We may similarly define sheaf of sets, sheaf of modules, etc.
Then it is easy to see that
satisfies the sheaf axiom and that
From the previous Lemma, we only need to prove locality and gluing lemma for
open sets of the form . That means, in proving the properties
(1) and (2) of Definition 7.19, we may assume that
for some elements
.
Furthermore, in doing so we may use the identification
.
By replacing
by
, this means that we may assume that
.
To sum up, we may assume
To simplify the notation, in the rest of the proof, we shall denote by
(PU) | ![]() |
Let be elements such that
Gluing lemma:
Let
be given
such that they satisfy
Now we put
Then since for any
Now, take any other open set
from the covering
.
is again a finite open
covering of
.
We thus know from the argument above that there exists an element
of
such that