Vektorbündel/A^1/Punktierte affine Fläche/Beispiel/en

Let ${}(R,{\mathfrak {m}})$ denote a two-dimensional local noetherian domain and let ${}f$ and ${}g$ be two parameters in ${}R$, i.e. elements which generate the maximal ideal ${}{\mathfrak {m}}$ up to radical. Then the punctured spectrum is

${}U=D({\mathfrak {m}})=D(f,g)=D(f)\cup D(g)\,$

and every cohomology class ${}c\in H^{1}(U,{\mathcal {O}}_{X})$ can be represented by a Čech cohomology class

$c={\frac {h}{f^{i}g^{j}}}$

with some ${}h\in R$ ($i,j\geq 1$). If ${}R$ is normal then the cohomology class is ${}0$ if and only if ${}h\in (f^{i},g^{j})$. To see this, we work with ${}i=j=1$, which does not make a difference as powers of powers are again parameters. We note that under the normality assumption the syzygy module ${}\operatorname {Syz} {\left(f,g\right)}$ is free of rank one ($(g,-f)$ is a generator). Then we look at the short exact sequence on ${}U$,

$0\longrightarrow {\mathcal {O}}_{U}\cong \operatorname {Syz} {\left(f,g\right)}\longrightarrow {\mathcal {O}}_{U}^{2}{\stackrel {f,g}{\longrightarrow }}{\mathcal {O}}_{U}\longrightarrow 0$

and its corresponding long exact sequence of cohomology,

$0\longrightarrow R\longrightarrow R^{2}{\stackrel {f,g}{\longrightarrow }}R{\stackrel {\delta }{\longrightarrow }}H^{1}(U,{\mathcal {O}})\longrightarrow \ldots .$

Here, the connecting homomorphisms ${}\delta$ for an element ${}h\in R$ works in the following way. On both open subsets ${}D(f)$ and ${}D(g)$ we get the local representatives ${}({\frac {h}{f}},0)$ and ${}(0,{\frac {h}{g}})$ and their difference, considered in ${}\Gamma (D(fg),{\mathcal {O}}_{U})$, defines the cohomology class. This difference is just ${}{\frac {h}{fg}}$. By the exactness of the long cohomology sequence, ${}c=\delta (h)={\frac {h}{fg}}=0$ if and only if ${}h$ comes from the left, which is true if and only if ${}h$ belongs to the ideal generated by ${}(f,g)$.

If we want to realize the geometric torsor corresponding to such a cohomology class, we start with two affine lines over ${}D(f)$ and ${}D(g)$, which we write as ${}\operatorname {Spec} {\left(R_{f}[V]\right)}$ and ${}\operatorname {Spec} {\left(R_{g}[W]\right)}$. According to Fakt these have to be glued with the identification ${}W=V+{\frac {h}{fg}}$.