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

[2608.27660] Acyclic and totally acyclic objects in the $Q$-shaped derived category Skip to main content Search Submit Donate Log in Mathematics > Representation Theory arXiv:2608.27660 (math) [Submitted on 27 Aug 2026] Title: Acyclic and totally acyclic objects in the $Q$-shaped derived category Authors: Henrik Holm , Peter Jorgensen View PDF HTML (experimental) 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. Comments: 28 pages Subjects: Representation Theory (math.RT) ; Rings and Algebras (math.RA) MSC classes: 13D09, 13H10, 18G80, 18N40 Cite as: arXiv:2608.27660 [math.RT] (or arXiv:2608.27660v1 [math.RT] for this version) https://doi.org/10.48550/arXiv.2608.27660 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Peter Jorgensen [ view email ] [v1] Thu, 27 Aug 2026 19:47:24 UTC (33 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2026-08 Change to browse by: math math.RA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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