20/09/05 08:19:29.38 YJxrx+O5.net
>>151
つづき
Another easy example of a sheaf is the skyscraper sheaf. Let X be a space, and p be a
point on that space. Now, associate sections S(U) to open sets as follows:
・ If U contains p, then S(U) = C.
・ If U does not contain p, then S(U) = {0}.
It is easy to check that this is a sheaf. Moreover, the support of the sheaf S (meaning, the
subset of X over which the sheaf is nonzero) is only the point p.
Skyscraper sheaves are the simplest examples of “vector bundles on submanifolds” alluded to earlier.
Given a continuous map i : Y → X and a sheaf S on Y , we can form a sheaf denoted
i*S on X, defined as follows: i*S(U) ≡ S(i-1(U)).
For example, given a vector bundle E (or rather, the associated sheaf) on a submanifold
S of a manifold X, with inclusion map i : S → X, we can form the sheaf i*E on X. It is easy
to check that i*E only has support on S ? if an open set in X does not intersect S, then there
are no sections associated to that open set. The sheaf i*E is the more precise meaning of the
phrase “vector bundle on a submanifold,” and is an example of what is known technically
as a torsion sheaf. Notice that a skyscraper sheaf can also be put in the form i*E, where
i : p → X and E is the rank one line bundle on the point p, and so skyscraper sheaves are
also examples of torsion sheaves.
In particular, later we shall study the extent to which we can describe a D-brane on a
complex submanifold S with holomorphic vector bundle E by the sheaf i*E, for example by
comparing open string spectra to Ext groups (which will be discussed momentarily).
Another example of a sheaf is the sheaf of maps into Z, that assigns, to every open set
U, the set of continuous maps U → Z. This sheaf comes up in considerations of cohomology,
but is less useful for modelling D-branes.
つづく