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presheaves

We first define presheaves.

DEFINITION 06.2   Let $ X$ be a topological space. We say ``a presheaf $ \mathcal F$ of rings over $ X$ is given'' if we are given the following data.
  1. For each open set $ U\subset X$ , a ring denoted by $ \mathcal F(U)$ . (which is called the ring of sections of $ \mathcal F$ on $ U$ .)
  2. For each pair $ U,V$ of open subsets of $ X$ such that $ V\subset U$ , a ring homomorphism (called restriction)

    $\displaystyle \rho_{ V U}: \mathcal F(U)\to \mathcal F(V).
$

with the properties
  1. $ \mathcal F(\emptyset)=0$ .
  2. We have $ \rho_{U,U}=$identity for any open subset $ U\subset X$ .
  3. We have

    % latex2html id marker 1840
$\displaystyle \rho_{W V} \rho_{V U}=\rho_{W V} \qquad
$

    for any open sets $ U,V,W\subset X$ such that $ W\subset V\subset U$ .



2017-07-21