## 86.13 Flat morphisms

In this section we define flat morphisms of locally Noetherian formal algebraic spaces.

Lemma 86.13.1. The property $P(\varphi )=$“$\varphi$ is flat” on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark 85.17.5.

Proof. Let us recall what the statement signifies. First, $\textit{WAdm}^{Noeth}$ is the category whose objects are adic Noetherian topological rings and whose morphisms are continuous ring homomorphisms. Consider a commutative diagram

$\xymatrix{ B \ar[r] & (B')^\wedge \\ A \ar[r] \ar[u]^\varphi & (A')^\wedge \ar[u]_{\varphi '} }$

satisfying the following conditions: $A$ and $B$ are adic Noetherian topological rings, $A \to A'$ and $B \to B'$ are étale ring maps, $(A')^\wedge = \mathop{\mathrm{lim}}\nolimits A'/I^ nA'$ for some ideal of definition $I \subset A$, $(B')^\wedge = \mathop{\mathrm{lim}}\nolimits B'/J^ nB'$ for some ideal of definition $J \subset B$, and $\varphi : A \to B$ and $\varphi ' : (A')^\wedge \to (B')^\wedge$ are continuous. Note that $(A')^\wedge$ and $(B')^\wedge$ are adic Noetherian topological rings by Formal Spaces, Lemma 85.17.1. We have to show

1. $\varphi$ is flat $\Rightarrow \varphi '$ is flat,

2. if $B \to B'$ faithfully flat, then $\varphi '$ is flat $\Rightarrow \varphi$ is flat, and

3. if $A \to B_ i$ is flat for $i = 1, \ldots , n$, then $A \to \prod _{i = 1, \ldots , n} B_ i$ is flat.

We will use without further mention that completions of Noetherian rings are flat (Algebra, Lemma 10.97.2). Since of course $A \to A'$ and $B \to B'$ are flat, we see in particular that the horizontal arrows in the diagram are flat.

Proof of (1). If $\varphi$ is flat, then the composition $A \to (A')^\wedge \to (B')^\wedge$ is flat. Hence $A' \to (B')^\wedge$ is flat by More on Flatness, Lemma 38.2.3. Hence we see that $(A')^\wedge \to (B')^\wedge$ is flat by applying More on Algebra, Lemma 15.27.5 with $R = A'$, with ideal $I(A')$, and with $M = (B')^\wedge = M^\wedge$.

Proof of (2). Assume $\varphi '$ is flat and $B \to B'$ is faithfully flat. Then the composition $A \to (A')^\wedge \to (B')^\wedge$ is flat. Also we see that $B \to (B')^\wedge$ is faithfully flat by Formal Spaces, Lemma 85.15.14. Hence by Algebra, Lemma 10.39.9 we find that $\varphi : A \to B$ is flat.

Proof of (3). Omitted. $\square$

Lemma 86.13.2. Denote $P$ the property of arrows of $\textit{WAdm}^{Noeth}$ defined in Lemma 86.13.1. Denote $Q$ the property defined in Lemma 86.12.1 viewed as a property of arrows of $\textit{WAdm}^{Noeth}$. Denote $R$ the property defined in Lemma 86.11.1 viewed as a property of arrows of $\textit{WAdm}^{Noeth}$. Then

1. $P$ is stable under base change by $Q$ (Formal Spaces, Remark 85.17.10), and

2. $P + R$ is stable under base change (Formal Spaces, Remark 85.17.9).

Proof. The statement makes sense as each of the properties $P$, $Q$, and $R$ is a local property of morphisms of $\textit{WAdm}^{Noeth}$. Let $\varphi : B \to A$ and $\psi : B \to C$ be morphisms of $\textit{WAdm}^{Noeth}$. If either $Q(\varphi )$ or $Q(\psi )$ then we see that $A \widehat{\otimes }_ B C$ is Noetherian by Formal Spaces, Lemma 85.4.16. Since $R$ implies $Q$ (Lemma 86.12.4), we find that this holds in both cases (1) and (2). This is the first thing we have to check. It remains to show that $C \to A \widehat{\otimes }_ B C$ is flat.

Proof of (1). Fix ideals of definition $I \subset A$ and $J \subset B$. By Lemma 86.12.5 the ring map $B \to C$ is topologically of finite type. Hence $B \to C/J^ n$ is of finite type for all $n \geq 1$. Hence $A \otimes _ B C/J^ n$ is Noetherian as a ring (because it is of finite type over $A$ and $A$ is Noetherian). Thus the $I$-adic completion $A \widehat{\otimes }_ B C/J^ n$ of $A \otimes _ B C/J^ n$ is flat over $C/J^ n$ because $C/J^ n \to A \otimes _ B C/J^ n$ is flat as a base change of $B \to A$ and because $A \otimes _ B C/J^ n \to A \widehat{\otimes }_ B C/J^ n$ is flat by Algebra, Lemma 10.97.2 Observe that $A \widehat{\otimes }_ B C/J^ n = (A \widehat{\otimes }_ B C)/J^ n(A \widehat{\otimes }_ B C)$; details omitted. We conclude that $M = A \widehat{\otimes }_ B C$ is a $C$-module which is complete with respect to the $J$-adic topology such that $M/J^ nM$ is flat over $C/J^ n$ for all $n \geq 1$. This implies that $M$ is flat over $C$ by More on Algebra, Lemma 15.27.4.

