Let $X$ be $n$-dimensional scheme of finite type over an algebraically closed field. In the proof of Proposition 7.4.11 in Gathmann, the first paragraph reads
If $\Omega_X$ is locally free of rank $n$ then its fibers at any point $P$, i.e. the cotangent spaces $T_{X,P}^\vee$, have dimension $n$.
Here $\Omega_X$ is the sheaf of relative differentials on $X$ (relative to $\operatorname{Spec} k$).
I don't see why this is true. My understanding is that (a) the cotangent space $T^\vee_{X,p}$ is the Zariski cotangent space of $\mathscr O_{X,p}$ (i.e. $\mathfrak m/ \mathfrak m^2$) and (b) the fiber of a locally free sheaf $\mathscr F$ at the point $P$ is given by $i^\ast \mathscr F$, where $i : P \hookrightarrow X$ is inclusion. I can't see why these are isomorphic.