## Lifting of projective representations of a torus over a valuation ring

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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.

-

## 1 Answer

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 $H^1(S,\underline{\mathbb{Z}})=0$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8915850520133972, "perplexity_flag": "head"}