I am going over the proof of Serre duality for a coherent sheaf on a projective k-scheme. First, I am trying to understand it for the case where $X = \mathbb{P}_{k}^{n}$. I am following Hartshorne III Proposition 7.1. Let $\mathcal{F}$ be a coherent sheaf on $X$ and let $\omega_{X} = \mathcal{O}(-n-1) $. I want to exhibit an isomorphism, $$ \text{Hom}(\mathcal{F}, \omega_{X}) \stackrel{\simeq}{\longrightarrow} H^{n}(X, \mathcal{F})^{*} $$ where $*$ denotes the vector space dual. Fixing a particular isomorphism $H^{n}(X,\omega_{X}) \simeq k$, this is equivalent to finding an isomorphism, $$ \text{Hom}(\mathcal{F}, \omega_{X}) \stackrel{\simeq}{\longrightarrow} H^{n}(X, \mathcal{F})^{*}. $$ The problem is that it is not completely clear to me what these $\text{Hom}$ sets even are. Indeed we can view the cohomology object $H^{n}(X, \mathcal{F})$ in three ways: An abelian group (module over $\mathbb{Z}$), a $\Gamma(X, \mathcal{O}_{X})$-module, or a $k$-vector space. Now in my particular case, $\Gamma(X, \mathcal{O}_{X}) = k$, but for a general projective scheme my understanding is that this could be a finite extension of $k$?
In any case, if I proceed as follows, $$ \text{Hom}_{\mathcal{O}_{X}}(\mathcal{F}, \omega_{X}) \longrightarrow \text{Hom}(H^{n}(X, \mathcal{F}), H^{n}(X, \omega_{X})) $$ where a morphism of $\mathcal{O}_{X}$-modules induces a morphism of cohomology groups. So the Hom set on the right is of abelian groups. But we want it as a morphism of vector spaces. But there is no reason to think the extension of scalars $\mathbb{Z} \rightarrow k$ will be fully faithful. This problem is worse when we are dealing with a projective scheme, since we also have modules over $\Gamma(X, \mathcal{O}_{X})$.
So my question can really be summarised as: What is the correct subscript on each of the Hom sets involved here? Hartshorne just seems to gloss over all this.