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I am looking for a stronger notion of generic flatness. Let $A$ be a Noetherian ring, $M$ a finitely generated module over $A$. Suppose there exists a maximal ideal $m$ of $A$ such that $M_m$ (the localization of $M$ at $m$) is $A_m$-flat (here $A_m$ is the localization of $A$ at $m$). Does there exist an open neighbourhood $U$ of $\mathrm{Spec}(A)$ containing the point corresponding to $m$ such that for all point (not necessarily closed) $u \in U$, the localization of $\tilde{M}$ (the coherent sheaf corresponding to the module $M$) at $u$ is $\mathcal{O}_{U,u}$-flat i.e., $\tilde{M}|_U$ is $\mathcal{O}_U$-flat?

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  • $\begingroup$ We don't need that $m$ is maximal. Any prime ideal works. $\endgroup$ – Martin Brandenburg Dec 2 '14 at 19:12
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$M$ is finitely presented and $M_m$ is free of finite rank (over a local ring a finitely presented module flat is already free!). Hence, it is locally free in a neighborhood of $m$:

Lemma: Let $X$ be a ringed space and $M$ an $\mathcal{O}_X$-module of finite presentation. If $x \in X$ is a point such that $M_x$ is a free of finite rank, then there is an open neighborhood $U$ of $x$ such that $M|_U$ is free of finite rank.

Proof: Since $M$ is of finitely presentation, we have $\underline{\hom}(M,N)_x = \hom(M_x,N_x)$ for all $N$, and the same holds for $\mathcal{O}_X^n$. Hence, the isomorphism $M_x \cong \mathcal{O}_{X,x}^n$ lifts to some open neighborhood of $x$. $\checkmark$

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