Models for chain homotopy category of relative acyclic complexes

Jiangsheng Hu; Wei Ren; Xiaoyan Yang; Hanyang You; Jiangsheng Hu; Wei Ren; Xiaoyan Yang; Hanyang You

doi:10.3934/era.2026038

Electronic Research Archive

2026, Volume 34, Issue 2: 830-843. doi: 10.3934/era.2026038

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Research article

Models for chain homotopy category of relative acyclic complexes

1.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
3.
School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received: 06 November 2025 Revised: 04 January 2026 Accepted: 12 January 2026 Published: 23 January 2026

Let $ (\mathcal{X}, \mathcal{Y}) $ be a balanced pair in an abelian category $ \mathcal{A} $. Denote by $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X}) $ the chain homotopy category of right $ \mathcal{X} $-acyclic complexes with all items in $ \mathcal{X} $, and dually by $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y}) $ the chain homotopy category of left $ \mathcal{Y} $-acyclic complexes with all items in $ \mathcal{Y} $. We establish realizations of $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X}) $ and $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y}) $ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further present applications to Gorenstein projective and Gorenstein injective modules.
- balanced pair,
- model category,
- exact category,
- chain homotopy category,
- recollement
Citation: Jiangsheng Hu, Wei Ren, Xiaoyan Yang, Hanyang You. Models for chain homotopy category of relative acyclic complexes[J]. Electronic Research Archive, 2026, 34(2): 830-843. doi: 10.3934/era.2026038

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Abstract

Let $ (\mathcal{X}, \mathcal{Y}) $ be a balanced pair in an abelian category $ \mathcal{A} $. Denote by $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X}) $ the chain homotopy category of right $ \mathcal{X} $-acyclic complexes with all items in $ \mathcal{X} $, and dually by $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y}) $ the chain homotopy category of left $ \mathcal{Y} $-acyclic complexes with all items in $ \mathcal{Y} $. We establish realizations of $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{X}) $ and $ {\bf K}_{\mathcal{E}\text{-}{\rm ac}}(\mathcal{Y}) $ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further present applications to Gorenstein projective and Gorenstein injective modules.

