×

zbMATH — the first resource for mathematics

Reducing invariants and total reflexivity. (English) Zbl 1441.13039
Let \(R\) denote a commutative Noetherian local ring with unique maximal ideal \(\mathfrak{m}\) and residue field \(k\). In the paper under review, the authors introduce reducing versions of homological invariants (in particular those of homological dimensions) of \(R\)-modules. They show that a module can have finite reducing Gorenstein dimension, even if it has infinite Gorenstein dimension and they give such examples. As a main result the authors prove the following.
Theorem. Let \(M\) be an \(R\)-module which has finite reducing Gorenstein dimension. Then \(\mathrm{G-dim}_R (M) = sup\{i\in \mathbb{Z}| \mathrm{Ext}^i_R(M,R)\neq 0\}\).
MSC:
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13C13 Other special types of modules and ideals in commutative rings
13C14 Cohen-Macaulay modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] T. Araya, R. Takahashi, and Y. Yoshino, Homological invariants associated to semi-dualizing bimodules, J. Math. Kyoto Univ. 45 (2005), no. 2, 287-306. · Zbl 1096.16001
[2] T. Araya and Y. Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), no. 11, 3793-3806. · Zbl 0906.13002
[3] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. (1969), no. 94. · Zbl 0204.36402
[4] L. L. Avramov, “Infinite free resolutions” in Six Lectures on Commutative Algebra (Bellaterra, 1996), Progr. Math. 166, Birkhäuser, Basel, 1998, 1-118. · Zbl 0934.13008
[5] L. L. Avramov, V. N. Gasharov, and I. V. Peeva, Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 67-114. · Zbl 0918.13008
[6] P. A. Bergh, Modules with reducible complexity, J. Algebra 310 (2007), 132-147. · Zbl 1117.13016
[7] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993. · Zbl 0788.13005
[8] L. W. Christensen, Gorenstein Dimensions, Lecture Notes in Math. 1747, Springer, Berlin, 2000. · Zbl 0965.13010
[9] E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Ser. 106, Cambridge Univ. Press, Cambridge, 1985.
[10] H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1279-1283. · Zbl 1062.16008
[11] F. Ischebeck, Eine dualität zwischen den funktoren Ext und Tor, J. Algebra 11 (1969), 510-531. · Zbl 0191.01306
[12] D. A. Jorgensen and G. J. Leuschke, On the growth of the Betti sequence of the canonical module, Math. Z. 256 (2007), no. 3, 647-659. · Zbl 1141.13011
[13] D. A. Jorgensen and L. M. Şega, Independence of the total reflexivity conditions for modules, Algebr. Represent. Theory 9 (2006), no. 2, 217-226. · Zbl 1101.13021
[14] G. J. Leuschke and R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys Monogr. 181, Amer. Math. Soc., Providence, RI, 2012.
[15] V. Maşiek, Gorenstein dimension and torsion of modules over commutative Noetherian rings, Comm. Algebra 28 (2000), no. 12, 5783-5811. · Zbl 1002.13005
[16] S. Sather-Wagstaff, Semidualizing modules and the divisor class group, Illinois J. Math. 51 (2007), no. 1, 255-285. · Zbl 1127.13007
[17] R. Takahashi, On G-regular local rings, Comm. Algebra 36 (2008), no. 12, 4472-4491. · Zbl 1156.13009
[18] S. Yassemi, G-dimension, Math. Scand. 77 (1995), 161-174. · Zbl 0864.13010
[19] Y. Yoshino, Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0745.13003
[20] Y. Yoshino, “Modules of G-dimension zero over local rings with the cube of maximal ideal being zero” in Commutative Algebra, Singularities and Computer Algebra (Sinaia, 2002), NATO Sci. Ser. II Math. Phys. Chem. 115, Kluwer, Dordrecht, 2003, 255-273. · Zbl 1071.13505
[21] Y. Yoshino, Homotopy categories of unbounded complexes of projective modules, preprint, arXiv:1805.05705 [math.AC].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.