Representation embeddings and the third Brauer--Thrall conjecture
Abstract
Let A be a representation-infinite finite-dimensional algebra over an algebraically closed field k. We prove that for every infinite cardinal κ there is an indecomposable A-module M with dimk M=κ. Over algebraically closed fields, this settles Simson's conjecture, also called the third Brauer--Thrall conjecture, that every representation-infinite finite-dimensional algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend Bongartz's representation embedding from finite-dimensional Kronecker modules without a simple injective summand to such modules of arbitrary dimension. The extended functor commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by Göbel and Simson yields the required A-modules.
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BibTeX
@misc{enomoto2026representation,
title = {{Representation embeddings and the third Brauer--Thrall conjecture}},
author = {Haruhisa Enomoto},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06325},
eprint = {2611.06325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06325},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Representation embeddings and the third Brauer--Thrall conjecture
AU - Haruhisa Enomoto
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06325
UR - https://arxiv.org/abs/2611.06325
AB - Let A be a representation-infinite finite-dimensional algebra over an algebraically closed field k. We prove that for every infinite cardinal κ there is an indecomposable A-module M with dim_k M=κ. Over algebraically closed fields, this settles Simson's conjecture, also called the third Brauer--Thrall conjecture, that every representation-infinite finite-dimensional algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend Bongartz's representation embedding from finite-dimensional Kronecker modules without a simple injective summand to such modules of arbitrary dimension. The extended functor commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by Göbel and Simson yields the required A-modules.
ER -