Let $R$ be a valuation ring. We don't assume it to be discrete or have a finite residue field. Let $T$ be a split torus over $R$, so $T\cong {\mathbb G}_m^r$ for some $r$. Left there be given a homomorphism of group schemes $T\to {\rm PGL}_n$, defined over $R$ ,does there exist a lift to a morphism $T\to {\rm GL}_n$ defined over $R$? I expect this to be somewhere in the literature, but I couldn't locate it.
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The obstruction to lifting is given by a central extension $E$ of $T$ by $\mathbb{G}_m$. Such an extension must be commutative because the commutator induces a bi-homomorphism $T\times T\to\mathbb{G}_m$, and every such map is trivial. So $E$ is a torus, and the extension is dual to an extension of the constant group scheme $\underline{\mathbb{Z}}$ by $\underline{\mathbb{Z}}^r$, hence trivial. The argument works over any base scheme $S$ satisfying $H^1(S,\underline{\mathbb{Z}})=0$. |
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