Lemma 74.8.1. 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 $P$.
74.8 Descent of finiteness and smoothness properties of morphisms
The following type of lemma is occasionally useful.
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 67.30.1, Definition 67.23.1, and Definition 67.28.1) 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 67.31.5.)
Lemma 74.8.2. Let $S$ be a scheme. Let 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).
\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]
Proof. Since $\{ X \to Y\} $ is an fppf covering, it induces a surjection of fppf sheaves (Topologies on Spaces, Lemma 73.7.5) and the lemma is a special case of Lemma 74.8.1. 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 67.23.4 and 67.28.4). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 67.30.6), 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 74.8.3. Let $S$ be a scheme. Let be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that
$f$ is surjective, and syntomic (resp. smooth, resp. étale),
$p$ is syntomic (resp. smooth, resp. étale).
\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]
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 67.36.4, 67.37.4, and 67.39.2). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 67.30.6), 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 74.8.4. Let $S$ be a scheme. Let be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that
$f$ is surjective, flat, and locally of finite presentation,
$p$ is smooth (resp. étale).
\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]
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 67.37.4 and 67.39.2). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 67.30.6), 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 74.8.5. Let $S$ be a scheme. Let be a commutative diagram of morphisms of algebraic spaces over $S$. Assume that
$f$ is surjective, flat, and locally of finite presentation,
$p$ is syntomic.
\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & B } \]
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 67.36.4). Hence we may assume that $B$ and $Y$ are affine schemes. Since $|X| \to |Y|$ is open (Morphisms of Spaces, Lemma 67.30.6), 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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)