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 $x\in X(k)$ . 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 $r=rk(V)$ ).

Notes

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

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