Schema/Geometrisches Vektorbündel/Als Schema/en/Definition

Geometric vector bundle

Let $X$ denote a scheme. A scheme $V$ equipped with a morphism

p : V ⟶ X

is called a geometric vector bundle of rank $r$ over $X$ if there exists an open covering $X = ⋃_{i \in I} U_{i}$ and $U_{i}$ -isomorphisms

ψ_{i} : U_{i} \times 𝔸_{}^{r} = 𝔸_{U_{i}}^{r} ⟶ V |_{U_{i}} = p^{- 1} (U_{i})

such that for every open affine subset $U \subseteq U_{i} \cap U_{j}$ the transition mappings

ψ_{j}^{- 1} \circ ψ_{i} : 𝔸_{U_{i}}^{r} |_{U} ⟶ 𝔸_{U_{j}}^{r} |_{U}

are linear automorphisms, i.e. they are induced by an automorphism of the polynomial ring $Γ (U, 𝒪_{X}) [T_{1}, \dots, T_{r}]$ given by $T_{i} \mapsto \sum_{j = 1}^{r} a_{i j} T_{j}$ .