The Stacks Project

Tag: 06DP

This tag has label spaces-descent-section-descending-properties-spaces, it is called Descending properties of spaces in the Stacks project and it points to

The corresponding content:

51.8. Descending properties of spaces

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

Lemma 51.8.1. 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 45.27.8 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 9.36.17). $\square$

Lemma 51.8.2. 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 44.7). The case of schemes is Descent, Lemma 31.15.1. $\square$

Lemma 51.8.3. 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 44.7). In the case of schemes the result follows from Descent, Lemma 31.12.1. $\square$

Lemma 51.8.4. 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 51.8.3 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 24.9.2. 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 9.105.8. $\square$

\section{Descending properties of spaces}
\label{section-descending-properties-spaces}

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

\begin{lemma}
\label{lemma-descend-unibranch}
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 $f(x)$.
\end{lemma}

\begin{proof}
Consider the map of \'etale local rings
$\mathcal{O}_{Y, f(\overline{x})} \to \mathcal{O}_{X, \overline{x}}$.
By
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}
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 \ref{algebra-lemma-flat-going-down}).
\end{proof}

\begin{lemma}
\label{lemma-descend-reduced}
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.
\end{lemma}

\begin{proof}
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$.
Choose a scheme $U$ and a surjective \'etale 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 \ref{spaces-properties-section-types-properties}).
The case of schemes is
Descent, Lemma \ref{descent-lemma-descend-reduced}.
\end{proof}

\begin{lemma}
\label{lemma-descend-locally-Noetherian}
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.
\end{lemma}

\begin{proof}
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$.
Choose a scheme $U$ and a surjective \'etale 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 \ref{spaces-properties-section-types-properties}).
In the case of schemes the result follows from
Descent, Lemma \ref{descent-lemma-Noetherian-local-fppf}.
\end{proof}

\begin{lemma}
\label{lemma-descend-regular}
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.
\end{lemma}

\begin{proof}
By
Lemma \ref{lemma-descend-locally-Noetherian}
we know that $Y$ is locally Noetherian.
Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$.
It suffices to prove that the local rings of $V$ are all regular local
rings, see
Properties, Lemma \ref{properties-lemma-characterize-regular}.
Choose a scheme $U$ and a surjective \'etale 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 \ref{algebra-lemma-flat-under-regular}.
\end{proof}


To cite this tag (see How to reference tags), use:

\cite[\href{http://stacks.math.columbia.edu/tag/06DP}{Tag 06DP}]{stacks-project}


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 lower-right corner).