Essentially finite vector bundle

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 $r=rk(V)$).

Definition

Let X be a scheme and E a vector bundle on X. For $f=a_{0}+a_{1}x+\ldots +a_{n}x^{n}\in \mathbb {Z} _{\geq 0}[x]$ an integral polynomial with nonnegative coefficients define

$f(E):={\mathcal {O}}_{X}^{\oplus a_{0}}\oplus E^{\oplus a_{1}}\oplus \left(E^{\otimes 2}\right)^{\oplus a_{2}}\oplus \ldots \oplus \left(E^{\otimes n}\right)^{\oplus a_{n}}$

A vector bundle E is called finite if there are two distinct polynomials f, g for which f(E) is isomorphic to g(E). A bundle is essentially finite if it's a subquotient of a finite vector bundle.