Type of vector bundle
In mathematics, a Nori semistable vector bundle is a particular type of vector bundle whose first definition has been first implicitly suggested by Madhav V. Nori, as one of the main ingredients for the construction of the fundamental group scheme. The original definition given by Nori was obviously not called "Nori semistable". Also, Nori's definition was different from the one suggested nowadays. The category of Nori semistable vector bundles contains the Tannakian category of essentially finite vector bundles, whose naturally associated group scheme is the fundamental group scheme formula_1.
Definition.
Let formula_2 be a scheme over a field formula_3 and formula_4 a vector bundle on formula_2. It is said that formula_4 is "Nori semistable" if for any smooth and proper curve formula_7 over formula_8 and any morphism formula_9 the pull back formula_10 is semistable of degree 0.
Difference with Nori's original definition.
Nori semistable vector bundles were called by Nori "semistable" causing a lot of confusion with the already existing definition of semistable vector bundles. More importantly Nori simply said that the restriction of formula_4 to any curve in formula_2 had to be semistable of degree 0. Then for instance in positive characteristic a morphism formula_13 like the Frobenius morphism was not included in Nori's original definition. The importance of including it is that the above definition makes the category of Nori semistable vector bundles tannakian and the group scheme associated to it is the formula_14-fundamental group scheme formula_15. Instead, Nori's original definition didn't give rise to a Tannakian category but only to an abelian category.