Let $X$ be a Noetherian scheme and $F$ be a pure coherent sheaf of $O_X$-modules, that is any non-zero subsheaf $E\subset F$ satisfies $dim(E)=dim(F)$ (by dimension of a sheaf, I mean the dimension of its support).
It is known that a $F$ is pure if and only if all its associated points have the same dimension (the dimension of $F$). A proof can be found in Alain Leytem thesis (https://orbilu.uni.lu/handle/10993/23380), Theorem 3.1.11. I understand the proof in the local case, but I'm wondering how to go from the global to the local case. More precisely:
Is a sheaf $F$ pure if and only if $F|_U$ is pure for all affine subset $U\subset X$ ?
The if part is clear. Now, assume $F$ is pure. If it admits a subsheaf $K\subset F|_U$ on some affine open $U\subset X$, is it possible to extend $K$ to a subsheaf $K^{'}\subset F$ on the whole scheme $X$ ? Maybe a better way to proceed is to say that the generic point $\eta$ of $Supp(K)$ is associated to $F|_U$, and therefore $\eta$ is also associated to $F$. Then one needs to show how to associate a subsheaf of $F$ to an associated point of $F$.