In mathematics, an essentially finite vector bundle is a particular type of vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

 defined by Madhav Nori, as the main tool in the construction of the fundamental group scheme

Fundamental group scheme

In mathematics, the fundamental group scheme is a group scheme canonically associated to a scheme over a Dedekind scheme . It is a generalisation of the étale fundamental group...

. 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

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. So before recalling the definition we give this characterization:

Characterization

Let be a reduced and connected scheme

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

 over a perfect field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

  endowed with a section . Then a vector bundle over is essentially finite if and only if there exists a finite -group scheme

Group scheme

In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...

  and a -torsor such that becomes trivial over (i.e. , where ).