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Essentially finite vector bundle

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In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav 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. So before recalling the definition we give this characterization:

Contents

Characterization

Let X be a reduced and connected scheme over a perfect field k endowed with a section x ∈ X ( k ) . Then a vector bundle V over X is essentially finite if and only if there exists a finite k -group scheme G and a G -torsor p : P → X such that V becomes trivial over P (i.e. p ∗ ( V ) ≅ O P ⊕ r , where r = r k ( V ) ).

Definition

Let X be a scheme and E a vector bundle on X. For f = a 0 + a 1 x + … + a n x n ∈ Z ≥ 0 [ x ] an integral polynomial with nonnegative coefficients define

f ( E ) := O X ⊕ a 0 ⊕ E ⊕ a 1 ⊕ ( E ⊗ 2 ) ⊕ a 2 ⊕ … ⊕ ( E ⊗ n ) ⊕ a n

A vector bundle E is called finite if there are two distinct polynomials f, g for which f(E) is isomorphic to g(E). A bundle is essentially finite if it's a subquotient of a finite vector bundle.

References

Essentially finite vector bundle Wikipedia
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