Lemma 88.12.7 (0GC7)—The Stacks project

Lemma 88.12.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent

for every commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 88.12.1,
there exists a covering $\{ Y_ j \to Y\} $ as in Formal Spaces, Definition 87.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Formal Spaces, Definition 87.11.1 such that each $X_{ji} \to Y_ j$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 88.12.1,
there exist a covering $\{ X_ i \to X\} $ as in Formal Spaces, Definition 87.11.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 88.12.1, and
the morphism $f_{red} : X_{red} \to Y_{red}$ is locally of finite type.

Proof. The equivalence of (1), (2), and (3) follows from Lemma 88.12.1 and an application of Formal Spaces, Lemma 87.21.3. Let $Y_ j$ and $X_{ji}$ be as in (2). Then

The families $\{ Y_{j, red} \to Y_{red}\} $ and $\{ X_{ji, red} \to X_{red}\} $ are étale coverings by affine schemes. This follows from the discussion in the proof of Formal Spaces, Lemma 87.12.1 or directly from Formal Spaces, Lemma 87.12.3.
If $X_{ji} \to Y_ j$ corresponds to the morphism $B_ j \to A_{ji}$ of $\textit{WAdm}^{count}$, then $X_{ji, red} \to Y_{j, red}$ corresponds to the ring map $B_ j/\mathfrak b_ j \to A_{ji}/\mathfrak a_{ji}$ where $\mathfrak b_ j$ and $\mathfrak a_{ji}$ are the ideals of topologically nilpotent elements. This follows from Formal Spaces, Example 87.12.2. Hence $X_{ji, red} \to Y_{j, red}$ is locally of finite type if and only if $B_ j \to A_{ji}$ satisfies the property defined in Lemma 88.12.1.

The equivalence of (2) and (4) follows from these remarks because being locally of finite type is a property of morphisms of algebraic spaces which is étale local on source and target, see discussion in Morphisms of Spaces, Section 67.23. $\square$

The Stacks project

Comments (0)

Post a comment