Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-10T10:03:36.688Z Has data issue: false hasContentIssue false
  • English
  • Français

G-dimensions for DG-modules over commutative DG-rings

Part of: Commutative algebra: Homological methods Homological methods

Published online by Cambridge University Press:  01 August 2025

Jiangsheng Hu ,
Xiaoyan Yang Open the ORCID record for Xiaoyan Yang [Opens in a new window]  and
Rongmin Zhu Open the ORCID record for Rongmin Zhu [Opens in a new window]
Jiangsheng Hu
Affiliation:
School of Mathematics, Hangzhou Normal University, Hangzhou, P. R. China
Xiaoyan Yang*
Affiliation:
School of Science, Zhejiang University of Science and Technology, Hangzhou, P. R. China
Rongmin Zhu
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, P. R. China
*
Corresponding author: Xiaoyan Yang, email: yangxy@zust.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We define and study a notion of G-dimension for DG-modules over a non-positively graded commutative noetherian DG-ring A. Some criteria for the finiteness of the G-dimension of a DG-module are given by applying a DG-version of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021) 409-454]. Moreover, it is proved that the finiteness of G-dimension characterizes the local Gorenstein property of A. Applications go in three directions. The first is to establish the connection between G-dimensions and the little finitistic dimensions of A. The second is to characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of DG-modules. The third is to extend the classical Buchweitz-Happel Theorem and its inverse from commutative noetherian local rings to the setting of commutative noetherian local DG-rings. Our method is somewhat different from classical commutative ring.

Keywords

Buchwtweiz-Happel TheoryG-dimensionGorenstein DG-algebralittle finitistic dimensionmaximal local-Cohen-Macaulay DG-module

MSC classification

Primary: 13D05: Homological dimension
Secondary: 16E45: Differential graded algebras and applications 13D09: Derived categories

