# Generic flatness and closed subscheme

Let $f:X\rightarrow S$ be a finite type morphism of noetherian schemes and let $S$ be integral. Let $F$ be a coherent sheaf on $X$.

Let $T$ be an integral closed subscheme of $S$. Then we can apply generic flatness to $f^*F$ on $X_T\rightarrow T$. Then there is an open dense set $V\subset T$ such that $f^*F|_V$ is flat over $V$.

Can we find an open set $U$ of $S$ such that $F|_U$ is flat over $U$ and $V\subset U$?

Edit:

I had another question about flat families that I am adding to this. If $W$ is the maximal subset of $S$ such that $F|_W$ is $W$-flat. Then is $W$ open in general? I think not. But what if $f$ is a faithfully flat morphism?

• No. Let $S$ be $\text{Spec}\ k[y]$. Let $X$ equal $S$. Let $f$ be the identity map. Let $F$ be the coherent sheaf associated to the $k[y]$-module $k[y]/\langle y \rangle$. Let $T$ be the closed subscheme associated to the prime ideal $\langle y \rangle$. Then $f^*F$ is flat on $T$, which is a singleton set. Yet the maximal open subscheme $U$ of $S$ over which $F$ is flat is the open complement of $T$. Having said that, the notion of "flattening stratification" almost does what you want. – Jason Starr Mar 30 '17 at 11:56
• @JasonStarr, thanks. Can you please elaborate on the comment -the notion of flattening stratification almost does what you want. – user349424 Mar 30 '17 at 12:55
• When $f$ is proper, the flattening stratification of $F$ is a finite type, separated morphism $i:\Sigma \to S$ that is bijective on points and is a disjoint union of locally closed subschemes such that (i) $i^*F$ is $\Sigma$-flat on $X_\Sigma\to \Sigma$, and (ii) for every $f:T\to S$, $f^*F$ is $T$-flat on $X_T$ if and only if $f$ factors through $i$ (in which case the factorization is unique). In particular, if $T$ is integral, then there exists a unique stratum $U$ of $\Sigma$ such that $f^{-1}(U)$ contains a dense open subset $V$ of $T$. – Jason Starr Mar 30 '17 at 14:12