Essentially finite vector bundle

In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav Nori,^[1][2] 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 ${\displaystyle x\in X(k)}$. 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 ${\displaystyle p:P\to X}$ such that $V$ becomes trivial over $P$ (i.e. ${\displaystyle p^{*}(V)\cong O_{P}^{\oplus r}}$, where ${\displaystyle r=rk(V)}$).

Notes

1. M. V. Nori On the Representations of the Fundamental Group, Compositio Mathematica, Vol. 33, Fasc. 1, (1976), p. 29–42
2. T. Szamuely Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, Vol. 117 (2009)