Formal scheme....

Question

on studying the Harshorne book, I have some question for the formal scheme...

Let $X$ be a noetherian scheme and let $Y$ be a closed subschme defined by a sheaf of ideals $\mathcal{I}$. Suppose that $\widehat{X}$ be the formal completion of $X$ along $Y$ i.e. $(\widehat{X},\mathcal{O}_{\widehat{X}})=(Y, \varprojlim \mathcal{O}_X/\mathcal{I}^n)$.

1. I don't understand that the stalk of the sheaf $\mathcal{O}_{\widehat{X}}$ is a local ring...

2. If $Y=\{p\}$ is a closed point, why $\mathcal{O}_{\widehat{X}}$ is the completion of $\widehat{\mathcal{O}}_p$ of the local ring of $p$...?

Answer 1

1. It is a general fact, and easy to prove, that a directed limit of local rings with surjective transition maps is again a local ring. You can write down the maximal ideal explicitly and show that everything outside of it is a unit. Hence it's the unique maximal ideal.

2. This follows from the general fact / definition (depending how you set up the theory) that $\lim_n A/p^n A$ is the $p$-adic completion of $A$.