## 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. $\square$

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