On the F-fundamental group scheme

Abstract

Let X be a smooth projective variety defined over a perfect field k of positive characteristic, and let F X be the absolute Frobenius morphism of X. For any vector bundle E → X , and any polynomial g with non-negative integer coefficients, define the vector bundle g ˜ ( E ) using the powers of F X and the direct sum operation. We construct a neutral Tannakian category using the vector bundles with the property that there are two distinct polynomials f and g with non-negative integer coefficients such that f ˜ ( E ) = g ˜ ( E ) . We also investigate the group scheme defined by this neutral Tannakian category.