The Stacks project

Lemma 85.16.8. Let $S$ be a scheme. Let $X \to Y$ and $Z \to Y$ be morphisms of formal algebraic space over $S$. Then

  1. If $X$ and $Z$ are locally countably indexed, then $X \times _ Y Z$ is locally countably indexed.

  2. If $X$ and $Z$ are locally adic*, then $X \times _ Y Z$ is locally adic*.

  3. If $X$ and $Z$ are locally Noetherian and $X_{red} \to Y_{red}$ is locally of finite type, then $X \times _ Y Z$ is locally Noetherian.

Proof. Choose a covering $\{ Y_ j \to Y\} $ as in Definition 85.7.1. For each $j$ choose a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 85.7.1. For each $j$ choose a covering $\{ Z_{jk} \to Y_ j \times _ Y Z\} $ as in Definition 85.7.1. Observe that $X_{ji} \times _{Y_ j} Z_{jk}$ is an affine formal algebraic space by Lemma 85.12.4. Hence

\[ \{ X_{ji} \times _{Y_ j} Z_{jk} \to X \times _ Y Z\} \]

is a covering as in Definition 85.7.1. Thus it suffices to prove (1) and (2) in case $X$, $Y$, and $Z$ are affine formal algebraic spaces.

Assume $X$, $Y$, and $Z$ are affine formal algebraic spaces and McQuillan. Then we can write $X = \text{Spf}(A)$, $Y = \text{Spf}(B)$, $Z = \text{Spf}(C)$ for some weakly admissible topological rings $A$, $B$, and $C$ and the morphsms $X \to Y$ and $Z \to Y$ are given by continuous ring maps $B \to A$ and $B \to C$, see Definition 85.5.7 and Lemma 85.5.10 By Lemma 85.12.4 we see that $X \times _ Y Z = \text{Spf}(A \widehat{\otimes }_ B C)$ and that $A \widehat{\otimes }_ B C$ is a weakly admissible topological ring. This reduces cases (1) and (2) of our lemma to parts (3) and (4) of Lemma 85.4.16.

To deduce case (3) from Lemma 85.4.16 part (5) we need to match the hypotheses. First, we observe that taking the reduction corresponds to dividing by the ideal of topologically nilpotent elements (Example 85.8.2). Second, if $X_{red} \to Y_{red}$ is locally of finite type, then $(X_{ji})_{red} \to (Y_ j)_{red}$ is locally of finite type (and hence correspond to finite type homomorphisms of rings). This follows Morphisms of Spaces, Lemma 65.23.4 and the fact that in the commutative diagram

\[ \xymatrix{ (X_{ji})_{red} \ar[d] \ar[r] & (Y_ j)_{red} \ar[d] \\ X_{red} \ar[r] & Y_{red} } \]

the vertical morphisms are ├ętale. Namely, we have $(X_{ji})_{red} = X_{ij} \times _ X X_{red}$ and $(Y_ j)_{red} = Y_ j \times _ Y Y_{red}$ by Lemma 85.8.3. $\square$

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