# Flattening techniques of Raynaud and Gruson

Suppose $R$ is an adic valuation ring with a finitely generated ideal of definition. Let $A$ be an $R$-algebra of topologically finite type, i.e. $A$ is isomorphic to $R<\zeta_1,\zeta_2,...,\zeta_n>/\mathfrak{a}$, where $\mathfrak{a}$ is an ideal of the restricted power series $R$-algebra $R<\zeta_1,\zeta_2,...,\zeta_n>$.

Claim: If $A$ is flat over $R$, then $A$ is in fact of topologically finite presentation, i.e. we can assume $\mathfrak{a}$ to be finitely generated.

What would be the idea of proving this? How to understand the flatness condition here?

Thanks!

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Have you looked in the early parts of the paper "Formal and Rigid Geometry I" by Bosch and L\"utkebohmert, or perhaps the part II sequel? It is addressed in one or both of those papers for sure. –  user28172 May 28 '13 at 23:29