I am trying to prove Theorem 11.3.7 (no proof or reference is left in the text) of Edixhoven and Taelman lectures on Algebraic Geometry. The main goal of these lectures is to introduce algebraic geometry by proving Weil's conjectures for curves.
Here's some context: Let $\mathbb{F}_q$ be the field with $q$ elements and let $\mathbb{F}_q \to \mathbb{F}$ be an algebraic closure. Let $X$ be a smooth irreducible projective curve over $\mathbb{F}_q$, which for us means that $X$ is an irreducible smooth algebraic set in some $\mathbb{P}^m(\mathbb{F})$ of dimension 1 defined by some homogeneous polynomials in $\mathbb{F}_q[x_0,\dots,x_m]$. Then the theorem asserts that for any divisor $D$ on $X$
$$ \dim_{\mathbb{F}_q} H^0(X,D)^\sigma = \dim_{\mathbb{F}}H^0(X,D). $$
where $$ H^0(X,D) = \lbrace f \in K(X) : f=0 \text{ or } \operatorname{div}(f)+D \geq 0\rbrace $$ and $$ H^0(X,D)^\sigma = \lbrace f \in K(X) : f\text{ has coefficients in } \mathbb{F}_q \rbrace \cap H^0(X,D).$$
Is this trivial? Am I missing something? I am not very familiar with the language of schemes or cohomology. Thanks in advance!