On reducing homological dimensions over noetherian rings
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- by Tokuji Araya and Ryo Takahashi PDF
- Proc. Amer. Math. Soc. 150 (2022), 469-480 Request permission
Abstract:
Let $\Lambda$ be a left and right Noetherian ring. First, for $m,n\in \mathbb {N}\cup \{\infty \}$, we give equivalent conditions for a given $\Lambda$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate totally reflexive modules and reducing Gorenstein dimension. Next, we introduce homological invariants for $\Lambda$-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when $\Lambda$ is commutative and local. Using it, we consider how upper reducing projective dimension relates to reducing projective dimension, and the complete intersection and AB properties of a commutative Noetherian local ring.References
- Tokuji Araya and Olgur Celikbas, Reducing invariants and total reflexivity, Illinois J. Math. 64 (2020), no. 2, 169–184. MR 4092954, DOI 10.1215/00192082-8303469
- Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
- L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), no. 1, 71–101. MR 981738, DOI 10.1007/BF01393971
- Luchezar L. Avramov, Infinite free resolutions [MR1648664], Six lectures on commutative algebra, Mod. Birkhäuser Class., Birkhäuser Verlag, Basel, 2010, pp. 1–118. MR 2641236, DOI 10.1007/978-3-0346-0329-4_{1}
- Luchezar L. Avramov and Ragnar-Olaf Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318. MR 1794064, DOI 10.1007/s002220000090
- Luchezar L. Avramov and Alex Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393–440. MR 1912056, DOI 10.1112/S0024611502013527
- Petter Andreas Bergh, Modules with reducible complexity, J. Algebra 310 (2007), no. 1, 132–147. MR 2307785, DOI 10.1016/j.jalgebra.2006.09.008
- Lars Winther Christensen, Janet Striuli, and Oana Veliche, Growth in the minimal injective resolution of a local ring, J. Lond. Math. Soc. (2) 81 (2010), no. 1, 24–44. MR 2580452, DOI 10.1112/jlms/jdp058
- Hailong Dao and Ryo Takahashi, The radius of a subcategory of modules, Algebra Number Theory 8 (2014), no. 1, 141–172. MR 3207581, DOI 10.2140/ant.2014.8.141
- Hailong Dao and Oana Veliche, Comparing complexities of pairs of modules, J. Algebra 322 (2009), no. 9, 3047–3062. MR 2567409, DOI 10.1016/j.jalgebra.2008.08.011
- Edgar E. Enochs and Overtoun M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611–633. MR 1363858, DOI 10.1007/BF02572634
- E. Graham Evans and Phillip Griffith, Syzygies, London Mathematical Society Lecture Note Series, vol. 106, Cambridge University Press, Cambridge, 1985. MR 811636, DOI 10.1017/CBO9781107325661
- Craig Huneke and David A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand. 93 (2003), no. 2, 161–184. MR 2009580, DOI 10.7146/math.scand.a-14418
- Osamu Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), no. 1, 22–50. MR 2298819, DOI 10.1016/j.aim.2006.06.002
- Sean Sather-Wagstaff, Bass numbers and semidualizing complexes, Commutative algebra and its applications, Walter de Gruyter, Berlin, 2009, pp. 349–381. MR 2640315
- Sean Sather-Wagstaff, Tirdad Sharif, and Diana White, Stability of Gorenstein categories, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 481–502. MR 2400403, DOI 10.1112/jlms/jdm124
Additional Information
- Tokuji Araya
- Affiliation: Department of Applied Science, Faculty of Science, Okayama University of Science, Ridaicho, Kitaku, Okayama 700-0005, Japan
- MR Author ID: 639398
- ORCID: 0000-0001-7309-080X
- Email: araya@das.ous.ac.jp
- Ryo Takahashi
- Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
- MR Author ID: 674867
- ORCID: 0000-0001-6287-8941
- Email: takahashi@math.nagoya-u.ac.jp
- Received by editor(s): November 1, 2020
- Published electronically: November 19, 2021
- Additional Notes: The second author was partly supported by JSPS Grant-in-Aid for Scientific Research 19K03443
- Communicated by: Jerzy Weyman
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 469-480
- MSC (2020): Primary 13D05, 13H10, 16E10
- DOI: https://doi.org/10.1090/proc/15785
- MathSciNet review: 4356161