In this section we prove some properties of rig-étale homomorphisms of adic Noetherian topological rings which are needed to introduce rig-étale morphisms of locally Noetherian algebraic spaces.
Lemma 88.19.1. 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-étale 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-étale 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-étale over $(A, I)$ if and only if $B$ is rig-étale over $(A, I')$. This follows from Definition 88.8.1, 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. $\square$
Definition 88.19.2. Let $\varphi : A \to B$ be a continuous ring homomorphism between adic Noetherian topological rings, i.e., $\varphi $ is an arrow of $\textit{WAdm}^{Noeth}$. We say $\varphi $ is rig-etale if the equivalent conditions of Lemma 88.19.1 hold.
This defines a local property.
Lemma 88.19.3. The property $P(\varphi )=$“$\varphi $ is rig-étale” on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark 87.21.5.
Proof. This proof is exactly the same as the proof of Lemma 88.17.3. Let us recall what the statement signifies. First, $\textit{WAdm}^{Noeth}$ is the category whose objects are adic Noetherian topological rings and whose morphisms are continuous ring homomorphisms. Consider a commutative diagram
satisfying the following conditions: $A$ and $B$ are adic Noetherian topological rings, $A \to A'$ and $B \to B'$ are étale ring maps, $(A')^\wedge = \mathop{\mathrm{lim}}\nolimits A'/I^ nA'$ for some ideal of definition $I \subset A$, $(B')^\wedge = \mathop{\mathrm{lim}}\nolimits B'/J^ nB'$ for some ideal of definition $J \subset B$, and $\varphi : A \to B$ and $\varphi ' : (A')^\wedge \to (B')^\wedge $ are continuous. Note that $(A')^\wedge $ and $(B')^\wedge $ are adic Noetherian topological rings by Formal Spaces, Lemma 87.21.1. We have to show
$\varphi $ is rig-étale $\Rightarrow \varphi '$ is rig-étale,
if $B \to B'$ faithfully flat, then $\varphi '$ is rig-étale $\Rightarrow \varphi $ is rig-étale, and
if $A \to B_ i$ is rig-étale for $i = 1, \ldots , n$, then $A \to \prod _{i = 1, \ldots , n} B_ i$ is rig-étale.
The equivalent conditions of Lemma 88.11.1 satisfy conditions (1), (2), and (3). Thus in verifying (1), (2), and (3) for the property “rig-étale” we may already assume our ring maps satisfy the equivalent conditions of Lemma 88.11.1 in each case.
Pick an ideal of definition $I \subset A$. By the remarks above the topology on each ring in the diagram is the $I$-adic topology and $B$, $(A')^\wedge $, and $(B')^\wedge $ are in the category (88.2.0.2) for $(A, I)$. Since $A \to A'$ and $B \to B'$ are étale the complexes $\mathop{N\! L}\nolimits _{A'/A}$ and $\mathop{N\! L}\nolimits _{B'/B}$ are zero and hence $\mathop{N\! L}\nolimits _{(A')^\wedge /A}^\wedge $ and $\mathop{N\! L}\nolimits _{(B')^\wedge /B}^\wedge $ are zero by Lemma 88.3.2. Applying Lemma 88.3.5 to $A \to (A')^\wedge \to (B')^\wedge $ we get isomorphisms
Thus $\mathop{N\! L}\nolimits _{(B')^\wedge /A}^\wedge \to \mathop{N\! L}\nolimits _{(B')^\wedge /(A')^\wedge }$ is a quasi-isomorphism. The ring maps $B/I^ nB \to B'/I^ nB'$ are étale and hence are local complete intersections (Algebra, Lemma 10.143.2). Hence we may apply Lemmas 88.3.5 and 88.3.6 to $A \to B \to (B')^\wedge $ and we get isomorphisms
We conclude that $\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B (B')^\wedge \to \mathop{N\! L}\nolimits _{(B')^\wedge /A}^\wedge $ is a quasi-isomorphism. Combining these two observations we obtain that
in $D((B')^\wedge )$. With these preparations out of the way we can start the actual proof.
Proof of (1). Assume $\varphi $ is rig-étale. Then there exists a $c \geq 0$ such that multiplication by $a \in I^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge $ in $D(B)$. This property is preserved under base change by $B \to (B')^\wedge $, see More on Algebra, Lemmas 15.84.6. By the isomorphism above we find that $\varphi '$ is rig-étale. This proves (1).
