# 《代数几何》33页注释

$\mathcal{O}_{X,P}\to\mathcal{O}_{X,\xi}$
This homomorphism is injective and it identifies $\mathcal{O}_{X,\xi}$ with the fraction field of $\mathcal{O}_{X,P}$ for any point $P\in X$.

$\mathcal{O}_X(U)\to\mathcal{O}_{X,P}\to\mathcal{O}_{X,\xi}$
$\because\mathcal{O}_X(U)=A$, we get
$A\to\mathcal{O}_{X,P}\to\mathcal{O}_{X,\xi}$
The two homomorphisms are both injective and induce
$A_{(0)}\to(\mathcal{O}_{X,P})_{(0)}\to\mathcal{O}_{X,\xi}$
$\because A_{(0)}\cong\mathcal{O}_{X,\xi}$
$\therefore(\mathcal{O}_{X,P})_{(0)}\cong\mathcal{O}_{X,\xi}$