Hostname: page-component-6587cd75c8-cpvbf Total loading time: 0 Render date: 2025-04-24T04:45:12.387Z Has data issue: false hasContentIssue false
  • English
  • Français

On the reducing projective dimension over local rings

Part of: Commutative algebra: Homological methods Theory of modules and ideals Local rings and semilocal rings

Published online by Cambridge University Press:  31 October 2023

Olgur Celikbas ,
Souvik Dey Open the ORCID record for Souvik Dey [Opens in a new window] ,
Toshinori Kobayashi  and
Hiroki Matsui
Olgur Celikbas
Affiliation:
School of Mathematical and Data Sciences, West Virginia University, Morgantown, WV, USA
Souvik Dey
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS, USA Faculty of Mathematics and Physics, Department of Algebra, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic
Toshinori Kobayashi
Affiliation:
School of Science and Technology, Meiji University, Kawasaki-shi, Kanagawa, Japan
Hiroki Matsui*
Affiliation:
Department of Mathematical Sciences, Faculty of Science and Technology, Tokushima University, Tokushima, Japan
*
Corresponding author: Hiroki Matsui; Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

Keywords

G-regular ringsminimal multiplicityreducing projective and reducing Gorenstein dimension

MSC classification

Primary: 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13C12: Torsion modules and ideals 13D05: Homological dimension 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 104 - 118
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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

Footnotes

Souvik Dey was partly supported by Charles University Research Center program No.UNCE/SCI/022 and a grant GA ČR 23-05148S from the Czech Science Foundation; Toshinori Kobayashi was partly supported by JSPS Grant-in-Aid for JSPS Fellows 18J20660; Hiroki Matsui was partly supported by JSPS Grant-in-Aid for Early-Career Scientists 22K13894.

References

Araya, T. and Celikbas, O., Reducing invariants and total reflexivity, Ill. J. Math. 64(2) (2020), 169–184.Google Scholar
Araya, T., Celikbas, O., Cook, J. and Kobayashi, T., Modules with finite reducing Gorenstein dimension, Beitr. Algebra Geom. (2023). https://doi.org/10.1007/s13366-023-00687-x CrossRefGoogle Scholar
Araya, T. and Takahashi, R., On reducing homological dimensions over Noetherian rings, Proc. Am. Math. Soc. 150(2) (2022), 469–480.Google Scholar
Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, vol. 94 (American Mathematical Society, Providence, RI, 1969).Google Scholar
Avramov, L. L., Infinite free resolutions, in Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., vol. 166 (Birkhäuser, Basel, 1998), 1–118.Google Scholar
Avramov, L. L. and Foxby, H.-B., Locally Gorenstein homomorphisms, Am. J. Math. 114(5) (1992), 1007–1047.CrossRefGoogle Scholar
Avramov, L. L. and Foxby, H.-B., Ring homomorphisms and finite Gorenstein dimension, Proc. Lond. Math. Soc. 75(2) (1997), 241–270.CrossRefGoogle Scholar
Avramov, L. L., Gasharov, V. N. and Peeva, I. V., Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67–114.CrossRefGoogle Scholar
Bergh, P. A., Modules with reducible complexity, J. Algebra 310 (2007), 132–147.CrossRefGoogle Scholar
Bergh, P. A., Modules with reducible complexity II, Commun. Algebra 37(6) (2009), 1908–1920.CrossRefGoogle Scholar
Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993).Google Scholar
Celikbas, O., Dey, S., Kobayashi, T. and Matsui, H., Some characterizations of local rings via reducing dimensions, arXiv:2212.05220.Google Scholar
Celikbas, O. and Takahashi, R., On the second rigidity theorem of Huneke and Wiegand, Proc Am. Math. Soc. 147(7) (2019), 2733–2739.CrossRefGoogle Scholar
Christensen, L. W., Gorenstein dimensions, Lecture Notes in Math., vol. 1747 (Springer-Verlag, Berlin, 2000).CrossRefGoogle Scholar
Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Am. Math. Soc. 353 (2001), 1839–1883.CrossRefGoogle Scholar
Dao, H. and Takahashi, R., Classification of resolving subcategories and grade consistent functions, Int. Math. Res. Not. 2015(1) (2015), 119–149.CrossRefGoogle Scholar
Goto, S., Ozeki, K., Takahashi, R., Watanabe, K.-I. and Yoshida, K.-I., Ulrich ideals and modules, Math. Proc. Camb. Philos. Soc. 156 (2014), 137–166.CrossRefGoogle Scholar
Gulliksen, T. H., On the deviations of a local ring, Math. Scand. 47(1) (1980), 5–20.CrossRefGoogle Scholar
Iyama, O. and Wemyss, M., The classification of special Cohen-Macaulay modules, Math. Z. 265(1) (2010), 41–83.CrossRefGoogle Scholar
Jorgensen, D. A. and Şega, L. M., Independence of the total reflexivity conditions for modules, Algebr. Represent. Theory 9(2) (2006), 217–226.CrossRefGoogle Scholar
Kobayashi, T. and Takahashi, R., Rings whose ideals are isomorphic to trace ideals, Math. Nachr. 292(10) (2019), 2252–2261.CrossRefGoogle Scholar
Leuschke, G. J. and Wiegand, R., Cohen-Macaulay representations, Math. Surveys Monogr., vol. 181 (American Mathematical Society, Providence, RI, 2012).Google Scholar
Nasseh, S. and Takahashi, R., Local rings with quasi-decomposable maximal ideal, Math. Proc. Camb. Philos. Soc. 168(2) (2020), 305–322.CrossRefGoogle Scholar
Sally, J. D., Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra 56(1) (1979), 168–183.CrossRefGoogle Scholar
Takahashi, R., On G-regular local rings, Commun. Algebra 36(12) (2008), 4472–4491.CrossRefGoogle Scholar
Yassemi, S., G-dimension, Math. Scand. 77(2) (1995), 161–174.CrossRefGoogle Scholar