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

Definition

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

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

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.[3]

Notes

1. Nori, Madhav V. (1976). "On the Representations of the Fundamental Group". Compositio Mathematica. 33.1: 29–42. MR 0417179.
2. Szamuely, T. (2009). Galois Groups and Fundamental Groups. 117. Cambridge Studies in Advanced Mathematics.
3. Esnault, Hélène; Hai, Phùng Hô (2010). "Two small remarks on Nori fundamental group scheme". Adv. Stud. Pure Math. 60: 237–243. MR 2761929.