Essentially finite vector bundle

In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav V. Nori, 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:

Finite vector bundles

Let be a scheme and a vector bundle on . For an integral polynomial with nonnegative coefficients define

Then is called finite if there are two distinct polynomials for which is isomorphic to .

Definition

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.

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.

Properties

• 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.

Notes