PreprintCC BY 4.0arXiv
A Gaschütz Lemma for linear algebraic groups with applications to the Riemann-Hilbert correspondence
Abstract
The classical Gaschütz Lemma states that for a surjective morphism G→ H of finite groups, generators h1,…,hd of H can be lifted to generators g1,…,gd of G, provided that G can be generated by d elements. We prove an analog of this statement for linear algebraic groups. This has applications to the algebraic-geometric theory of linear differential equations: We show that every regular singular differential embedding problem over the rational function field has a regular singular solution. We also establish a purely algebraic variant of the Riemann-Hilbert correspondence on the punctured projective line that works over any countable algebraically closed field of characteristic zero.
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BibTeX
@misc{bell2026gaschütz,
title = {{A Gaschütz Lemma for linear algebraic groups with applications to the Riemann-Hilbert correspondence}},
author = {Jason Bell and Michael Wibmer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38919},
eprint = {2610.38919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38919},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Gaschütz Lemma for linear algebraic groups with applications to the Riemann-Hilbert correspondence
AU - Jason Bell
AU - Michael Wibmer
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38919
UR - https://arxiv.org/abs/2610.38919
AB - The classical Gaschütz Lemma states that for a surjective morphism G→ H of finite groups, generators h_1,…,h_d of H can be lifted to generators g_1,…,g_d of G, provided that G can be generated by d elements. We prove an analog of this statement for linear algebraic groups. This has applications to the algebraic-geometric theory of linear differential equations: We show that every regular singular differential embedding problem over the rational function field has a regular singular solution. We also establish a purely algebraic variant of the Riemann-Hilbert correspondence on the punctured projective line that works over any countable algebraically closed field of characteristic zero.
ER -