Global obstruction property and its applications
Abstract
We introduce the global obstruction property (GOP) for Artin algebras: for every finitely generated module of infinite projective dimension, two distinct syzygies have a common nonprojective indecomposable direct summand. We prove that GOP implies the generalized Auslander--Reiten condition, and hence the Auslander--Reiten conjecture. We give criteria for GOP to hold for pullbacks over semisimple algebras and construct an example which is not syzygy-finite. For each Artin algebra A, we construct an algebra 𝒢(A) satisfying GOP by using triangular matrix extensions. We show that fin.dim𝒢(A)=fin.dim A+2 and that A and 𝒢(A) are singularly equivalent. Finally, we give a pair of derived equivalent finite-dimensional algebras for which GOP holds on only one side.
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BibTeX
@misc{xia2026global,
title = {{Global obstruction property and its applications}},
author = {Weiheng Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12583},
eprint = {2610.12583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12583},
}
RIS
TY - UNPB
TI - Global obstruction property and its applications
AU - Weiheng Xia
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12583
UR - https://arxiv.org/abs/2610.12583
AB - We introduce the global obstruction property (GOP) for Artin algebras: for every finitely generated module of infinite projective dimension, two distinct syzygies have a common nonprojective indecomposable direct summand. We prove that GOP implies the generalized Auslander--Reiten condition, and hence the Auslander--Reiten conjecture. We give criteria for GOP to hold for pullbacks over semisimple algebras and construct an example which is not syzygy-finite. For each Artin algebra A, we construct an algebra 𝒢(A) satisfying GOP by using triangular matrix extensions. We show that fin.dim𝒢(A)=fin.dim A+2 and that A and 𝒢(A) are singularly equivalent. Finally, we give a pair of derived equivalent finite-dimensional algebras for which GOP holds on only one side.
ER -