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.