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Is there anything in the literature on generalizing Cartier duality to non-finite group schemes? Pointers would be welcome.
I have seen some reference days ago, but having read so much last one month it is difficult to remember. I think it was about the case of pro-finite groups.
Thanks. I’d need it specifically for elliptic curves over rings. I see that there is a book titled “Elliptic Curves and Arithmetic Invariants” which has a section 6.1.6 that sounds like it should talk about Cartier duality of elliptic curves over rings. But I cannot open that book right now (am on a train with shaky and slow connection…)
Sorry, I am sure my reading did not cover that case…you need some expert…
Theorem 4.63 of that book discusses Cartier duality for locally free commutative group schemes (see also here in the author’s other book, which I suspect is a copy/paste on the part of the author), but I suspect this doesn’t cover the case you need.
Thanks for offering help, David!
Yeah, this doesn’t cover the case I was looking for, this is still finite group schemes (over $Spec A$).
Hm, maybe what I was hoping for does not exist…
The page Dual abelian variety has the statement
The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other.
but again that’s not what you want.
I see now that Cartier duality for groups more general than finite group schemes is discussed here:
Generalization beyond finite group schemes is discussed in
and in
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