# Reference request for Cartier Duality of algebraic tori

Hi,

I need a reference for the following result:

Let $S$ be a scheme and let $X$ be an algebraic torus over $S$. Then the functor $F_X :S'\mapsto Hom_{S'}(X\times S',\mathbb{G}_M\times S')$ is representable by an $S$- group scheme $Y$ which is locally etale isomorphic to $\mathbb{Z}^n$. Furthermore, the similarly defined functor $F_Y$ is represented by $X$.

This result can probably be found somewhere in SGA7. However I have very basic understanding of French and the style in which SGA is written makes it practically impossible for me to find the exact reference. Are there any other references (in English or in simpler French) which one could use? If not, do you know exactly which statements in SGA7 imply the Cartier duality theorem?

Thanks in advance, your help will be very much appreciated.

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## 1 Answer

I believe you are looking for SGA3 Exp. X, Corollary 5.7.

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That is exactly what I was looking for. Thanks a lot! –  Tzanko Matev May 31 '12 at 8:54
SGA3 does sound more plausible than SGA7. Note that the lightly edited Gille-Polo version of SGA3 is conveniently available online: people.math.jussieu.fr/~polo/SGA3 –  Jim Humphreys May 31 '12 at 10:38
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