The Stacks project

73.8 Descent of finiteness and smoothness properties of morphisms

The following type of lemma is occasionally useful.

Lemma 73.8.1. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphism of algebraic spaces. Let $P$ be one of the following properties of morphisms of algebraic spaces over $S$: flat, locally finite type, locally finite presentation. Assume that $X \to Z$ has $P$ and that $X \to Y$ is a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$. Then $Y \to Z$ is $P$.

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By assumption we can find an fppf covering $\{ V_ i \to V\} $ and lifts $V_ i \to X$ of the morphism $V_ i \to Y$. Since $U \to X$ is surjective étale we see that over the members of the fppf covering $\{ V_ i \times _ X U \to V\} $ we have lifts into $U$. Hence $U \to V$ induces a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$. By our definition of what it means to have property $P$ for a morphism of algebraic spaces (see Morphisms of Spaces, Definition 66.30.1, Definition 66.23.1, and Definition 66.28.1) we see that $U \to W$ has $P$ and we have to show $V \to W$ has $P$. Thus we reduce the question to the case of morphisms of schemes which is treated in Descent, Lemma 35.14.8. $\square$

A more standard case of the above lemma is the following. (The version with “flat” follows from Morphisms of Spaces, Lemma 66.31.5.)

Lemma 73.8.2. Let $S$ be a scheme. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]

be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that $f$ is surjective, flat, and locally of finite presentation and assume that $p$ is locally of finite presentation (resp. locally of finite type). Then $q$ is locally of finite presentation (resp. locally of finite type).

Proof. Since $\{ X \to Y\} $ is an fppf covering, it induces a surjection of fppf sheaves (Topologies on Spaces, Lemma 72.7.5) and the lemma is a special case of Lemma 73.8.1. On the other hand, an easier argument is to deduce it from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas 66.23.4 and 66.28.4). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 66.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.3. $\square$

Lemma 73.8.3. Let $S$ be a scheme. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]

be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that

  1. $f$ is surjective, and syntomic (resp. smooth, resp. étale),

  2. $p$ is syntomic (resp. smooth, resp. étale).

Then $q$ is syntomic (resp. smooth, resp. étale).

Proof. We deduce this from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas 66.36.4, 66.37.4, and 66.39.2). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 66.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.4. $\square$

Actually we can strengthen this result as follows.

Lemma 73.8.4. Let $S$ be a scheme. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]

be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that

  1. $f$ is surjective, flat, and locally of finite presentation,

  2. $p$ is smooth (resp. étale).

Then $q$ is smooth (resp. étale).

Proof. We deduce this from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemmas 66.37.4 and 66.39.2). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 66.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.5. $\square$

Lemma 73.8.5. Let $S$ be a scheme. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]

be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that

  1. $f$ is surjective, flat, and locally of finite presentation,

  2. $p$ is syntomic.

Then both $q$ and $f$ are syntomic.

Proof. We deduce this from the analogue for schemes. Namely, the problem is étale local on $B$ and $Y$ (Morphisms of Spaces, Lemma 66.36.4). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 66.30.6), we can choose an affine scheme $U$ and an étale morphism $U \to X$ such that the composition $U \to Y$ is surjective. In this case the result follows from Descent, Lemma 35.14.7. $\square$


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