I'm reading Bhatt and Scholze's, ["Prisms and prismatic cohomology"](https://arxiv.org/pdf/1905.08229.pdf), and I have a few questions about Lemma 3.1.


1. Why can they choose a finite number of elements generating $A$ such that $IA[1/g\_i]$ is principal? I don't understand how this follows from the fact that $I$ is locally principal.
2. What do they mean by the Zariski localization of a ring along a closed subset of its spectrum?
From Def. 2.1.6 in Bhatt and Scholze's, ["The pro-étale cohomology for schemes"](http://www.math.uni-bonn.de/people/scholze/proetale.pdf), I understand that the **localization of $Spec(A)$ along $V(I)$** should be something like
$$\cup\_{\mathfrak p \in V(I)} Spec(A\_{\mathfrak p}),$$ 
which can be seen as a subset of $Spec(A)$. On the other hand, seeing Def. 2.1.12, ibid., regarding **Zariski localizations of spectral spaces**, I imagine it should be something like 
$$\sqcup\_{\mathfrak p \in V(I)} Spec(A\_{\mathfrak p})$$ 
endowed with the natural map to $Spec(A)$.   

And so the Zariski localization of $A$ along $V(I)$ should be something like 
$$\prod\_{\mathfrak p \in V(I)} A\_{\mathfrak p},$$ 
but I don't know how this fits Def. 2.2.1, ibid., regarding **Zariski localizations of rings**.



Definition of Zariski localization along a closed subset
Regarding question 2:


My interpretation of the phrase (which fits the proof, i.e. makes the proof of the Lemma work) is the following:



> 
> Given a ring $B$ and an ideal $I \subseteq B$, the localization of $B$ along $V(I)$ is $S^{-1}B$ where $S=B \setminus \bigcup V(I)=B \setminus \bigcup\_{\mathfrak{p}\in V(I)} \mathfrak{p}$ (note that $S$ is multiplicatively closed).
> 
> 
> 


This is, firstly, ind-Zariski localization of the ring $B$ as per the mentioned Definition 2.2.1. 


Secondly, it does what is claimed in the proof of Lemma 3.1, namely $IS^{-1}B$ is contained in the radical:


Indeed, primes of $S^{-1}B$ correspond to primes $\mathfrak{p}$ of $B$ that are contained in $\bigcup V(I)$. Given such a prime, the inclusion $\mathfrak{p}+I \subseteq \bigcup V(I)(=\bigcup\_{I \subseteq \mathfrak{q} \in \mathrm{Spec}\,B}\mathfrak{q})$ shows that $\mathfrak{p}+I$ does not contain $1$, so there is a maximal ideal $\mathfrak{m}$ of $B$ with $\mathfrak{p}, I \subseteq \mathfrak{m}$. This shows that every prime ideal of $S^{-1}B$ is contained in a maximal ideal coming from $V(I)$, so in particular, every maximal ideal of $S^{-1}B$ contains $I$. 


In fact, this argument shows that $\mathrm{Spec}\,S^{-1}B=\bigcup\_{\mathfrak{p}\in V(I)}\mathrm{Spec}\,A\_{\mathfrak{p}}$ as you mention.


