# Definition of Zariski localization along a closed subset

I'm reading Bhatt and Scholze's, "Prisms and prismatic cohomology", 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", 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.

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.

• It is worthwhile to mention that there is a neater (depends on taste) description of $S$, that is, $S=1+I$ with the usual assumption of Axiom of Choice. See Atiyah-Macdonald, Exercise 3.7. – Yai0Phah Mar 20 at 17:32

Let me answer question 1. The short answer is: "this is essentially by definition of locally prinicipal, and the facts that $${\rm Spec}A$$ is quasi-compact, and principal opens form a basis for its topology".

The extended version of an answer is: Let $$\mathscr{I}$$ denote the ideal sheaf on $${\rm Spec} A$$ corresponding to $$I$$. The fact that $$I$$ is locally principal means precisely that that locally on $${\rm Spec} A$$, $$\mathscr{I}$$ is generated by one global section. THus there is an open covering $${\rm Spec} A = \bigcup_{i \in I} U_i$$ such that $$\mathscr{I}|_{U_i}$$ is generated -- as an $$\mathcal{O}_{U_i}$$-module -- by one global section. Note that this property remains true after restricting to an open subset of $$U_i$$. Thus, and as the principal open subsets form a basis for topology of $${\rm Spec} A$$, we can refine our covering to one by principal opens $$D(g_i) \cong {\rm Spec}A[g_i^{-1}] \subseteq {\rm Spec} A$$. Moreover, as $${\rm Spec} A$$ is quasi-compact, we can pick a finite subcovering and so we have finitely many $$g_i$$'s such that $${\rm Spec} A = \bigcup_i D(g_i)$$, which is equivalent to the claim that the unit ideal of $$A$$ is generated by the $$g_i$$'s. Now $$\mathscr{I}|_{D(g_i)}$$ is the ideal sheaf on $$D(g_i) = {\rm Spec}A[g_i^{-1}]$$ corresponding to the $$A[g_i^{-1}]$$-module $$IA[g_i^{-1}]$$, and and it is generated by one global section by construction. This implies that $$IA[g_i^{-1}] = \Gamma(D(g_i), \mathscr{I})$$ is generated (as $$A[g_i^{-1}]$$-module) by one element.

• Thank you for the detailed explanation! This is clear now. – Yiiwa Nov 29 '19 at 11:07