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

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

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

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

$L^1$ 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

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

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

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

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

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

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 $\mathcal G(U)$ .

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

Then we may easily see that

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

holds.

Then we define

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

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.)