20/09/05 08:21:37.88 YJxrx+O5.net
>>154
つづき
For most of this paper, we shall only be interested in sheaf cohomology on complex
manifolds with coefficients in holomorphic sheaves. However, there are some amusing games
one can play with more general sheaves, which we wish to take a moment to describe.
In general, any map of sheaves S → T induces a map between sheaf cohomology groups
Hn(X, S) → Hn(X, T ), and also, a short exact sequence of sheaves
0 -→ S -→ T -→ U -→ 0
induces a long exact sequence of sheaf cohomology groups
・ ・ ・ -→ Hn(X, S) -→ Hn(X, T ) -→ Hn(X, U) -→ Hn+1(X, S) -→ ・ ・ ・
Using this fact we can quickly derive the topological classification of principal U(1) bundles
by first Chern classes, as follows. There is a short exact sequence of sheaves
0 -→ Z -→ C∞(R) -→ C∞(U(1)) -→ 0
which induces a long exact sequence as above.
However, Hn(X, C∞(R)) = 0 for n > 0, so,
for example, we find immediately that
H1(X, C∞(U(1))) ~= H2(X,Z).
As we have already seen, the group on the left classifies principal U(1) bundles,
and their images in H2(X,Z) under the induced map in the long exact sequence are just their first
Chern classes. Another relationship that can be immediately derived from the associated
long exact sequence is that
H2(X, C∞(U(1))) ~= H3(X,Z)
which will play an important role when we discuss B fields.
つづく