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example of presheaves and sheafification

To proceed our theory further, we need to study a bit more about presheaves. Unfortunately, a sheaf of modules $ \tilde M$ on an affine schemes are ``too good''. Namely, in terms of cohomology (which we study later,) we have always

$\displaystyle H^i (\operatorname{Spec}(A),\tilde {M} )=0 \qquad ($if $\displaystyle i>0).
$

So to study some important problems on sheaf theory (which we will sure to encounter when we deal with non-affine schemes,) we need to study some examples from other mathematical areas.

A first example is a presheaf which satisfies the ``locality'' of sheaf axiom, but which fails to obey ``gluing lemma''.

EXAMPLE 1.25   Let $ X=\mathbb{R}$ be the (usual) real line with the usual Lebesgue measure. Then we have a presheaf of $ L^1$ -functions given by

$\displaystyle L^1(U)=\{ f:U\to \mathbb{C}; \vert f\vert$ is integrable$\displaystyle \}.
$

$ L^1$ is a presheaf which satisfies the ``locality'' of sheaf axiom, but which fails to obey ``gluing lemma''. Indeed, Let $ \{U_n=(-n, n)\}$ be an open covering of $ \mathbb{R}$ and define a section $ f_n$ on $ U_n$ by

$\displaystyle f_n(x)=1 \qquad(x\in U_n).
$

Then we see immediately that $ \{f_n\}$ is a family of sections which satisfies the assumption of ``gluing lemma''. The function which should appear as the ``glued function'' is the constant function $ 1$ , which fails to be integrable on the whole of $ \mathbb{R}$ .

We may ``sheafificate'' the presheaf $ L^1$ above. Instead of $ L^1$ -functions we consider functions which are locally $ L^1$ . Namely, for any open subset $ U\subset \mathbb{R}$ , we consider

\begin{equation*}
L^1_{\operatorname{loc}}(U)=
\left\{f: U\to \mathbb{C};
\begin...
...that $\vert f\vert$ is integrable on $V$ }
\end{aligned}\right\}
\end{equation*}

The presheaf so defined is a sheaf, which we may call ``the sheaf of locally $ L^1$ -functions''.

EXAMPLE 1.26   Similarly, we may consider a presheaf $ U\mapsto$   Bdd$ (U)$ of bounded functions on a topological space $ X$ . We may sheafificate this example and the sheaf so created is the sheaf of locally bounded functions.

EXAMPLE 1.27   It is psychologically a bit difficult to give an example of a presheaf which does not satisfy the locality axiom of a sheaf. But there are in fact a lot of them.

For any differentiable ($ C^\infty$ ) manifold $ X$ (students which are not familiar with the manifolds may take $ X$ as an open subset of $ \mathbb{R}^n$ for an example.), we define a presheaf $ \mathcal G$ on $ X$ defined as follows

$\displaystyle \mathcal G(U)=C^{\infty}(U\times U)=
\{$complex valued $C^&infin#infty;$-functions on $U×U$$\displaystyle \}.
$

The restriction is defined in an obvious manner. It is an easy exercise to see that the presheaf does not satisfy the locality axiom of a sheaf.

To sheafificate this, we first need to introduce an equivalence relation on $ \mathcal G(U)$ .

\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 may easily see that

\begin{equation*}
f \sim g  \iff \
\left(
\begin{aligned}
&\text{there exists ...
...mes U$ } \\
&\text{such that $f=g$ on $V$}
\end{aligned}\right)
\end{equation*}

holds.

Then we define

$\displaystyle \mathcal F(U)=\mathcal G(U)/\sim.
$

It is now an easy exercise again to verify that $ \mathcal F$ so defined is a sheaf. (Readers who are familiar with the theory of jets may notice that the sheaf is related to the sheaf of jets. In other words, there is a sheaf homomorphism from this sheaf to the sheaf of jets.)


next up previous
Next: sheafification of a sheaf Up: (Usual) affine schemes Previous: homomorphisms of (pre)sheaves
2007-12-11