Timeline for Galois descent of a Hopf algebra
Current License: CC BY-SA 4.0
10 events
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| Mar 9, 2021 at 19:43 | comment | added | Mikhail Borovoi | The idea of Galois cohomology is as follows: If you have a $k$-object: a Lie algebra, or Hopf algebra, etc. $X$ over $k$, then you consider $X_L:=X\otimes _k L$, and you consider the automorphism group $A={\rm Aut}_L(X):={\rm Aut}_L(X_L)$. The Galois group $\Gamma:={\rm Gal}(L/k)$ acts on $A$. Then the set of isomorphism classes of $k$-objects $Y$ such that $Y_L\simeq X_L$ is in a canonical bijection with the Galois cohomology set $H^1(\Gamma,A)$. | |
| Mar 9, 2021 at 19:31 | comment | added | Mikhail Borovoi | Concerning Galois cohomology: See Section III.1.1 of Serre's excellent book: Jean-Pierre Serre, Galois Cohomology, Springer-Verlag, Berlin, 1997. | |
| Mar 9, 2021 at 19:20 | comment | added | Mikhail Borovoi | See also this my question | |
| Mar 9, 2021 at 19:13 | comment | added | Mikhail Borovoi | 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. | |
| Mar 9, 2021 at 13:18 | history | edited | JamalS | CC BY-SA 4.0 |
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| Mar 8, 2021 at 18:32 | comment | added | Achim Krause | 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 $\mathbb{R}[X]/(X^2\pm 1)$. | |
| Mar 8, 2021 at 18:24 | comment | added | JamalS | @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. | |
| Mar 8, 2021 at 18:19 | comment | added | Achim Krause | 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. | |
| Mar 8, 2021 at 18:17 | comment | added | Achim Krause | 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... | |
| Mar 8, 2021 at 15:46 | history | asked | JamalS | CC BY-SA 4.0 |