Schema/Lokal freie Garbe/Torsor/Beschreibung mit Ext und projektiven Bündeln/en/Bemerkung

Let $𝒮$ denote a locally free sheaf on a scheme $X$. For a cohomology class  $c\in H^1(X,𝒮)$  one can construct a geometric object: Because of  $H^1(X,𝒮)\cong {\mathrm{Ext}}^1({𝒪}_X,𝒮)$,  the class defines an extension

$0⟶𝒮⟶{𝒮}^{\prime }⟶{𝒪}_X⟶0.$

This extension is such that under the connecting homomorphism of cohomology,  $1\in \Gamma (X,{𝒪}_X)$  is sent to  $c\in H^1(X,𝒮)$.  The extension yields a projective subbundle

$\mathbb{P}({𝒮}^{\vee })\subset \mathbb{P}({{𝒮}^{\prime }}^{\vee }).$

If $V$ is the corresponding geometric vector bundle of $𝒮$, one may think of $\mathbb{P}({𝒮}^{\vee })$ 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 $\mathbb{P}(V^{\prime })$, 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

$T=\mathbb{P}({{𝒮}^{\prime }}^{\vee })\setminus \mathbb{P}({𝒮}^{\vee })$

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.