{-n_1},\dots,y^{-n_r})$ by multiplying it with an
invertible matrix over $k[y]$ from the right and an invertible matrix
over $k[y^{-1}]$ from the left. This yields an isomorphism
${\mathcal F}\xrightarrow{\sim}{\mathcal O}(n_1)\oplus \dots\oplus{\mathcal O}(n_r)$.
\end{proof}

A sheaf $(M',M'',\mu)$ in $\coh\mathbb P^1_k$ is called
\index{sheaf!torsion} \emph{torsion} if $M'$ and $M''$ are torsion
modules over $k[y]$ and $k[y^{-1}]$ respectively. We denote the full
subcategory of torsion sheaves in $\coh \mathbb P^1_k$ by $\coh_0
\mathbb P^1_k$. Note that each sheaf ${\mathcal F}=(M',M'',\mu)$ admits a unique
maximal subobject $\tor{\mathcal F}$ in $\coh\mathbb P^1_k$ that is torsion. One
obtains $\tor{\mathcal F}$ by taking the torsion parts of $M'$ and $M''$
respectively. Clearly, ${\mathcal F}/\tor{\mathcal F}$ is locally free.

\begin{prop}
Each coherent sheaf ${\mathcal F}$ on $\mathbb P^1_k$ admits an essentially unique
decomposition ${\mathcal F}={\mathcal F}'\oplus{\mathcal F}''$ such that ${\mathcal F}'$ is torsion and ${\mathcal F}''$
is locally free. The torsion sheaves are precisely the objects of
finite length in $\coh\mathbb P^1_k$.
\end{prop}

\begin{proof}
We use elementary facts about finitely generated modules over $k[y]$
and $k[y^{-1}]$ respectively.  Take ${\mathcal F}'=\tor{\mathcal F}$ and
${\mathcal F}''={\mathcal F}/\tor{\mathcal F}$. Serre duality implies
$\Ext^1({\mathcal F}'',{\mathcal F}')=0$, since there are no non-zero morphisms
from torsion to locally free sheaves. Thus the canonical exact
sequence $0\to{\mathcal F}'\to{\mathcal F}\to{\mathcal F}''\to 0$ splits.

Let ${\mathcal F}=(M',M'',\mu)$ be a coherent sheaf. If ${\mathcal F}$ is torsion, then it
is a finite length object in $\coh\mathbb P^1_k$, since the corresponding
torsion modules $M'$ and $M''$ have finite length. On the other hand,
the structure sheaf has infinite