Spherical sector

In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav Nori, as the main tool in the construction of the fundamental group scheme. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural objects to study in algebraic geometry. So before recalling the definition we give this characterization:

Characterization

Let $X$ be a reduced and connected scheme over a perfect field $k$ endowed with a section $x\in X(k)$. Then a vector bundle $V$ over $X$ is essentially finite if and only if there exists a finite $k$-group scheme $G$ and a $G$-torsor $p:P\to X$ such that $V$ becomes trivial over $P$ (i.e. $p^{*}(V)\cong O_{P}^{\oplus r}$, where $r=rk(V)$).