Skip to content
DraconiticResearch
Search Sources API
186,123 studies31 sourcesUpdated 24 min ago
PreprintarXiv licencearXiv

Acyclic and totally acyclic objects in the Q-shaped derived category

  • Henrik Holm
  • Peter Jorgensen
Open at sourcePDFDOICite

Abstract

This paper concerns homological algebra based on Q-shaped diagrams, where Q belongs to a certain class of small categories. We will show that the Gorenstein property of noetherian commutative rings is characterised by the condition that acyclicity coincides with total acyclicity for Q-shaped diagrams of suitable classes of modules. Many special cases occur as corollaries, for instance N-complexes. This generalises a classic result by Iyengar and Krause to the programme of Q-shaped derived categories, which builds on an insight of Iyama and Minamoto. We prove our result using an adjoint triple of functors relating Q-shaped diagrams to chain complexes. One of the functors was introduced by Jasso, and the whole triple generalises the compression, cocompression, and expansion functors for differential modules introduced by Avramov, Buchweitz, and Iyengar and by Nkansah. We consider the adjoint triple to be the main contribution of this paper.

Related

  1. On the totally acyclic condition and the Auslander-Reiten Conjecture
    Justin Lyle
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Extremal numbers of acyclic matrices
    Lior Gishboliner, Xiangyu Li
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Hamiltonicity of graphs of acyclic orientations and acyclic polynomials
    Leonie Mühlherr, Germain Poullot
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Acyclic Dichromatic Number of Tournaments: these are the Champions
    Pierre Aboulker, Pierre Charbit, Samuel Coulomb +2
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. The flipping principle for boundedly acyclic covers and colorings
    Pietro Capovilla, Alberto Casali, Marco Moraschini
    arXivPreprintarXiv licenceFull textAlgebra & number theoryGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{holm2026acyclic,
  title = {{Acyclic and totally acyclic objects in the $Q$-shaped derived category}},
  author = {Henrik Holm and Peter Jorgensen},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27660},
  eprint = {2608.27660},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27660},
}
RIS
TY  - UNPB
TI  - Acyclic and totally acyclic objects in the Q-shaped derived category
AU  - Henrik Holm
AU  - Peter Jorgensen
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27660
UR  - https://arxiv.org/abs/2608.27660
AB  - This paper concerns homological algebra based on Q-shaped diagrams, where Q belongs to a certain class of small categories. We will show that the Gorenstein property of noetherian commutative rings is characterised by the condition that acyclicity coincides with total acyclicity for Q-shaped diagrams of suitable classes of modules. Many special cases occur as corollaries, for instance N-complexes. This generalises a classic result by Iyengar and Krause to the programme of Q-shaped derived categories, which builds on an insight of Iyama and Minamoto. We prove our result using an adjoint triple of functors relating Q-shaped diagrams to chain complexes. One of the functors was introduced by Jasso, and the whole triple generalises the compression, cocompression, and expansion functors for differential modules introduced by Avramov, Buchweitz, and Iyengar and by Nkansah. We consider the adjoint triple to be the main contribution of this paper.
ER  -