Proof of (2). In this case $B \to A$ is adic and hence we have just $A \widehat{\otimes }_ B C = \mathop{\mathrm{lim}}\nolimits A \otimes _ B C/J^ n$. The rings $A \otimes _ B C/J^ n$ are Noetherian by an application of Formal Spaces, Lemma 85.4.16 with $C$ replaced by $C/J^ n$. We conclude in the same manner as before. $\square$

Lemma 86.13.3. Denote $P$ the property of arrows of $\textit{WAdm}^{Noeth}$ defined in Lemma 86.13.1. Then $P$ is stable under composition (Formal Spaces, Remark 85.17.14).

Proof. This is true because compositions of flat ring maps are flat. $\square$

Definition 86.13.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. We say $f$ is flat if 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 a flat map of adic Noetherian topological rings.

Let us prove that we can check this condition étale locally on the source and target.

Lemma 86.13.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent

1. $f$ is flat,

2. 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 a flat map in $\textit{WAdm}^{Noeth}$,

3. there exists a covering $\{ Y_ j \to Y\}$ as in Formal Spaces, Definition 85.7.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\}$ as in Formal Spaces, Definition 85.7.1 such that each $X_{ji} \to Y_ j$ corresponds to a flat map in $\textit{WAdm}^{Noeth}$, and

4. there exist a covering $\{ X_ i \to X\}$ as in Formal Spaces, Definition 85.7.1 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 a flat map in $\textit{WAdm}^{Noeth}$.

Proof. The equivalence of (1) and (2) is Definition 86.13.4. The equivalence of (2), (3), and (4) follows from the fact that being flat is a local property of arrows of $\text{WAdm}^{Noeth}$ by Lemma 86.13.1 and an application of the variant of Formal Spaces, Lemma 85.17.3 for morphisms between locally Noetherian algebraic spaces mentioned in Formal Spaces, Remark 85.17.5. $\square$

Lemma 86.13.6. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally Noetherian formal algebraic spaces over $S$.

1. If $f$ is flat and $g_{red} : Z_{red} \to Y_{red}$ is locally of finite type, then the base change $X \times _ Y Z \to Z$ is flat.

2. If $f$ is flat and locally of finite type, then the base change $X \times _ Y Z \to Z$ is flat and locally of finite type.

Lemma 86.13.7. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ and $g$ are flat, then so is $g \circ f$.

Proof. Combine Formal Spaces, Remark 85.17.14 and Lemma 86.13.3. $\square$

Lemma 86.13.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ is representable by algebraic spaces and flat in the sense of Bootstrap, Definition 78.4.1, then $f$ is flat in the sense of Definition 86.13.4.

Proof. This is a sanity check whose proof should be trivial but isn't quite. We urge the reader to skip the proof. Assume $f$ is representable by algebraic spaces and flat in the sense of Bootstrap, Definition 78.4.1. Consider a 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. Then the morphism $U \to V$ corresponds to a taut map $B \to A$ of $\textit{WAdm}^{Noeth}$ by Formal Spaces, Lemma 85.18.2. Observe that this means $B \to A$ is adic (Formal Spaces, Lemma 85.19.2) and in particular for any ideal of definition $J \subset B$ the topology on $A$ is the $J$-adic topology and the diagrams

$\xymatrix{ \mathop{\mathrm{Spec}}(A/J^ nA) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(B/J^ n) \ar[d] \\ U \ar[r] & V }$

are cartesian.

Let $T \to V$ is a morphism where $T$ is a scheme. Then

\begin{align*} X \times _ Y T \to T\text{ is flat} & \Rightarrow U \times _ Y T \to T\text{ is flat} \\ & \Rightarrow U \times _ V V \times _ Y T \to T\text{ is flat} \\ & \Rightarrow U \times _ V V \times _ Y T \to V \times _ Y T\text{ is flat} \\ & \Rightarrow U \times _ V T \to T\text{ is flat} \end{align*}

The first statement is the assumption on $f$. The first implication because $U \to X$ is étale and hence flat and compositions of flat morphisms of algebraic spaces are flat. The second impliciation because $U \times _ Y T = U \times _ V V \times _ Y T$. The third implication by More on Flatness, Lemma 38.2.3. The fourth implication because we can pullback by the morphism $T \to V \times _ Y T$. We conclude that $U \to V$ is flat in the sense of Bootstrap, Definition 78.4.1. In terms of the continuous ring map $B \to A$ this means the ring maps $B/J^ n \to A/J^ nA$ are flat (see diagram above).

Finally, we can conclude that $B \to A$ is flat for example by More on Algebra, Lemma 15.27.4. $\square$

## 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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GC8. Beware of the difference between the letter 'O' and the digit '0'.