The tangent bundle of an abelian variety $A/K$ is trivial. Mumford's proof of this fact goes something like this: given the (non-zero) value of a section in a fiber, one can determine the value in all of the fiber by the group law. This is the same as the proof for Lie groups.
But this seems somewhat incomplete to me, since in the algebraic geometry setting one specifies a section by stalks, rather than values in the residue field at each stalk. So my question is, how does one actually show that knowing all these values gives you a section with those values?