Essentially finite vector bundle

Let ${\displaystyle X}$ be a scheme and ${\displaystyle V}$ a vector bundle on ${\displaystyle X}$. 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

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

Then ${\displaystyle V}$ is called finite if there are two distinct polynomials ${\displaystyle f,g\in \mathbb {Z} _{\geq 0}[x]}$ for which ${\displaystyle f(V)}$ is isomorphic to ${\displaystyle g(V)}$.

The following two definitions coincide whenever ${\displaystyle X}$ is a reduced, connected and proper scheme over a perfect field.

A vector bundle is essentially finite if it is the kernel of a morphism ${\displaystyle u:F_{1}\to F_{2}}$ where ${\displaystyle F_{1},F_{2}}$ are finite vector bundles. ^[3]

A vector bundle is essentially finite if it is a subquotient of a finite vector bundle in the category of Nori-semistable vector bundles.^[1]

• 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 ${\displaystyle r=rk(V)}$).

• When ${\displaystyle X}$ is a reduced, connected and proper scheme over a perfect field with a point ${\displaystyle x\in X(k)}$ then the category ${\displaystyle EF(X)}$ of essentially finite vector bundles provided with the usual tensor product ${\displaystyle \otimes _{{\mathcal {O}}_{X}}}$, the trivial object ${\displaystyle {\mathcal {O}}_{X}}$ and the fiber functor ${\displaystyle x^{*}}$ is a Tannakian category.

• The ${\displaystyle k}$-affine group scheme ${\displaystyle \pi _{1}(X,x)}$ naturally associated to the Tannakian category ${\displaystyle EF(X)}$ is called the fundamental group scheme.