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
Madhav Vithal Nori is an Indian mathematician. In 1980 he has received the INSA Medal for Young Scientists.
Career
Nori was awarded his PhD in mathematics in 1981 from the University of Mumbai. He studies within the fields of algebraic g ...
, 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
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...
of
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 char ...
s, whose naturally associated group scheme is the
fundamental group scheme .
Definition
Let
be a scheme over a field
and
a vector bundle on
. It is said that
is ''Nori semistable'' if for any smooth and proper curve
over
and any morphism
the pull back
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
to any curve in
had to be semistable of degree 0. Then for instance in positive characteristic a morphism
like the
Frobenius morphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism ma ...
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
-fundamental group scheme
. Instead, Nori's original definition didn't give rise to a Tannakian category but only to an
abelian category.
Notes
Scheme theory
Topological methods of algebraic geometry
{{Algebraic-geometry-stub