In a question here on why the classification of commutative Hopf algebras requires the field to be algebraically closed, a brilliant answer is given discussing the notion of Galois descent.
As I understand (having no algebraic number theory background beyond knowing Galois theory and group cohomology separately), a Hopf algebra over $L$ with $L/k$ an extension of $k$ can be obtained by starting with the Hopf algebra over $k$ and then tensoring up to $L$. However, the subtlety is that there can exist algebras over $k$ such that going up to $L$ gives something isomorphic to $H$ over $L$.
I understand this as analogous to the fact that $M_2(\mathbb C) = M_2(\mathbb R) \otimes_{\mathbb R} \mathbb C$ but we also have $M_2(\mathbb C) \cong \mathbb H \otimes_{\mathbb R} \mathbb C$, where $\mathbb H$ are Hamilton's quaternions.
As such my questions are:
- Is there a cohomology that classifies for a given $H$ over $L$, all the objects on $k$ that "tensor up" to $H$?
- Is there a (gentle) reference that explores these ideas of Galois descent for Hopf algebras?
I should say that my background in terms of category theory (should anyone use this to give an answer) is limited to that of Hartshorne's algebraic geometry, and not a category theory textbook.