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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.

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  • $\begingroup$ The general idea is "Thing over $k$" $\simeq$ "Thing over $L$ together with a $Gal(L/K)$-semilinear $Gal(L/K)$-Action." For example, the category of $\mathbb{R}$ vector spaces is equivalent to the category of $\mathbb{C}$ vector spaces with complex-antilinear involution, and this equivalence is symmetric-monoidal, thus carries over to equivalences on Lie algebras, Hopf algebras, group representations... $\endgroup$
    – Achim Krause
    Commented Mar 8, 2021 at 18:17
  • $\begingroup$ So in general the question whether something descends from your big field to the smaller field is about existence of such an action (the "descent datum"), and different ways to descend correspond to nonequivalent such choices. I don't have a reference handy (I learned this from homotopy theorists :) ) and so I leave this as comment, other people are probably better suited to give a full answer. $\endgroup$
    – Achim Krause
    Commented Mar 8, 2021 at 18:19
  • $\begingroup$ @AchimKrause so roughly speaking, depending on how I choose the $\mathrm{Gal}(L/K)$ action, I will descent to something different? I appreciate you taking the time to comment. $\endgroup$
    – JamalS
    Commented Mar 8, 2021 at 18:24
  • $\begingroup$ Yes, toy example: if you descend algebras over $\mathbb{C}$ to $\mathbb{R}$, you can put two different such descent data on the algebra $\mathbb{C} [X] /(X^2-1)$, corresponding to $\mathbb{R}[X]/(X^2\pm 1)$. $\endgroup$
    – Achim Krause
    Commented Mar 8, 2021 at 18:32
  • $\begingroup$ The best reference on Galois descent that I know: Jörg Jahnel, The Brauer-Severi variety associated with a central simple algebra: a survey, Preprint server: Linear Algebraic Groups and Related Structures, no. 52, 2000. $\endgroup$
    – Mikhail Borovoi
    Commented Mar 9, 2021 at 19:13

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