presheaves

We first define presheaves.

DEFINITION 07.18   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)
   $$\rho_{ V U}: \mathcal F(U)\to \mathcal F(V).$$

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
   $$\rho_{W V} \rho_{V U}=\rho_{W V} \qquad$$
   for any open sets $U,V,W\subset X$ such that $W\subset V\subset U$.