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:
The following two definitions coincide whenever
is a reduced, connected and proper scheme over a perfect field.
According to Borne and Vistoli
A vector bundle is essentially finite if it is the kernel of a morphism
where
are finite vector bundles. [3]
The original definition of Nori
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
be a reduced and connected scheme over a perfect field
endowed with a section
. Then a vector bundle
over
is essentially finite if and only if there exists a finite
-group scheme
and a
-torsor
such that
becomes trivial over
(i.e.
, where
).
- When
is a reduced, connected and proper scheme over a perfect field with a point
then the category
of essentially finite vector bundles provided with the usual tensor product
, the trivial object
and the fiber functor
is a Tannakian category.
- The
-affine group scheme
naturally associated to the Tannakian category
is called the fundamental group scheme.
Szamuely, T. (2009). Galois Groups and Fundamental Groups. Vol. 117. Cambridge Studies in Advanced Mathematics.
N. Borne, A. Vistoli The Nori fundamental gerbe of a fibered category, J. Algebr. Geom. 24, No. 2, 311-353 (2015)