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 Mat. Sb., 2001, Volume 192, Number 8, Pages 79–94 (Mi msb587)

On homological dimensions

A. A. Gerko

M. V. Lomonosov Moscow State University

Abstract: For finite modules over a local ring the general problem is considered of finding an extension of the class of modules of finite projective dimension preserving various properties. In the first section the concept of a suitable complex is introduced, which is a generalization of both a dualizing complex and a suitable module. Several properties of the dimension of modules with respect to such complexes are established. In particular, a generalization of Golod's theorem on the behaviour of $G_K$-dimension with respect to a suitable module $K$ under factorization by ideals of a special kind is obtained and a new form of the Avramov–Foxby conjecture on the transitivity of $G$-dimension is suggested. In the second section a class of modules containing modules of finite CI-dimension is considered, which has some additional properties. A dimension constructed in the third section characterizes the Cohen–Macaulay rings in precisely the same way as the class of modules of finite projective dimension characterizes regular rings and the class of modules of finite CI-dimension characterizes complete intersections.

DOI: https://doi.org/10.4213/sm587

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English version:
Sbornik: Mathematics, 2001, 192:8, 1165–1179

Bibliographic databases:

UDC: 512.717
MSC: 13D05, 13C15

Citation: A. A. Gerko, “On homological dimensions”, Mat. Sb., 192:8 (2001), 79–94; Sb. Math., 192:8 (2001), 1165–1179

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Christensen L.W., Foxby H.-B., Frankild A., “Restricted homological dimensions and Cohen-Macaulayness”, J. Algebra, 251:1 (2002), 479–502
2. Veliche O., “Construction of modules with finite homological dimensions”, J. Algebra, 250:2 (2002), 427–449
3. Miller C., “The Frobenius endomorphism and homological dimensions”, Commutative Algebra: Interactions with Algebraic Geometry, Contemporary Mathematics Series, 331, 2003, 207–234
4. A. A. Gerko, “The structure of the set of semidualizing complexes”, Russian Math. Surveys, 59:5 (2004), 954–955
5. Gerko A., “On the structure of the set of semidualizing complexes”, Illinois J. Math., 48:3 (2004), 965–976
6. Asadollahi J., Salarian Sh., “Buchsbaum and monomial conjecture dimension”, Comm. Algebra, 32:10 (2004), 3969–3979
7. Takahashi R., Yoshino Yu., “Characterizing Cohen-Macaulay local rings by Frobenius maps”, Proc. Amer. Math. Soc., 132:11 (2004), 3177–3187
8. Sazeedeh R., “Strongly torsion free, copure flat and Matlis reflexive modules”, J. Pure Appl. Algebra, 192:1-3 (2004), 265–274
9. Sather-Wagstaff S., “Complete intersection dimensions for complexes”, J. Pure Appl. Algebra, 190:1-3 (2004), 267–290
10. Asadollahi J., Salarian Sh., “Generalized Cohen-Macaulay dimension”, J. Algebra, 273:1 (2004), 384–394
11. Asadollahi J., Salarain Sh., “Lower and upper bounds for Cohen-Macaulay dimension”, Bull. Austral. Math. Soc., 71:2 (2005), 337–346
12. Asadollahi J., “Cohen-Macaulay dimension of modules over Noetherian rings”, Rocky Mountain J. Math., 35:4 (2005), 1069–1076
13. Araya T., Takahashi R., Yoshino Yu., “Homological invariants associated to semi-dualizing bimodules”, J. Math. Kyoto Univ., 45:2 (2005), 287–306
14. Sazeedeh R., “Strongly torsion-free modules and local cohomology over Cohen-Macaulay rings”, Comm. Algebra, 33:4 (2005), 1127–1135
15. Asadollahi J., Puthenpurakal T.J., “An analogue of a theorem due to Levin and Vasconcelos”, Commutative Algebra and Algebraic Geometry, Contemporary Mathematics Series, 390, 2005, 9–15
16. Salarian S., Sather-Wagstaff S., Yassemi S., “Characterizing local rings via homological dimensions and regular sequences”, J. Pure Appl. Algebra, 207:1 (2006), 99–108
17. Veliche O., “Gorenstein projective dimension for complexes”, Trans. Amer. Math. Soc., 358:3 (2006), 1257–1283
18. Holm H., White D., “Foxby equivalence over associative rings”, J. Math. Kyoto Univ., 47:4 (2007), 781–808
19. Holm H., Jørgensen P., “Cohen-macaulay homological dimensions”, Rend. Semin. Mat. Univ. Padova, 117 (2007), 87–112
20. Takahashi R., Watanabe Kei-ichi, “Totally reflexive modules constructed from smooth projective curves of genus $g\ge 2$”, Arch. Math., 89:1 (2007), 60–67
21. Azami J., “Weakly $G_K$-perfect and integral closure of ideals”, Comm. Algebra, 36:12 (2008), 4500–4508
22. Sahandi P., Yassemi S., “Filter rings under flat base change”, Algebra Colloq., 15:3 (2008), 463–470
23. Sather-Wagstaff S., Sharif T., White D., “Stability of Gorenstein categories”, J. Lond. Math. Soc. (2), 77:2 (2008), 481–502
24. Bergh P.A., “Modules with reducible complexity. II”, Comm. Algebra, 37:6 (2009), 1908–1920
25. Sazeedeh R., “Gorenstein injectivity of the section functor”, Forum Mathematicum, 22:6 (2010), 1117–1127
26. Sather-Wagstaff S., Sharif T., White D., “Tate cohomology with respect to semidualizing modules”, Journal of Algebra, 324:9 (2010), 2336–2368
27. Takahashi R., “Classifying thick subcategories of the stable category of Cohen-Macaulay modules”, Advances in Mathematics, 225:4 (2010), 2076–2116
28. Tokuji Araya, Kei-Ichiro Iima, Ryo Takahashi, “On the Left Perpendicular Category of the Modules of Finite Projective Dimension”, Communications in Algebra, 40:8 (2012), 2693
29. Yoshizawa T., “On Gorenstein Injectivity of TOP Local Cohomology Modules”, Proc. Amer. Math. Soc., 140:6 (2012), 1897–1907
30. Nasseh S., Sather-Wagstaff S., “Cohen Factorizations: Weak Functoriality and Applications”, J. Pure Appl. Algebr., 219:3, SI (2015), 622–645
31. Liang L., Yang G., “Lower complete intersection dimension over local homomorphisms”, Indian J. Pure Appl. Math., 47:4 (2016), 673–686
32. Sadeghi A., J. Pure Appl. Algebr., 221:6 (2017), 1344–1365
33. Sanders W., “Classifying resolving subcategories”, Pac. J. Math., 286:2 (2017), 401–438
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