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The Stacks project

Lemma 88.13.1. The property P(\varphi )=\varphi is flat” on arrows of \textit{WAdm}^{Noeth} is a local property as defined in Formal Spaces, Remark 87.21.5.

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 87.21.1. 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 87.19.14. Hence by Algebra, Lemma 10.39.9 we find that \varphi : A \to B is flat.

Proof of (3). Omitted. \square


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