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sheafification of a sheaf

In the preceding subsection, we have not been explained what ``sheafification'' really means. Here is the definition.

LEMMA 1.28   Let $ \mathcal G$ be a presheaf on a topological space $ X$ . Then there exists a sheaf $ \operatorname{sheaf}(\mathcal G)$ and a presheaf morphism

$\displaystyle \iota_{\mathcal G}:\mathcal G \to \operatorname{sheaf}(\mathcal G)
$

such that the following property holds.
  1. If there is another sheaf $ \mathcal F$ with a presheaf morphism

    $\displaystyle \alpha:\mathcal G \to \mathcal F,
$

    then there exists a unique sheaf homomorphism

    $\displaystyle \tilde{\alpha}: \operatorname{sheaf}(\mathcal G) \to \mathcal F
$

    such that

    $\displaystyle \alpha=\tilde{\alpha}\circ \iota_{\mathcal G}
$

    holds.

Furthermore, such $ \operatorname{sheaf}(\mathcal G), \iota_{\mathcal G}$ is unique.

DEFINITION 1.29   The sheaf $ \operatorname{sheaf}(\mathcal G)$ (together with $ \iota_{\mathcal G}$ ) is called the sheafification of $ \mathcal G$ .

The proof of Lemma 1.28 is divided in steps.

The first step is to know the uniqueness of such sheafification. It is most easily done by using universality arguments. ([1] has a short explanation on this topic.)

Then we divide the sheafification process in two steps.

LEMMA 1.30 (First step of sheafification)   Let $ \mathcal G$ be a presheaf on a topological space $ X$ . Then for each open set $ U\subset X$ , we may define a equivalence relation on $ \mathcal G(U)$ by

\begin{equation*}
f \sim g \iff
\left(
\begin{aligned}
&\text{there exists an o...
...ambda,U}g$ } \\
&\text{for any $\lambda$.}
\end{aligned}\right)
\end{equation*}

Then we define

$\displaystyle \mathcal G^{(1)}(U)=\mathcal G(U)/\sim.
$

Then $ \mathcal G^{(1)}$ is a presheaf that satisfies the locality axiom of a sheaf. There is also a presheaf homomorphism from $ \mathcal G$ to $ \mathcal G^{(1)}$ . Furthermore, $ \mathcal G^{(1)}$ is universal among such.

LEMMA 1.31 (Second step of sheafification)   Let $ \mathcal G$ be a presheaf on a topological space $ X$ which satisfies the locality axiom of a sheaf. Then we define a presheaf $ \mathcal G^{(2)}$ in the following manner. First for any open covering $ \{U_\lambda\}$ of an open set $ U\subset X$ , we define

\begin{equation*}
\mathcal G^{(2)}(U; \{U_\lambda\})
=\left\{
\{r_\lambda \} \in...
...&\text{for any $\lambda, \mu\in \Lambda$.}
\end{aligned}\right\}
\end{equation*}

Then we define

$\displaystyle \mathcal G^{(2)}(U)=
\varinjlim_{\{U_\lambda\}}
\mathcal G^{(2)}(U; \{U_\lambda\})
$

Then we may see that $ G^{(2)}$ is a sheaf and that there exists a homomorphism from $ G$ to $ G^{(2)}$ . Furthermore, $ G^{(2)}$ is universal among such.

Proofs of the above two lemma are routine work and are left to the reader.

Finish of the proof of Lemma 1.28: We put

$\displaystyle \operatorname{sheaf}(\mathcal G)=((\mathcal G)^{(1)})^{(2)}
$

$ \qedsymbol$

ARRAY(0x92743dc)


next up previous
Next: stalk of a presheaf Up: (Usual) affine schemes Previous: example of presheaves and
2007-12-11