The Stacks project

73.9 Descending properties of spaces

In this section we put some results of the following kind.

Lemma 73.9.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. If $f$ is flat at $x$ and $X$ is geometrically unibranch at $x$, then $Y$ is geometrically unibranch at $f(x)$.

Proof. Consider the map of étale local rings $\mathcal{O}_{Y, f(\overline{x})} \to \mathcal{O}_{X, \overline{x}}$. By Morphisms of Spaces, Lemma 66.30.8 this is flat. Hence if $\mathcal{O}_{X, \overline{x}}$ has a unique minimal prime, so does $\mathcal{O}_{Y, f(\overline{x})}$ (by going down, see Algebra, Lemma 10.39.19). $\square$

slogan

Lemma 73.9.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is flat and surjective and $X$ is reduced, then $Y$ is reduced.

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to X \times _ Y V$. As $f$ is surjective and flat, the morphism of schemes $U \to V$ is surjective and flat. In this way we reduce the problem to the case of schemes (as reducedness of $X$ and $Y$ is defined in terms of reducedness of $U$ and $V$, see Properties of Spaces, Section 65.7). The case of schemes is Descent, Lemma 35.19.1. $\square$

Lemma 73.9.3. Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is locally Noetherian, then $Y$ is locally Noetherian.

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to X \times _ Y V$. As $f$ is surjective, flat, and locally of finite presentation the morphism of schemes $U \to V$ is surjective, flat, and locally of finite presentation. In this way we reduce the problem to the case of schemes (as being locally Noetherian for $X$ and $Y$ is defined in terms of being locally Noetherian of $U$ and $V$, see Properties of Spaces, Section 65.7). In the case of schemes the result follows from Descent, Lemma 35.16.1. $\square$

Lemma 73.9.4. Let $f : X \to Y$ be a morphism of algebraic spaces. If $f$ is locally of finite presentation, flat, and surjective and $X$ is regular, then $Y$ is regular.

Proof. By Lemma 73.9.3 we know that $Y$ is locally Noetherian. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. It suffices to prove that the local rings of $V$ are all regular local rings, see Properties, Lemma 28.9.2. Choose a scheme $U$ and a surjective étale morphism $U \to X \times _ Y V$. As $f$ is surjective and flat the morphism of schemes $U \to V$ is surjective and flat. By assumption $U$ is a regular scheme in particular all of its local rings are regular (by the lemma above). Hence the lemma follows from Algebra, Lemma 10.110.9. $\square$

Lemma 73.9.5. Let $f : X \to Y$ be a smooth morphism of algebraic spaces. If $Y$ is reduced, then $X$ is reduced. If $f$ is surjective and $X$ is reduced, then $Y$ is reduced.

Proof. Choose a commutative diagram

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

where $U$ and $V$ are schemes, the vertical arrows are surjective and étale, and $U \to X \times _ Y V$ is surjective étale. Observe that $X$ is a reduced algebraic space if and only if $U$ is a reduced scheme by our definition of reduced algebraic spaces in Properties of Spaces, Section 65.7. Similarly for $Y$ and $V$. The morphism $U \to V$ is a smooth morphism of schemes, see Morphisms of Spaces, Lemma 66.37.4. Since being reduced is local for the smooth topology for schemes (Descent, Lemma 35.18.1) we see that $U$ is reduced if $V$ is reduced. On the other hand, if $X \to Y$ is surjective, then $U \to V$ is surjective and in this case if $U$ is reduced, then $V$ is reduced. $\square$


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