Hostname: page-component-6bf8c574d5-gr6zb Total loading time: 0 Render date: 2025-02-24T06:17:31.138Z Has data issue: false hasContentIssue false
  • English
  • Français

Local rings with quasi-decomposable maximal ideal

Published online by Cambridge University Press:  26 September 2018

SAEED NASSEH  and
RYO TAKAHASHI
SAEED NASSEH
Affiliation:
Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia 30460, U.S.A. e-mail: snasseh@georgiasouthern.edu
RYO TAKAHASHI
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan. e-mail: takahashi@math.nagoya-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Let (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 is a direct summand of (a direct sum of) syzygies of M. Applying this result to the case where 𝔪 is quasi-decomposable, we obtain several classifications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 305 - 322
Copyright
Copyright © Cambridge Philosophical Society 2018 

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.)

Footnotes

†

Partly supported by JSPS Grants-in-Aid for Scientific Research 16K05098.

References

REFERENCES

[1] Auslander, M. and Buchweitz, R.-O. The homological theory of maximal Cohen–Macaulay approximations. Mém. Soc. Math. France (N.S.), No. 38 (1989), 5–37.Google Scholar
[2] Avramov, L. L., Iyengar, S. B., Nasseh, S. and Sather–Wagstaff, S. Persistence of homology over commutative noetherian rings, in preparation.Google Scholar
[3] Bruns, W. and Herzog, J. Cohen–Macaulay rings, revised edition. Camb. Stud. in Adv. Math. 39 (Cambridge University Press, Cambridge, 1998).Google Scholar
[4] Buchweitz, R.-O. Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings. Unpublished paper written in 1986, available at http://hdl.handle.net/ 1807/16682.Google Scholar
[5] Christensen, L. W., Striuli, J. and Veliche, O. Growth in the minimal injective resolution of a local ring. J. Lond. Math. Soc. (2) 81 (2010), no. 1, 24–44.Google Scholar
[6] Dao, H. and Takahashi, R. Classification of resolving subcategories and grade consistent functions. Int. Math. Res. Not. IMRN (2015) no. 1, 119–149.Google Scholar
[7] Dress, A. and Krämer, H. Bettireihen von Faserprodukten lokaler Ringe Math. Ann. 215 (1975), 79–82.Google Scholar
[8] Fossum, R., Foxby, H.-B., Griffith, P. and Reiten, I. Minimal injective resolutions with applications to dualising modules and Gorenstein modules. Inst. Hautes Études Sci. Publ. Math. 45 (1975), 193–215.Google Scholar
[9] Goto, S., Takahashi, R. and Taniguchi, N. Almost Gorenstein rings - towards a theory of higher dimension J. Pure Appl. Algebra 219 (2015), no. 7, 2666–2712.Google Scholar
[10] Huneke, C. and Watanabe, K.-I. Upper bound of multiplicity of F-pure rings. Proc. Amer. Math. Soc. 143 (2015), no. 12, 5021–5026.Google Scholar
[11] Krause, H. and Stevenson, G. A note on thick subcategories of stable derived categories. Nagoya Math. J. 212 (2013), 87–96.Google Scholar
[12] Matsui, H. and Takahashi, R. Singularity categories and singular equivalences for resolving subcategories. Math. Z. 285 (2017), no. 1–2, 251–286.Google Scholar
[13] Matsumura, H. Commutative ring theory. Translated from the Japanese by Reid, M., Second edition. Camb. Stud. in Adv. Math. 8, (Cambridge University Press, Cambridge, 1989).Google Scholar
[14] Matsuoka, N. The defining ideal of an almost Gorenstein numerical semigroup ring. In preparation.Google Scholar
[15] Moore, W. F. Cohomology over fiber products of local rings. J. Algebra 321 (2009), no. 3, 758–773.Google Scholar
[16] Nasseh, S. and Sather–Wagstaff, S. Vanishing of Ext and Tor over fiber products. Proc. Amer. Math. Soc. 145 (2017), no. 11, 4661–4674.Google Scholar
[17] Ogoma, T. Existence of dualizing complexes. J. Math. Kyoto Univ. 24 (1984), no. 1, 27–48.Google Scholar
[18] Orlov, D. O. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc. Steklov Inst. Math. 246 (2004), no. 3, 227–248.Google Scholar
[19] Schoutens, H. Projective dimension and the singular locus. Comm. Algebra 31 (2003), no. 1, 217–239.Google Scholar
[20] Takahashi, R. Classifying thick subcategories of the stable category of Cohen–Macaulay modules. Adv. Math. 225 (2010), no. 4, 2076–2116.Google Scholar
[21] Takahashi, R. Classifying resolving subcategories over a Cohen–Macaulay local ring. Math. Z. 273 (2013), no. 1–2, 569–587.Google Scholar
[22] Takahashi, R. Thick subcategories over Gorenstein local rings that are locally hypersurfaces on the punctured spectra. J. Math. Soc. Japan 65 (2013), no. 2, 357–374.Google Scholar