Vector bundle and its sheaf of sections

A vector bundle over a scheme is a scheme with a projection map in which locally is has the form , i.e there is an isomorphism over . The local descriptions have to be compatible in the following sense :

if we have an overlapping then induces a map of into inself, the dual of which is required to be linear ( i.e., if is the ring of section on , then the dual map of into itself is a linear isomorphism on the variables ).

Then we have the following properties :

- To give a vector bundle over is the same as to give the associated sheaf of its section, which is locally free of rank . To see this, note that to give a section over is the same as to give an -algebra-morphism from to , this is the same as to give elements in , i.e, an element of . Now we need to see what happen at overlaps . We have an automorphism to itself, and two linear morphisms (on variables ) to , which both send to a basis. Then the two linear morphisms extend to corresponding isomorphisms of algebras to the symmetric power of . Now the transition functions for locally free sheaf are linear, hence the linear automorphisms of will force to be linear.

- is the cone over with the corresponding sheaf of is the symmetric algebra of the dual of its sheaf of section (*).

To see why (*) is true, note that if we denote to be the sheaf of sections, then on each open set where it is trivial, can be identified with the linear map from to , hence its dual sheaf is itself and hence if we take the symmetric algebra of this sheaf we recover the structure sheaf on each open set of the vector bundle.

- To give a homomorphism of vector bundles is the same as to a morphism of the sheaves of sections, as can be seen by composing a section of with we get a section of . Hence to give such homomorphism is the same as to give a section of the bundle

Thus, a vector bundle can be identify with its sheaf of sections, although its structure sheaf comes from taking the symmetric algebra of the dual sheaf of its sheaf of sections.

Projective completion of a cone over a scheme

A cone over a scheme is the Spec of the sheaf of graded - algebra , is generated by as an algebra.

The projective completion is Proj where we give the variable grade 1. Then is a global section of the canonical on the projective completion, the zero scheme of which is precisely Proj , and whose complement is exactly ( Check on local base : if we localize at , then Proj , and Spec via the isomorphism of algebras sending each to )

What is this weblog about?

This weblog is intended to be the place where I discuss about mathematics. My interests are Intersection Theory , Elliptic Curves, Algebraic Topology, Knots, and Graph Theory.