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 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!

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!