The Q-shaped derived category of a ring
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The Q-shaped derived category of a ring. / Holm, Henrik; Jørgensen, Peter.
In: Journal of the London Mathematical Society, Vol. 106, No. 4, 2022, p. 3263-3316.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - The Q-shaped derived category of a ring
AU - Holm, Henrik
AU - Jørgensen, Peter
N1 - Publisher Copyright: © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
PY - 2022
Y1 - 2022
N2 - 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).
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).
UR - http://www.scopus.com/inward/record.url?scp=85134175604&partnerID=8YFLogxK
U2 - 10.1112/jlms.12662
DO - 10.1112/jlms.12662
M3 - Journal article
AN - SCOPUS:85134175604
VL - 106
SP - 3263
EP - 3316
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
SN - 0024-6107
IS - 4
ER -
ID: 317817886