A structure theorem for subgroups of 𝐺𝐿_{𝑛} over complete local Noetherian rings with large residual image

Manoharmayum, Jayanta

doi:10.1090/S0002-9939-2015-12306-4

A structure theorem for subgroups of $GL_n$ over complete local Noetherian rings with large residual image
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by Jayanta Manoharmayum

Proc. Amer. Math. Soc. 143 (2015), 2743-2758

DOI: https://doi.org/10.1090/S0002-9939-2015-12306-4

Published electronically: March 11, 2015

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Abstract:

Given a complete local Noetherian ring $(A,\mathfrak {m}_A)$ with finite residue field and a subfield $\boldsymbol {k}$ of $A/\mathfrak {m}_A$, we show that every closed subgroup $G$ of $GL_n(A)$ such that $G\mod {\mathfrak {m}_A}\supseteq SL_n(\boldsymbol {k})$ contains a conjugate of $SL_n(W(\boldsymbol {k})_A)$ under some small restrictions on $\boldsymbol {k}$. Here $W(\boldsymbol {k})_A$ is the closed subring of $A$ generated by the Teichmüller lifts of elements of the subfield $\boldsymbol {k}$.

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