Let $(R,m)$ be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero Cohen-Macaulay R-module of finite projective dimension or a nonzero finitely generated R-module of finite injective dimension. In this article, we will prove the complete intersection analogues of these facts. Also, by using complete intersection homological dimensions, we will characterize local rings which are either regular, complete intersection or Gorenstein.
N. Bourbaki, Commutative algebra, Chapter 1-7, Springer-Verlag, Berlin, 1998.
W. Bruns and J. Herzog, Cohen-Macaulay rings, in: Cambridge Studies in Advanced
Math. 39, 1993.
D. Bennis and N. Mahdou, First, second, and third change of rings theorems for
Gorenstein Homological dimensions, Comm. Algebra, 38 (10), 3837–3850, 2010.
K. Divaani-Aazar, F. Mohammadi Aghjeh Mashhad and M. Tousi, On the existence
of certain modules of finite Gorenstein homological dimensions, Comm. Algebra, 42,
1630–1643, 2014.
H-B. Foxby and A. Frankild, Cyclic modules of finite Gorenstein injective dimension
and Gorenstein rings, Illinois J. Math. 51 (1), 67–82, 2007.
H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc.
132, 1279–1283, 2004.
S. Iyengar, Depth for complexes, and intersection theorems, Math. Z. 230, 545–567,
1999.
C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applica-
tions à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck,
Publ. Math. Inst. Hautes Études Sci., 42, 47–119, 1973.
P. Roberts, Multiplicities and chern classes in local algebra, in: Cambridge Tracts in
Math. 133, Cambridge University Press, Cambridge, 1998.
P. Sahandi, T. Sharif and S. Yassemi, Homological flat dimensions,
arXiv:0709.4078v2.
J.P. Serre, Sur la dimension homologique des anneaux et des modules Noetheriens,
Proc. Intern. Symp., Tokyo-Nikko, 1955, Science Council of Japan, 175–189, 1956.
R. Takahashi, Some characterizations of Gorenstein local rings in terms of G-
dimension, Acta Math. Hungar. 104 (4), 315–322, 2004.
R. Takahashi, The existence of finitely generated modules of finite Gorenstein injective
dimension, Proc. Amer. Math. Soc. 134 (11), 3115–3121, 2006.
R. Takahashi, On G-regular local rings, Comm. Algebra, 36 (12), 4472–4491, 2008.
S.S. Wagstaff, Complete intersection dimensions for complexes, J. Pure Appl. Alge-
bra, 190, 267–290, 2004.
S.S. Wagstaff, Complete intersection dimensions and Foxby classes, J. Pure Appl.
Algebra, 212, 2594–2611, 2008.
S. Yassemi, A generalization of a theorem of Bass, Comm. Algebra, 35, 249–251,
2007.
Year 2019,
Volume: 48 Issue: 2, 359 - 364, 01.04.2019
N. Bourbaki, Commutative algebra, Chapter 1-7, Springer-Verlag, Berlin, 1998.
W. Bruns and J. Herzog, Cohen-Macaulay rings, in: Cambridge Studies in Advanced
Math. 39, 1993.
D. Bennis and N. Mahdou, First, second, and third change of rings theorems for
Gorenstein Homological dimensions, Comm. Algebra, 38 (10), 3837–3850, 2010.
K. Divaani-Aazar, F. Mohammadi Aghjeh Mashhad and M. Tousi, On the existence
of certain modules of finite Gorenstein homological dimensions, Comm. Algebra, 42,
1630–1643, 2014.
H-B. Foxby and A. Frankild, Cyclic modules of finite Gorenstein injective dimension
and Gorenstein rings, Illinois J. Math. 51 (1), 67–82, 2007.
H. Holm, Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc.
132, 1279–1283, 2004.
S. Iyengar, Depth for complexes, and intersection theorems, Math. Z. 230, 545–567,
1999.
C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale. Applica-
tions à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck,
Publ. Math. Inst. Hautes Études Sci., 42, 47–119, 1973.
P. Roberts, Multiplicities and chern classes in local algebra, in: Cambridge Tracts in
Math. 133, Cambridge University Press, Cambridge, 1998.
P. Sahandi, T. Sharif and S. Yassemi, Homological flat dimensions,
arXiv:0709.4078v2.
J.P. Serre, Sur la dimension homologique des anneaux et des modules Noetheriens,
Proc. Intern. Symp., Tokyo-Nikko, 1955, Science Council of Japan, 175–189, 1956.
R. Takahashi, Some characterizations of Gorenstein local rings in terms of G-
dimension, Acta Math. Hungar. 104 (4), 315–322, 2004.
R. Takahashi, The existence of finitely generated modules of finite Gorenstein injective
dimension, Proc. Amer. Math. Soc. 134 (11), 3115–3121, 2006.
R. Takahashi, On G-regular local rings, Comm. Algebra, 36 (12), 4472–4491, 2008.
S.S. Wagstaff, Complete intersection dimensions for complexes, J. Pure Appl. Alge-
bra, 190, 267–290, 2004.
S.S. Wagstaff, Complete intersection dimensions and Foxby classes, J. Pure Appl.
Algebra, 212, 2594–2611, 2008.
S. Yassemi, A generalization of a theorem of Bass, Comm. Algebra, 35, 249–251,
2007.
Mashhad, F. M. A. (2019). Characterizing local rings via complete intersection homological dimensions. Hacettepe Journal of Mathematics and Statistics, 48(2), 359-364.
AMA
Mashhad FMA. Characterizing local rings via complete intersection homological dimensions. Hacettepe Journal of Mathematics and Statistics. April 2019;48(2):359-364.
Chicago
Mashhad, