Lemma 88.13.8. 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 80.4.1, then f is flat in the sense of Definition 88.13.4.
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 80.4.1. 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 87.22.2. Observe that this means B \to A is adic (Formal Spaces, Lemma 87.23.1) 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 80.4.1. 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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