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 20
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Auslander, M., Anneaux de Gorenstein, et torsion en algèbre commutative. In SéMinaire d’algèbre Commutative dirigé par P. Samuel, SecréTariat mathéMatique, Paris: Séminaire Samuel. Algèbre commutative, 1967.Google Scholar
Auslander, M. and Bridger, M., Stable module theory, Mem. Amer. Math. Soc. 94 (1969).Google Scholar
Auslander, M. and Reiten, I, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), 111–152.10.1016/0001-8708(91)90037-8CrossRefGoogle Scholar
Auslander, M., Reiten, I and Smalø, S. O., Representation theory of Artin algebras. In Cambridge Studies in Advanced Mathematics, Volume 36, (Cambridge University Press, Cambridge, 1995).Google Scholar
Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. 85 (3): (2002), 393–440.10.1112/S0024611502013527CrossRefGoogle Scholar
Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (3): (1960), 466–488.10.1090/S0002-9947-1960-0157984-8CrossRefGoogle Scholar
Beligiannis, A., The homological theory of contravariantly finite subcategories: Gorenstein categories, Auslander-Buchweitz contexts and (co-)stabilization, Comm. Algebra 28 (10): (2000), 4547–4596.10.1080/00927870008827105CrossRefGoogle Scholar
Bergh, P. A., Jorgensen, D. A. and Oppermann, S., The Gorenstein defect category, Quart. J. Math. 66(2): (2015), 459–471.10.1093/qmath/hav001CrossRefGoogle Scholar
Bird, I, Shaul, L., Sridhar, P. and Williamson, J., Finitistic dimensions over commutative DG-rings, Math. Z. 309 (3): (2025), 1–29.10.1007/s00209-024-03617-2CrossRefGoogle Scholar
Christensen, L. W., Gorenstein Dimensions. In Lecture Notes in Math., Volume 1747, (Springer-Verlag, Berlin, 2000).Google Scholar
Christensen, L. W., Piepmeyer, G., Striuli, J. and Takahashi, R., Finite Gorenstein representation type implies simple singularity, Adv. Math. 218 (4): (2008), 1012–1026.10.1016/j.aim.2008.03.005CrossRefGoogle Scholar
Enochs, E. E. and Jenda, O. M. G., Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633.10.1007/BF02572634CrossRefGoogle Scholar
Frankild, A. F., Iyengar, S. and Jogensen, P., Dualizing differential graded modules and Gorenstein differential graded algebras, J. London Math. Soc. 68(2): (2003), 288–306.10.1112/S0024610703004496CrossRefGoogle Scholar
Frankild, A. F. and Jogensen, P., Gorenstein differential graded algebras, Israel J. Math. 135 (2003), 327–353.10.1007/BF02776063CrossRefGoogle Scholar
Happel, D., On Gorenstein Algebras, in Representation theory of finite groups and finite-dimensional algebras (Proc. Conf. at Bielefeld, 1991). Progress in Math. , Volume 95,Birkhäuser, Basel, 1991, .Google Scholar
Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.10.1016/j.jpaa.2003.11.007CrossRefGoogle Scholar
Iyama, O. and Yang, D., Silting reduction and Calabi-Yau reduction of triangulated categories, Trans. Amer. Math. Soc. 370 (11): (2018), 7861–7898.10.1090/tran/7213CrossRefGoogle Scholar
Iyengar, S. B. and Letz, J. C., Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, with appendices by Avramov, L. L., and Briggs, B., Math. Surveys and Monographs vol. 262, (Amer. Math. Soc., Providence, Rhode Island, 2021).Google Scholar
Jin, H. B., Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations, Adv. Math. 374 (2020), 107338.10.1016/j.aim.2020.107338CrossRefGoogle Scholar
Jorgensen, P., Amplitude inequalities for differential graded modules, Forum Math. 22 (2010), 941–948.10.1515/forum.2010.049CrossRefGoogle Scholar
Kong, F. and Zhang, P., From CM-finite to CM-free, J. Pure Appl. Algebra 220 (2016), 782–801.10.1016/j.jpaa.2015.07.016CrossRefGoogle Scholar
Minamoto, H., Homological identities and dualizing complexes of commutative differential graded algebras, Israel J. Math. 242 (2021), 1–36.10.1007/s11856-021-2095-3CrossRefGoogle Scholar
Minamoto, H., Resolutions and homological dimensions of DG-modules, Israel J. Math. 245 (2021), 409–454.10.1007/s11856-021-2230-1CrossRefGoogle Scholar
Orlov, D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 3(246): (2004), 227–248.Google Scholar
Ringel, C. M. and Zhang, P., Representations of quivers over the algebra of dual numbers, J. Algebra 475 (2017), 327–360.10.1016/j.jalgebra.2016.12.001CrossRefGoogle Scholar
Shaul, L., Injective DG-modules over non-positive DG-rings, J. Algebra 515 (2018), 102–156.10.1016/j.jalgebra.2018.07.040CrossRefGoogle Scholar
Shaul, L., Completion and torsion over commutative DG rings, Israel J. Math. 232 (2019), 531–588.10.1007/s11856-019-1866-6CrossRefGoogle Scholar
Shaul, L., The Cohen-Macaulay property in derived commatative algebra, Trans. Amer. Math. Soc. 373 (2020), 6095–6138.10.1090/tran/8099CrossRefGoogle Scholar
Shaul, L., Sequence-regular commutative DG-rings, J. Algebra 647 (2024), 400–435.10.1016/j.jalgebra.2024.02.034CrossRefGoogle Scholar
Wei, J. Q., Relative singularity categories, Gorenstein objects and silting theory, J. Pure Appl. Algebra 222 (2018), 2310–2322.10.1016/j.jpaa.2017.09.014CrossRefGoogle Scholar
Xi, C. C., On the finitistic dimension conjecture, III: Related to the pair $eAe\subseteq A$, J. Algebra 319 (2008), 3666–3688.10.1016/j.jalgebra.2008.01.021CrossRefGoogle Scholar
Yang, X. Y. and Li, Y. J., Local Cohen-Macaulay DG-Modules, Appl. Categor. Struct. 31(8): (2023), 10.1007/s10485-022-09703-yCrossRefGoogle Scholar
Yang, X. Y. and Wang, L., Homological invariants over non-positive DG-rings, J. Algebra Appl. 2050153 (2020), 2050153.10.1142/S0219498820501534CrossRefGoogle Scholar
Yassemi, S., G-dimension, Math. Scand. 77 (1995), 161–174.10.7146/math.scand.a-12557CrossRefGoogle Scholar
Yekutieli, A., Duality and tilting for commutative DG rings, arXiv:1312.6411v4, 2016.Google Scholar
Yekutieli, A., Derived categories, Volume 183, (Cambridge University Press, Cambridge, 2020) volume.Google Scholar
Yoshino, Y., Cohen-Macaulay Modules Over Cohen-Macaulay Rings. London Math. Soc. Lecture Note Ser., Volume 146, (Cambridge Univ. Press, Cambridge, 1990).Google Scholar