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The theory of \(L\)-complexes and weak liftings of complexes. (English) Zbl 0928.13003

The notion of a weak lifting of a finite module is introduced and studied by M. Auslander, S. Ding and Ø. Solberg [J. Algebra 156, No. 2, 273-317 (1993; Zbl 0778.13007)]. In this paper the author introduces the notion of an \(L\)-complex and studies lifting problems on \(L\)-complexes. Some conditions for a complex to be weakly liftable in terms of \(L\)-complexes are given. The author shows that the Maranda theorem holds for complexes, which generalizes some theorems of S. Ding and Ø. Solberg [Commun. Algebra 21, No. 4, 1161-1187 (1993; Zbl 0782.13019)]. In addition, the author shows how to construct the Eisenbud resolution for any given complex, and gives an application in the Cohen-Macaulay approximations, which can reproduce a result of S. Ding [see J. Algebra 153, No. 2, 271-288 (1992; Zbl 0793.13009)].
The last part in this paper is devoted to studying the virtual projective dimension and the complexity for a complex. These invariants are defined for modules, but naturally generalized to any complexes. The author shows that Avramov’s equality [see L. L. Avramov, Invent. Math. 96, No. 1, 71-101 (1989; Zbl 0677.13004)] is valid also for any bounded complex, and gives the duality theorem on the virtual injective dimension.

MSC:

13B02 Extension theory of commutative rings
13D25 Complexes (MSC2000)
13C11 Injective and flat modules and ideals in commutative rings
13C10 Projective and free modules and ideals in commutative rings
18G05 Projectives and injectives (category-theoretic aspects)
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References:

[1] Auslander, M.; Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.), 38, 5-37 (1989) · Zbl 0697.13005
[2] Auslander, M.; Ding, S.; Solberg, Ø., Liftings and weak liftings of modules, J. Algebra, 156, 273-317 (1993) · Zbl 0778.13007
[3] Avramov, L. L., Modules of finite virtual projective dimension, Invent. Math., 96, 71-101 (1989) · Zbl 0677.13004
[4] Ding, S., Cohen-Macaulay approximation and multiplicity, J. Algebra, 153, 271-288 (1992) · Zbl 0793.13009
[5] Ding, S.; Solberg, Ø., The Maranda theorem and liftings of modules, Comm. Algebra, 12, 1161-1187 (1993) · Zbl 0782.13019
[6] Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., 260, 35-64 (1980) · Zbl 0444.13006
[7] Foxby, H. B., Bounded complexes of flat modules, J. Pure Appl. Algebra, 15, 149-172 (1979) · Zbl 0411.13006
[8] Hartshorne, R., Residue and Duality. Residue and Duality, Springer Lecture Notes in Math., 20 (1966), Springer-Verlag: Springer-Verlag Berlin/New York
[9] Shamash, J., The Poincaré series of a local ring, J. Algebra, 12, 453-470 (1969) · Zbl 0189.04004
[10] Yoshino, Y., Cohen-Macaulay Modules over Cohen-Macaulay Rings. Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math. Soc. Lecture Note Series, 146 (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0745.13003
[11] Yoshino, Y., Maximal Buchsbaum modules of finite projective dimension, J. Algebra, 159, 240-264 (1993) · Zbl 0791.13009
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