20/09/05 08:16:20.61 YJxrx+O5.net
>>142 追加
下記は、物理屋さんに、層の講義をしているので、分り易い
層が抽象的で分かり難いと思う方、一読の価値がある
（物理の話”Higgs vevs”とかは、飛ばしながらね）
URLﾘﾝｸ(arxiv.org)
Lectures on D-branes and Sheaves 2003
Eric Sharpe
(抜粋)
P8
2 Overview of mathematics of sheaves and Ext groups
2.1 Complexes and exact sequences
The language of complexes and exact sequences, standard in algebraic topology, will be used
throughout these lectures. However, many physicists do not know this language, so in this
introductory section we shall review these concepts.
A complex of groups, rings, modules, sheaves, etc is a collection An of groups, rings, etc,
with maps φn : An → An+1 satisfying the important property that φn+1 ◯φn = 0. Complexes
are typically denoted as follows:
An exact sequence is a special kind of complex, namely one in which the image of each
map is the same as the kernel of the next map, not just a subset. This is a stronger statement
than merely φn+1 ◯ φn = 0. For example, for the complex
Aφ-→ B -→ 0
to be exact implies that φ is surjective (onto): the kernel of the right map is all of B, since
the right map sends all of B to zero, yet since the complex is exact, the kernel of each map
equals the image of the previous map, so the image of φ is all of B, hence φ is surjective.
Similarly, for the complex
0 -→ A φ -→ B
to be exact implies that φ is injective (one-to-one): the image of the left map is zero, but
since the complex is exact, the image of each map equals the kernel of the next, so the kernel
of φ is zero, hence φ is injective.
つづく