To prove (2) assume $B \to B'$ is faithfully flat and that $\varphi '$ is rig-étale. Then there exists a $c \geq 0$ such that multiplication by $a \in I^ c$ is zero on $\mathop{N\! L}\nolimits _{(B')^\wedge /(A')^\wedge }^\wedge $ in $D((B')^\wedge )$. By the isomorphism above we see that $a^ c$ annihilates the cohomology modules of $\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B (B')^\wedge $. The composition $B \to (B')^\wedge $ is faithfully flat by our assumption that $B \to B'$ is faithfully flat, see Formal Spaces, Lemma 87.19.14. Hence the cohomology modules of $\mathop{N\! L}\nolimits _{B/A}^\wedge $ are annihilated by $I^ c$. It follows from Lemma 88.8.2 that $\varphi $ is rig-étale. This proves (2).
To prove (3), setting $B = \prod _{i = 1, \ldots , n} B_ i$ we just observe that $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is the direct sum of the complexes $\mathop{N\! L}\nolimits _{B_ i/A}^\wedge $ viewed as complexes of $B$-modules. $\square$
Lemma 88.19.4. Consider the properties $P(\varphi )=$“$\varphi $ is rig-étale” and $Q(\varphi )$=“$\varphi $ is adic” on arrows of $\textit{WAdm}^{Noeth}$. Then $P$ is stable under base change by $Q$ as defined in Formal Spaces, Remark 87.21.10.
Proof. The statement makes sense by Lemma 88.19.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ and that $B \to A$ is rig-étale and $B \to C$ is adic (Formal Spaces, Definition 87.6.1). Then we can choose an ideal of definition $I \subset B$ such that the topology on $A$ and $C$ is the $I$-adic topology. In this situation it follows immediately that $A \widehat{\otimes }_ B C$ is rig-étale over $(C, IC)$ by Lemma 88.8.6. $\square$
Lemma 88.19.5. The property $P(\varphi )=$“$\varphi $ is rig-étale” on arrows of $\textit{WAdm}^{Noeth}$ is stable under composition as defined in Formal Spaces, Remark 87.21.14.
Proof. The statement makes sense by Lemma 88.19.1. To see that it is true assume we have rig-étale morphisms $A \to B$ and $B \to C$ in $\textit{WAdm}^{Noeth}$. Then we can choose an ideal of definition $I \subset A$ such that the topology on $C$ and $B$ is the $I$-adic topology. By Lemma 88.3.5 we obtain an exact sequence
There exists a $c \geq 0$ such that for all $a \in I$ multiplication by $a^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge $ in $D(B)$ and $\mathop{N\! L}\nolimits _{C/B}^\wedge $ in $D(C)$. Then of course multiplication by $a^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C$ in $D(C)$ too. Hence $H^0(\mathop{N\! L}\nolimits _{B/A}^\wedge ) \otimes _ A C$, $H^0(\mathop{N\! L}\nolimits _{C/B}^\wedge )$, $H^{-1}(\mathop{N\! L}\nolimits _{B/A}^\wedge \otimes _ B C)$, and $H^{-1}(\mathop{N\! L}\nolimits _{C/B}^\wedge )$ are annihilated by $a^ c$. From the exact sequence we obtain that multiplication by $a^{2c}$ is zero on $H^0(\mathop{N\! L}\nolimits _{C/A}^\wedge )$ and $H^{-1}(\mathop{N\! L}\nolimits _{C/A}^\wedge )$. It follows from Lemma 88.8.2 that $C$ is rig-étale over $(A, I)$ as desired. $\square$
Lemma 88.19.6. The property $P(\varphi )=$“$\varphi $ is rig-étale” on arrows of $\textit{WAdm}^{Noeth}$ has the cancellation property as defined in Formal Spaces, Remark 87.21.18.
Proof. The statement makes sense by Lemma 88.19.1. To see that it is true assume we have maps $A \to B$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ with $A \to C$ and $A \to B$ rig-étale. We have to show that $B \to C$ is rig-étale. Then we can choose an ideal of definition $I \subset A$ such that the topology on $C$ and $B$ is the $I$-adic topology. By Lemma 88.3.5 we obtain an exact sequence
There exists a $c \geq 0$ such that for all $a \in I$ multiplication by $a^ c$ is zero on $\mathop{N\! L}\nolimits _{B/A}^\wedge $ in $D(B)$ and $\mathop{N\! L}\nolimits _{C/A}^\wedge $ in $D(C)$. Hence $H^0(\mathop{N\! L}\nolimits _{B/A}^\wedge ) \otimes _ A C$, $H^0(\mathop{N\! L}\nolimits _{C/A}^\wedge )$, and $H^{-1}(\mathop{N\! L}\nolimits _{C/A}^\wedge )$ are annihilated by $a^ c$. From the exact sequence we obtain that multiplication by $a^{2c}$ is zero on $H^0(\mathop{N\! L}\nolimits _{C/B}^\wedge )$ and $H^{-1}(\mathop{N\! L}\nolimits _{C/B}^\wedge )$. It follows from Lemma 88.8.2 that $C$ is rig-étale over $(B, IB)$ as desired. $\square$
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