Lemma 88.17.1 (0GCI)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 88: Algebraization of Formal Spaces
Section 88.17: Rig-smooth homomorphisms
Lemma 88.17.1 (cite)

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Lemma 88.17.1. Let $A \to B$ be a morphism in $\textit{WAdm}^{Noeth}$ (Formal Spaces, Section 87.21). The following are equivalent:

$A \to B$ satisfies the equivalent conditions of Lemma 88.11.1 and there exists an ideal of definition $I \subset B$ such that $B$ is rig-smooth over $(A, I)$ , and
$A \to B$ satisfies the equivalent conditions of Lemma 88.11.1 and for all ideals of definition $I \subset A$ the algebra $B$ is rig-smooth over $(A, I)$ .

Proof. Let $I$ and $I'$ be ideals of definitions of $A$ . Then there exists an integer $c \geq 0$ such that $I^ c \subset I'$ and $(I')^ c \subset I$ . Hence $B$ is rig-smooth over $(A, I)$ if and only if $B$ is rig-smooth over $(A, I')$ . This follows from Definition 88.4.1, the inclusions $I^ c \subset I'$ and $(I')^ c \subset I$ , and the fact that the naive cotangent complex $\mathop{N\! L}\nolimits _{B/A}^\wedge$ is independent of the choice of ideal of definition of $A$ by Remark 88.11.2. $\square$

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