# Extending sections of quasi-coherent sheaves on locally Noetherian normal schemes across codimension 2 sets?

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.

• I recommend that you delete this question and repost an identical version on MathOverflow. This seems "research level" to me! – Pete L. Clark Dec 27 '15 at 7:48
• I can't see any reason for $M\subset M_P$, in general. (You have to consider torsion-free modules.) People are considering $M$ an $A$-lattice, and the condition you are looking for is a definition for divisorial lattices, so this condition doesn't hold even for such a restrictive case. – user26857 Dec 27 '15 at 8:56
• @user26857 It holds in the particular case I was thinking about, which is the $O_X$ submodule of $K(X)$ associated to a Weil divisor (on an integral normal locally Noetherian scheme). But you are right and I will edit that into the hypothesis explicitly. Thanks for the keyword. – Lorenzo Dec 27 '15 at 20:13
• @AreaMan, I can move it to MO, which is better than making copy. – Mariano Suárez-Álvarez Dec 27 '15 at 20:18
• @MarianoSuárez-Alvarez There was a pretty silly set-theoretic mistake in my argument (which I guess was generating my feelings of suspicion), but I think I see how to fix it in the f.g case (provided it is true) (we have a diagram of exact sequences $A^n_p \to A^m_p \to M_p \to 0$ for each $p$, then I would like to apply some general nonsense to the limit). I'd like some time to think about what exactly I want to ask. Thanks. – Lorenzo Dec 27 '15 at 20:47