Mumford in his book on Abelian Varieties gives the following argument to compute the sheaf of differentials on an Abelian Variety:
Let $X/k$ be an Abelian Variety over a field with identity $0$. Let $\Omega_0$ be the cotangent vector space at $0$, then we want to show that the map $\Omega_0 \otimes_k \mathscr O_X \to \Omega_X$ is an isomorphism.
He defines the map by saying: For each $\theta \in \Omega_0$, define the image to be the unique $1$-form $\omega_\theta$ defined by $(\omega_{\theta})_x = m_{-x}^*\theta$ where $m_{-x}$ is translation by $-x$. He says that it is easily checked that $\omega_{\theta}$ is regular and completes the argument by proving that it is an isomorphism on the fibers and appealing to Nakayama.
Questions:
What would be the modern, scheme theoretic translation of the way in which he defines the map/ $\omega_{\theta}$? In particular, under what conditions can we define a sheaf fibrewise as Mumford does here and be guaranteed that we do indeed get a sheaf?
Just to be clear about what I am asking: I have seen the result proved in other places in modern language and I understand these proofs. However, I would like to also understand how to translate the intuitive, differential geometry-ish proof of Mumford above into scheme theory language.