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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2013
   • (edited Nov 11th 2013)
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   Author: Urs
   Format: MarkdownItexIs there anything in the literature on generalizing [[Cartier duality]] to non-finite group schemes? Pointers would be welcome.

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

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeNov 11th 2013
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   Author: zskoda
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 11th 2013
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   Author: Urs
   Format: MarkdownItexThanks. 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...)

   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…)

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeNov 11th 2013
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   Author: zskoda
   Format: MarkdownItexSorry, I am sure my reading did not cover that case...you need some expert...

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

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 12th 2013
   • (edited Nov 12th 2013)
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   Author: DavidRoberts
   Format: MarkdownItexTheorem 4.63 of that book discusses Cartier duality for locally free commutative group schemes (see also [here](https://books.google.com.au/books?id=z1orIlsissgC&pg=PA45) 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 12th 2013
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   Author: Urs
   Format: MarkdownItexThanks 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...

   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…

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 12th 2013
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   Author: DavidRoberts
   Format: MarkdownItexThe page [Dual abelian variety](http://en.wikipedia.org/wiki/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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeNov 14th 2013
   • (edited Nov 14th 2013)
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   Author: Urs
   Format: MarkdownItexI 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](http://arxiv.org/abs/0709.3735)) and in * [[Dima Arinkin]], appendix in [[Ron Donagi]], [[Tony Pantev]], _Torus fibrations, gerbes, and duality_, ([arXiv:math/0306213](http://arxiv.org/abs/math/0306213))

   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

   • Dima Arinkin, appendix in Ron Donagi, Tony Pantev, Torus fibrations, gerbes, and duality, (arXiv:math/0306213)