Does $\mathfrak{S}\left[\frac{1}{u} \right]_{(p)}^{\wedge}$ and $\widehat{\mathfrak{S}\left[\frac{1}{u} \right]_{(p)}}$ mean the same thing?

Question

In page $24$ of this survey paper on $\text{$p$-adic Hodge Theory}$, by Brian Conrad, I am having confusion understanding the following notation:

He denoted by $\mathscr{O}_{\varepsilon}:=\mathfrak{S}\left[\frac{1}{u} \right]_{(p)}^{\wedge}$, where $p$ is the uniformizer, i.e., $(p)$ is the prime or maximal ideal. Clearly the symbol $\wedge$ means he is taking $p$-adic completion of $\mathfrak{S}\left[\frac{1}{u} \right]$ with respect to $(p)$.

I have attached the page as well:

My question:

$(1)$ Does he mean to take localisation of $\mathfrak{S} \left[ \frac{1}{u}\right]$ at the prime ideal $(p)$ and then taking completion $\wedge$ of $\mathfrak{S}\left[\frac{1}{u} \right]_{(p)}$ ?

$(2)$ Does $\mathfrak{S}\left[\frac{1}{u} \right]_{(p)}^{\wedge}$ and $\widehat{\mathfrak{S}\left[\frac{1}{u} \right]_{(p)}}$ mean the same thing?

Please help with these two questions.