sheaf of sections
0.1 Presheaf Definition
Consider a rank vector bundle , whose typical fibre is defined with respect to a field . Let constitute a cover for . Then, sections of the bundle over some are defined as continuous functions , which commute with the natural projection map ; . Denote the space of sections of the bundle over U to be . The space of sections is a vector space over the field by defining addition and scalar multiplication pointwise: for , and
0.2 Sheaf Axioms
It is easy to see that it satisfies the sheaf axioms: for open and a cover of ,
if and for all , then .
if for all , such that for each with , , then there is an with for all .
We may also see the vector bundle by applying associated sheaf construction to the presheaf . First though, we show that the stalk of the sheaf at a point is isomorphic to the fibre of the bundle at the point. Let be a germ at , and define a map by
First, we show that the map is a vector space homomorphism. Consider two germs and in . These map to and respectively. We add the germs by finding an open set and adding the restrictions of the sections;
Of course, , so we have , since the restriction maps are simply restriction of functions. Now, it is easy to show that is injective. Assume . Then
Now, we show that is surjective. For , let open be isomorphic to some subset of . Then, is the set of continuous maps , where is the typical fibre of ;
collection of germs of sections at points in .
Then, the associated sheaf to is the presheaf which assigns continuous maps to each open . These are maps where the preimage of is open. Clearly, this implies that . To go the other way, note that open sets of are the images of continuous maps . An open subset of may be written as a union of ; . Then, by single-valuedness of maps, a continuous map must map to for some , so we have .
|Title||sheaf of sections|
|Date of creation||2013-03-22 15:46:36|
|Last modified on||2013-03-22 15:46:36|
|Last modified by||guffin (12505)|
|Defines||Sheaf of Sections|