# Tight closure/Two-dimensional/Standard-graded/Syzygy bundle/Introduction/Section

Let $R$ be a two-dimensional standard-graded normal domain over an algebraically closed field $K$. Let $C=\operatorname {Proj} {\left(R\right)}$ be the corresponding smooth projective curve and let
    $I={\left(f_{1},\ldots ,f_{n}\right)}\,$
be an $R_{+}$-primary homogeneous ideal with generators of degrees $d_{1},\ldots ,d_{n}$. Then we get on $C$ the short exact sequence
    $0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0.$
Here $\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)$ is a vector bundle, called the syzygy bundle, of rank $n-1$ and of degree
    $((n-1)m-\sum _{i=1}^{n}d_{i})\operatorname {deg} \,(C).$
Recall that the degree of a vector bundle ${\mathcal {S}}$ on a projective curve is defined as the degree of the invertible sheaf $\bigwedge ^{r}{\mathcal {S}}$, where $r$ is the rank of ${\mathcal {S}}$. The degree is additive on short exact sequences.
A homogeneous element $f$ of degree $m$ defines an element in $\Gamma (C,{\mathcal {O}}_{C}(m))$ and thus a cohomology class $\delta (f)\in H^{1}(C,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))$, so this defines a torsor over the projective curve. We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.

## Remark

Let ${\mathcal {S}}$ denote a locally free sheaf on a scheme $X$. For a cohomology class $c\in H^{1}(X,{\mathcal {S}})$ one can construct a geometric object: Because of $H^{1}(X,{\mathcal {S}})\cong \operatorname {Ext} ^{1}({\mathcal {O}}_{X},{\mathcal {S}})$, the class defines an extension
    $0\longrightarrow {\mathcal {S}}\longrightarrow {{\mathcal {S}}'}\longrightarrow {\mathcal {O}}_{X}\longrightarrow 0.$
This extension is such that under the connecting homomorphism of cohomology, $1\in \Gamma (X,{\mathcal {O}}_{X})$ is sent to $c\in H^{1}(X,{\mathcal {S}})$. The extension yields a projective subbundle
    ${\mathbb {P} }({\mathcal {S}}^{\vee })\subset {\mathbb {P} }({{\mathcal {S}}'}^{\vee })\,.$
If $V$ is the corresponding geometric vector bundle of ${\mathcal {S}}$, one may think of ${\mathbb {P} }({\mathcal {S}}^{\vee })$ as ${\mathbb {P} }(V)$ which consists for every base point $x\in X$ of all the lines in the fiber $V_{x}$ passing through the origin. The projective subbundle ${\mathbb {P} }(V)$ has codimension one inside ${\mathbb {P} }(V')$, for every point it is a projective space lying (linearly) inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement
    $T={\mathbb {P} }({{\mathcal {S}}'}^{\vee })\setminus {\mathbb {P} }({\mathcal {S}}^{\vee })\,$
is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when $X$ is projective, in an entirely projective setting.
* ↑ ${\mathcal {S}}^{\vee }$ denotes the dual bundle. According to our convention, the geometric vector bundle corresponding to a locally free sheaf ${\mathcal {T}}$ is given by $\operatorname {Spec} {\left(\oplus _{k\geq 0}S^{k}({\mathcal {T}})\right)}$ and the projective bundle is $\operatorname {Proj} {\left(\oplus _{k\geq 0}S^{k}({\mathcal {T}})\right)}$, where $S^{k}$ denotes the $k$th symmetric power.