References

[1]	M. Hovey, Model Categories, American Mathematical Society, 63 (1999).
[2]	D. G. Quillen, Homotopical Algebra, in Lecture Notes in Mathematics, $1^{st}$ edition, Springer-Verlag, 43 (1967). https://doi.org/10.1007/BFb0097438
[3]	J. Gillespie, Abelian Model Category Theory, Cambridge University Press, 215 2025. https://doi.org/10.1017/9781009449489
[4]	M. Hovey, Cotorsion pairs, model category structures and representation theory, Math. Z., 241 (2002), 553–592. https://doi.org/10.1007/s00209-002-0431-9 doi: 10.1007/s00209-002-0431-9
[5]	J. Gillespie, Model structures on exact categories, J. Pure Appl. Algebra, 215 (2011), 2892–2902. https://doi.org/10.1016/j.jpaa.2011.04.010 doi: 10.1016/j.jpaa.2011.04.010
[6]	J. Šťovíček, Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves, Adv. Representation Theory Algebras, (2014), 297–367. https://doi.org/10.4171/125-1/10 doi: 10.4171/125-1/10
[7]	H. Becker, Models for singularity categories, Adv. Math., 254 (2014), 187–232. https://doi.org/10.1016/j.aim.2013.11.016 doi: 10.1016/j.aim.2013.11.016
[8]	D. Georgios, C. Psaroudakis, Lifting recollements of abelian categories and model structures, J. Algebra, 623 (2023), 395–446. https://doi.org/10.1016/j.jalgebra.2023.01.030 doi: 10.1016/j.jalgebra.2023.01.030
[9]	J. Gillespie, Cotorsion pairs and degreewise homological model structures, Homol. Homotopy Appl., 10 (2008), 283–304. https://doi.org/10.4310/HHA.2008.v10.n1.a12 doi: 10.4310/HHA.2008.v10.n1.a12
[10]	J. Gillespie, Hereditary abelian model categories, Bull. London Math. Soc., 48 (2016), 895–922. https://doi.org/10.1112/blms/bdw051 doi: 10.1112/blms/bdw051
[11]	J. Hu, H. Zhu, R. Zhu, Gluing and lifting exact model structures for the recollement of exact categories, Bull. Malays. Math. Sci. Soc., 48 (2025), 173. https://doi.org/10.1007/s40840-025-01945-y doi: 10.1007/s40840-025-01945-y
[12]	A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux Pervers, Paris: Société mathématique de France, 100 (1982).
[13]	H. Krause, The stable derived category of a Noetherian scheme, Compos. Math., 141 (2005), 1128–1162. https://doi.org/10.1112/S0010437X05001375 doi: 10.1112/S0010437X05001375
[14]	D. Murfet, The Mock Homotopy Category of Projectives and Grothendieck Duality, Ph.D Thesis, Australian National University, 2007. https://doi.org/10.25911/5D5156B769745
[15]	E. E. Enochs, O. M. G. Jenda, Relative Homological Algebra, in De Gruyter Expositions in Mathematics, New York: Walter De Gruyter, 30 (2000). https://doi.org/10.1515/9783110215212
[16]	X. W. Chen, Homotopy equivalences induced by balanced pairs, J. Algebra, 324 (2010), 2718–2731. https://doi.org/10.1016/j.jalgebra.2010.09.002 doi: 10.1016/j.jalgebra.2010.09.002
[17]	S. Estrada, M. A. Pérez, H. Zhu, Balanced pairs, cotorsion triplets and quiver representations, in Proc. Edinburgh Math. Soc., 63 (2020), 67–90. https://doi.org/10.1017/S0013091519000270
[18]	H. Li, Z. Huang, Relative singularity categories, J. Pure Appl. Algebra, 219 (2015), 4090–4104. https://doi.org/10.1016/j.jpaa.2015.02.009 doi: 10.1016/j.jpaa.2015.02.009
[19]	S. Sather-Wagstaff, T. Sharif, D. White, Stability of Gorenstein categories, J. London Math. Soc., 77 (2008), 481–502. https://doi.org/10.1112/jlms/jdm124 doi: 10.1112/jlms/jdm124
[20]	T. Bühler, Exact Categories, Expositiones Math., 28 (2010), 1–69. https://doi.org/10.1016/j.exmath.2009.04.004
[21]	J. Hu, W. Ren, X. Yang, H. You, Quillen equivalence for chain homotopy categories induced by balanced pairs, preprint, arXiv: 2503.01188. https://doi.org/10.48550/arXiv.2503.01188
[22]	J. Šťovíček, On purity and applications to coderived and singularity categories, preprint, arXiv: 1412.1615. https://doi.org/10.48550/arXiv.1412.1615
[23]	A. Neeman, The homotopy category of flat modules, and Grothendieck duality, Invent. Math., 174 (2008), 255–308. https://doi.org/10.1007/s00222-008-0131-0 doi: 10.1007/s00222-008-0131-0
[24]	J. Wang, S. Estrada, Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension, J. Algebra, 678 (2025), 769–808. https://doi.org/10.1016/j.jalgebra.2025.04.033 doi: 10.1016/j.jalgebra.2025.04.033
[25]	N. Gao, P. Zhang, Gorenstein derived categories, J. Algebra, 323 (2010), 2041–2057. https://doi.org/10.1016/j.jalgebra.2010.01.027 doi: 10.1016/j.jalgebra.2010.01.027
[26]	D. G. Quillen, Higher algebraic $K$-theory: I, in Lecture Notes in Mathematics, Springer, Berlin, 341 (1973). https://doi.org/10.1007/BFb0067053
[27]	A. Neeman, The derived category of an exact category, J. Algebra, 138 (1990), 388–394. https://doi.org/10.1016/0021-8693(90)90296-Z doi: 10.1016/0021-8693(90)90296-Z
[28]	N. Gao, X. Lu, P. Zhang, Chains of model structures arising from modules of finite Gorenstein dimension, preprint, arXiv: 2403.05232. https://doi.org/10.48550/arXiv.2403.05232
[29]	J. Gillespie, The flat model structure on Ch($R$), Trans. Am. Math. Soc. 356 (2004), 3369–3390. https://doi.org/10.1090/S0002-9947-04-03416-6 doi: 10.1090/S0002-9947-04-03416-6
[30]	J. Gillespie, How to construct a Hovey triple from two cotorsion pairs, Fundam. Math., 230 (2015), 281–289. https://doi.org/10.4064/fm230-3-4 doi: 10.4064/fm230-3-4
[31]	P. Y. Zhou, Hovey triples arising from two cotorsion pairs of extriangulated categories, Colloq. Math., 171 (2023), 211–230. https://doi.org/10.4064/cm8615-3-2022 doi: 10.4064/cm8615-3-2022
[32]	E. E. Enochs, O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z., 220 (1995), 611–633. https://doi.org/10.1007/BF02572634 doi: 10.1007/BF02572634

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