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Let $F$ be a non-archimedean local field, and $\mathscr O$ its ring of integers. Suppose $T$ is an $F$-split torus, i.e., $T = (\mathbb G_m)^r$ where $\mathbb G_m$ denotes the multiplicative group. Can someone explain to me what is meant by the following (and why it is true): there is a unique $\mathscr O$-torus structure on the $F$-torus $T$?

Thanks in advance, AYK.

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  • $\begingroup$ Is there a reference you are looking at? If so, please give the reference. Almost certainly this refers to flat group schemes over $\text{Spec}\mathcal{O}$ that are 'etale locally isomorphic to $\mathbb{G}_m^r$. $\endgroup$
    – Jason Starr
    Commented Jun 1, 2015 at 14:39
  • $\begingroup$ Yes there is actually, I was trying to understand certain statements in the notes below: math.stanford.edu/~conrad/JLseminar/Notes/L4.pdf (see beginning of p.3) $\endgroup$
    – AYK
    Commented Jun 1, 2015 at 14:42
  • $\begingroup$ Let $M$ and $N$ be smooth group schemes over $\text{Spec}\mathcal{O}$ that are each 'etale locally isomorphic to $\mathbb{G}_{\mathcal{O},m}^r$. Let $i_F:M_F\to N_F$ be a homomorphism of algebraic $F$-group schemes of the generic fibers. Check, 'etale locally, that there is a unique homomorphism $i:M\to N$ of algebraic group schemes over $\text{Spec}\mathcal{O}$ whose generic fiber is $i_F$. Now apply this to prove that every isomorphism of $M_F$ and $N_F$ extends uniquely to an isomorphism of $M$ and $N$. $\endgroup$
    – Jason Starr
    Commented Jun 1, 2015 at 15:04

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