Local coherence and recollement of abelian categories
Abstract
We study how local coherenAAce of abelian categories behaves under gluing. For AB5 locally finitely presentable abelian categories, we give conditions on the gluing functors under which local coherence is transferred along a gluing (π, β¬, π): from π and π to β¬, and conversely from β¬ to π or π. The coherence theorem for augmented homotopes of Bondal and Zhdanovskiy is recovered as a special case, the coherence theorem for trivial extensions of Fossum, Griffith, and Reiten is a companion result in the same direction. We also introduce quasiprojective pairs of adjoint functors and prove a coherence criterion that generalises the criterion of Gomez Pardo and Martinez Hernandez to arbitrary Grothendieck categories. Finally, we extend Stenstr{ΓΆ}m's characterisation of coherence via FP-injective modules to compactly assembled Grothendieck categories: such a category is coherently assembled if and only if the class of CM-injective objects, defined through compact morphisms, is closed under filtered colimits. Along the way we revisit the various equivalent conditions for the coherence of a ring, collecting them in a single place. The proofs make use of the compactly assembled categories of Efimov.
Related
- A Krause-type recollement for Q-shaped derived categories
- Homology and higher-order intermediate categories for proper abelian subcategories
- Spectra and Support Theory for Wide Subcategories in n-Abelian Categories
- Ternary semi-direct products in semi-abelian categories
- Transfer of abelian model structures to equivariant categories and homotopy squares
BibTeX
@misc{dusman2026local,
title = {{Local coherence and recollement of abelian categories}},
author = {Mikhail Dusman and Alexander Pavlov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14931},
eprint = {2610.14931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14931},
}
RIS
TY - UNPB
TI - Local coherence and recollement of abelian categories
AU - Mikhail Dusman
AU - Alexander Pavlov
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14931
UR - https://arxiv.org/abs/2610.14931
AB - We study how local coherenAAce of abelian categories behaves under gluing. For AB5 locally finitely presentable abelian categories, we give conditions on the gluing functors under which local coherence is transferred along a gluing (π, β¬, π): from π and π to β¬, and conversely from β¬ to π or π. The coherence theorem for augmented homotopes of Bondal and Zhdanovskiy is recovered as a special case, the coherence theorem for trivial extensions of Fossum, Griffith, and Reiten is a companion result in the same direction. We also introduce quasiprojective pairs of adjoint functors and prove a coherence criterion that generalises the criterion of Gomez Pardo and Martinez Hernandez to arbitrary Grothendieck categories. Finally, we extend Stenstr{ΓΆ}m's characterisation of coherence via FP-injective modules to compactly assembled Grothendieck categories: such a category is coherently assembled if and only if the class of CM-injective objects, defined through compact morphisms, is closed under filtered colimits. Along the way we revisit the various equivalent conditions for the coherence of a ring, collecting them in a single place. The proofs make use of the compactly assembled categories of Efimov.
ER -