The Q-shaped derived category of a ring

Henrik Holm; Peter Jørgensen

doi:10.1112/jlms.12662

The Q-shaped derived category of a ring

^*Corresponding author af dette arbejde

Institut for Matematik

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

For any ring (Formula presented.) and a small, pre-additive, Hom-finite, and locally bounded category (Formula presented.) that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors (Formula presented.) has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the (Formula presented.) -shaped derived category of (Formula presented.). The usual derived category of (Formula presented.) is one example; more general examples arise by taking (Formula presented.) to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and García-Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and Šťovíček (Math. Z. 287 (2017), no. 3-4, 1109–1155).

Originalsprog	Engelsk
Tidsskrift	Journal of the London Mathematical Society
Vol/bind	106
Nummer	4
Sider (fra-til)	3263-3316
Antal sider	54
ISSN	0024-6107
DOI	https://doi.org/10.1112/jlms.12662
Status	Udgivet - dec. 2022

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10.1112/jlms.12662

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abstract = "For any ring (Formula presented.) and a small, pre-additive, Hom-finite, and locally bounded category (Formula presented.) that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors (Formula presented.) has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the (Formula presented.) -shaped derived category of (Formula presented.). The usual derived category of (Formula presented.) is one example; more general examples arise by taking (Formula presented.) to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and Garc{\'i}a-Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and {\v S}{\v t}ov{\'i}{\v c}ek (Math. Z. 287 (2017), no. 3-4, 1109–1155).",

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AB - For any ring (Formula presented.) and a small, pre-additive, Hom-finite, and locally bounded category (Formula presented.) that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors (Formula presented.) has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the (Formula presented.) -shaped derived category of (Formula presented.). The usual derived category of (Formula presented.) is one example; more general examples arise by taking (Formula presented.) to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and García-Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and Šťovíček (Math. Z. 287 (2017), no. 3-4, 1109–1155).

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The Q-shaped derived category of a ring

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