Let $X$ be a regular, irreducible projective scheme, of finite type over an arbitrary field $k$. Both Weil divisors and Cartier divisors are defined on $X$ and naturally correspond. If $k$ is algebraically closed (but of arbitrary characteristic), then we know (Hartshorne, Prop II.7.7) that global sections of $\mathcal{O}\_X(D)$ correspond to effective divisors linearly equivalent to $D$. Is the same thing true for schemes such as $X$ above?


In particular, if $k=\bar{k}$ we can tell that $D$ is not effective if $\dim\_k \Gamma (\mathcal{O}\_X(D) = 0$. In general, do we at least know that $D$ is linearly equivalent to an effective divisor iff $\dim\_k \Gamma (\mathcal{O}\_X(D) \ne 0$.


It doesn't seem to be very hard to just repeat Hartshorne's proof to a more general setting, but I would also like to know what the most general versions of this correspondence are. For example, what if $X$ is only proper over $k$, or maybe over something like $\textrm{Spec } \mathbb{Z[1/2]}$. Is essentially the same thing true - or is it actually false? Any references will be very appreciated!



Linear systems over non-algebraically closed field
Dear LMN, I think the answer to all your questions is *yes*. Everything is true. 


See for example, Proposition 2.9 in Hartshorne's *Generalized divisors on Gorenstein schemes* and for some corrections, see Remark 2.9 Hartshorne's *Generalized divisors and biliaison*.


Here's a statement (more general versions can be found in those references). This version is basically 2.9 in the first reference.


**Theorem:** *if $X$ is a normal Noetherian separated scheme, then for any Weil divisor $D$ on $X$, there is a 1-to-1 correspondence between non-degenerate global sections* $s \in \Gamma(X, O\_X(D))$, *modulo sections of* $\Gamma(X, O\_X^\*)$, *and the set of effective divisors $D' \sim D$.*


Here *non-degenerate* means not being contained in any prime associated to a generic point of any irreducible component of $X$ if I recall correctly.


In particular, I think the reason why Hartshorne generally assumed that $k = \overline{k}$ is that in these cases, these global sections can be used to induce maps to various projective spaces easily.


