Question (global): Is it possible to extend sections of quasi-coherent sheaves on locally Noetherian normal schemes across codimension 2 sets? Specifically, if $X$ is a locally Noetherian normal scheme, and $F$ is a quasicoherent sheaf, and $s \in \Gamma(U,F)$, where $U^c$ has codimension $\geq 2$, does there exist a section $s\ \in F(X)$ that restricts to $s$?
I know it can be done for locally free sheaves on a normal locally Noetherian scheme $X$. Can it be done under these weaker conditions? I think it is equivalent to ask if partially defined morphisms between quasi-coherent sheaves over $X$ can be extended across codimension 2 sets, since on the one hand a section is the same as a map from the structure sheaf, and on the other hand a morphism is a section of the sheaf hom (when it is quasi-coherent).
Under the hypothesis that $F \to K(X) \otimes_{O_X} F$ (the rational sections of $F$) is injective, i.e. the extension is unique, it suffices to consider the affine case, and then glue together the extended sections.
Question (affine): Given a module $M$ over some integrally closed Noetherian domain $A$, so that $M \subset K(A) \otimes_A M$, is it true that $M = \cap M_P$, where $P$ runs over all codimension 1 primes of $A$?
(In the terminology that user26857 brought up, I think I am asking under what conditions $M$ is a divisorial $A$-lattice.)
(Vague thoughts: We are given a surjection $\phi : A^I \to M \to 0$. Tensoring is right exact, so we still have $\phi_P : A^I_P \to M_P \to 0$, which are all restrictions of $\phi_K : K(A) \otimes_A A^I \to K(A) \otimes_A M \to 0$, so $A^I = \cap_P (A^I_P)$ maps into $\cap_P M_P$. Can this be made into an onto map? If $M$ is assumed to be finitely generated and A Noetherian, then we have a diagram of exact sequences $A^n_P \to A^m_P \to M_P \to 0$ for each $P$. The intersections are the limits of each term inside the category of submodules of $K(A)^n$, $K(A)^m$ and $K(A) \otimes_A M$ respectively. I'm not sure under what conditions on can guarantee that taking such a limit respects right exactness. Certainly not in general, since that would imply some set theoretic fact about images of intersections that is not true. Maybe I am just running around in circles here...)
How can $0 \to F \to K(X) \otimes_{O_X} F$ be guaranteed? Is it necessary?
Edit: https://en.wikipedia.org/wiki/Reflexive_sheaf is relevant. In particular, "A coherent sheaf $F$ is said to be normal in the sense of Barth if $F(U) \to F(U \setminus Y)$ is bijective for every open set $U$ and closed subset $Y$ of $U$ of codimension at least $2$. A coherent sheaf on an integral normal scheme is reflexive iff it is torsion free and normal in the sense of Barth."
Here reflexive means that the natural map from a sheaf to its double dual is an isomorphism.