Timeline for Hopf-Galois Structure Maps
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 7, 2016 at 21:08 | comment | added | Theo Johnson-Freyd | Sure, but that might not be the same $A$-module structure as the one on $B \otimes H$ in which you just act on $B$. I mean, ignoring that these are algebras, if $B$ is an $A$-module and $H$ is a ground-ring-module, then $B\otimes H$ is an $A$-module in a canonical way. I assume you want that one for your application, but I don't know. | |
| Jan 3, 2016 at 21:45 | comment | added | Jonathan Beardsley | Hey Theo, sorry to bug you about this again, but I was thinking about it, and isn't the map $B\to B\otimes H$ always $A$-linear so long as we take the $A$-module structure on $B\otimes H$ given by the composition $A\to B\to B\otimes H$? | |
| Dec 29, 2015 at 4:28 | vote | accept | Jonathan Beardsley | ||
| Dec 15, 2015 at 5:07 | comment | added | Theo Johnson-Freyd | Coring structures like this are important, though, in the theory of "quantum groupoids". Probably this is what you're working in anyway, given the Hopf-Galois stuff that appears there. | |
| Dec 15, 2015 at 5:06 | comment | added | Theo Johnson-Freyd | Your coring structure on $B\otimes H$ is the coring structure on $H$ base-changed to $B$. Your "coring" structure on $B \otimes_A B$ is funny, though. You say it is a "$B$-coring structure", but you use two different $B$-module structures on $B \otimes_A B$. It is a coalgebra object in the monoidal category of $B$-$B$-bimodules, but not in the category of $B$-modules (which is monoidal if $B$ is commutative). The category of bimodules is never symmetric (except in trivial cases), and so for example it is not meaningful to ask that this coring be cocommutative. | |
| Dec 15, 2015 at 5:02 | comment | added | Theo Johnson-Freyd | So it really doesn't matter what coring structure you have in mind: either the map is a morphism, or it isn't, and in neither case is asking that it be an iso any better than asking that (it be a morphism and) it be a bijection. | |
| Dec 15, 2015 at 5:01 | comment | added | Theo Johnson-Freyd | @JonBeardsley Just find-change "vector space" for "object in your underlying category" (e.g. "abelian group") and you get the same statement --- for pretty much any world you might hope to work in, isomorphism of rings/modules/... is detected on underlying objects. | |
| Dec 14, 2015 at 3:55 | comment | added | Jonathan Beardsley | Honestly I guess I can write down a whole bunch of diagrams and say the necessary things about them (good catch on the $A$-linearity by the way), but I kind of wonder why they don't seem to be built into the definitions. Also, I'm not necessarily working over a field (re: your statement about vector spaces). | |
| Dec 14, 2015 at 3:51 | comment | added | Jonathan Beardsley | Thanks Theo! I'd like to use as few commutativity conditions as possible, but it seems like there's a bit of give and take. The coring structure I had in mind was that $B\otimes_AB$ is a $B$-coring by $B\otimes_A A\otimes_AB\to B\otimes_A B\otimes_AB$, and $B\otimes H$ should be a coring by the diagonal map of $H$, i.e. $B\otimes H\to B\otimes H\otimes H\cong B\otimes H\otimes_B B\otimes H$. | |
| Dec 14, 2015 at 3:44 | history | answered | Theo Johnson-Freyd | CC BY-SA 3.0 |