PreprintarXiv licencearXiv
A parity obstruction to completeness of object cotorsion pairs
Abstract
Fu, Guil Asensio, Herzog and Torrecillas asked whether a complete ideal cotorsion pair of object ideals in an exact category always induces a complete cotorsion pair of objects. We show that it need not, even in a Hom-finite, weakly idempotent complete Frobenius exact category. Our example consists of bounded complexes of finite-dimensional vector spaces with even total cohomology dimension. We characterize special object approximations by parity and identify the obstruction as a non-split idempotent in the stable category. For cohereditary pairs, a criterion of Saor\'ın and Šťov\'ıček gives an affirmative answer when the stable category is idempotent complete.
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BibTeX
@misc{ren2026parity,
title = {{A parity obstruction to completeness of object cotorsion pairs}},
author = {Junpeng Ren and Yucheng Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18681},
eprint = {2609.18681},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18681},
}
RIS
TY - UNPB
TI - A parity obstruction to completeness of object cotorsion pairs
AU - Junpeng Ren
AU - Yucheng Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2609.18681
UR - https://arxiv.org/abs/2609.18681
AB - Fu, Guil Asensio, Herzog and Torrecillas asked whether a complete ideal cotorsion pair of object ideals in an exact category always induces a complete cotorsion pair of objects. We show that it need not, even in a Hom-finite, weakly idempotent complete Frobenius exact category. Our example consists of bounded complexes of finite-dimensional vector spaces with even total cohomology dimension. We characterize special object approximations by parity and identify the obstruction as a non-split idempotent in the stable category. For cohereditary pairs, a criterion of Saor\'ın and Šťov\'ıček gives an affirmative answer when the stable category is idempotent complete.
ER -