flatness in the category of quasi coherent sheaves

Question

Let $X$ be an scheme and $F$ be a quasi coherent sheaf on $X$. There are three ways to consider flatness in the category of quasi coherent sheaves Qco($X$). 1- $F$ is called flat if $F_x$ is flat for each $x$ in $X$. (This definition is more usual) 2- $F$ is flat if $F(u)$ is a flat $O(U)$-module for each open affine subset $U$ of $X$. 3-(categorical definition) In the case that $X$ is semi-separated quasi-compact scheme, Qco($X$) is a locally finitely presented category. So one can think of flat objects via purity. $F$ is called flat if for each exact sequence $$0\to M\to N\to F\to 0$$ of quasi coherent sheaves, the induced sequence $$ 0\to Hom(P,M)\to Hom(P,N) \to Hom(P,F)\to 0$$ of abelian groups is exact for each finitely presentedquasi coherent sheaf $P$.

\remark: Note that when $X$ is semi separated and quasi compact, $P$ is finitely presented if and only if it is coherent.(Murfet notes, therisingsea.org)

IS there any relation between these classes of quasi-coherent sheaves?

Answer 1

It is well-known that flatness of usual modules over rings can be checked stalkwise. This easily implies that (1) and (2) are equivalent. Another quite natural condition concerns the structure of $\mathsf{Qcoh}(X)$ as an abelian symmetric monoidal category: (4) $F$ is flat if $F \otimes -$ is an exact functor. Using the compatibility between stalks and tensor products it is easily seen that (1) implies (4).

I claim that (4) implies (2) if $X$ is a quasi-separated scheme $X$. In fact, assume (4) and let $U \subseteq X$ be an open affine subset and $M \to N$ a monomorphism of quasi-coherent modules on $U$. Let $j : U \to X$ denote the open immersion. It is quasi-compact (since $X$ is quasi-separated) and quasi-separated, hence $j_{*} M \to j_{*} N$ is a monomorphism of quasi-coherent modules on $X$. It follows that $F \otimes j_* M \to F \otimes j_* N$ is a monomorphism. Applying $j^*$, we find that $F|_U \otimes M \to F|_U \otimes N$ is a monomorphism. Hence, $F|_U$ is flat (i.e. $F(U)$ is flat since $U$ is affine).

By the way, if $X$ is a quasi-compact quasi-separated scheme, then $\mathsf{Qcoh}(X)$ is locally finitely presentable, and "finitely presentable" defined categorically coincides with "of finite presentation" (this follows from results in EGA I (new), 6.9, I have proven this in my diploma thesis, should I make this public?). So we don't need that $X$ is semi-separated.

Probably one can also show that (2) and (4) coincide in this setting. Namely, (2) => (4) is trivial from the affine case and the converse uses, again, extension of quasi-coherent modules along open immersions.