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

This article is cited in 33 scientific papers (total in 33 papers)

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
Received: 24.08.2000

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
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    3. Miller C., “The Frobenius endomorphism and homological dimensions”, Commutative Algebra: Interactions with Algebraic Geometry, Contemporary Mathematics Series, 331, 2003, 207–234  crossref  mathscinet  zmath  isi
    4. A. A. Gerko, “The structure of the set of semidualizing complexes”, Russian Math. Surveys, 59:5 (2004), 954–955  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. Gerko A., “On the structure of the set of semidualizing complexes”, Illinois J. Math., 48:3 (2004), 965–976  mathscinet  zmath  isi  elib
    6. Asadollahi J., Salarian Sh., “Buchsbaum and monomial conjecture dimension”, Comm. Algebra, 32:10 (2004), 3969–3979  crossref  mathscinet  zmath  isi  scopus
    7. Takahashi R., Yoshino Yu., “Characterizing Cohen-Macaulay local rings by Frobenius maps”, Proc. Amer. Math. Soc., 132:11 (2004), 3177–3187  crossref  mathscinet  zmath  isi  scopus
    8. Sazeedeh R., “Strongly torsion free, copure flat and Matlis reflexive modules”, J. Pure Appl. Algebra, 192:1-3 (2004), 265–274  crossref  mathscinet  zmath  isi  scopus  scopus
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    11. Asadollahi J., Salarain Sh., “Lower and upper bounds for Cohen-Macaulay dimension”, Bull. Austral. Math. Soc., 71:2 (2005), 337–346  crossref  mathscinet  zmath  isi
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    14. Sazeedeh R., “Strongly torsion-free modules and local cohomology over Cohen-Macaulay rings”, Comm. Algebra, 33:4 (2005), 1127–1135  crossref  mathscinet  zmath  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    17. Veliche O., “Gorenstein projective dimension for complexes”, Trans. Amer. Math. Soc., 358:3 (2006), 1257–1283  crossref  mathscinet  zmath  isi  scopus  scopus
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    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  crossref  mathscinet  zmath  isi  scopus  scopus
    21. Azami J., “Weakly $G_K$-perfect and integral closure of ideals”, Comm. Algebra, 36:12 (2008), 4500–4508  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    22. Sahandi P., Yassemi S., “Filter rings under flat base change”, Algebra Colloq., 15:3 (2008), 463–470  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    27. Takahashi R., “Classifying thick subcategories of the stable category of Cohen-Macaulay modules”, Advances in Mathematics, 225:4 (2010), 2076–2116  crossref  mathscinet  zmath  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus
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  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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