Let $ A $ be a commutative ring with unity, and $ \mathrm{Spec} A $ the spectra of $ A $.
We define over $ \mathrm{Spec} A $ the presheaf of rings as follows :
If $ U $ is an open subset of $ \mathrm{Spec} A $, let put : $ S(U) = \{ \ a \in A \ | \ a \not \in \mathfrak{p} \ , \ \forall \mathfrak{p} \in U \ \} $ and, $ \mathcal{F} (U) = S (U)^{-1} A $.
If : $ V \subset U $, we take as the restriction morphism, the canonical morphism : $ S (U)^{-1} A \to S (V)^{-1} A $.
My questions are :
1) Why does : $ \mathcal{F}_x = A_{ \mathfrak{p} } $ such that $ x $ are the corresponding point of the ideal : $ \mathfrak{p} \in \mathrm{Spec} A $ ?.
2) Why does : $ S ( \mathrm{Spec} A ) ^{-1} A = ( A^* )^{-1} A $ ?
Thank you in advance for your help.