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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 11th 2013
• (edited Nov 11th 2013)

Is there anything in the literature on generalizing Cartier duality to non-finite group schemes? Pointers would be welcome.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeNov 11th 2013

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 11th 2013

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…)

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeNov 11th 2013

Sorry, I am sure my reading did not cover that case…you need some expert…

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeNov 12th 2013
• (edited Nov 12th 2013)

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 12th 2013

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…

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeNov 12th 2013

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 14th 2013
• (edited Nov 14th 2013)

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

• Amelia Álvarez Sánchez, Carlos Sancho de Salas, Pedro Sancho de Salas, _Functorial Cartier duality (arXiv:0709.3735)

and in