In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav V. 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.
The following notion of finite vector bundle is due to André Weil and will be needed to define essentially finite vector bundles: Let
a vector bundle on
an integral polynomial with nonnegative coefficients define Then
is called finite if there are two distinct polynomials
The following two definitions coincide whenever
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
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.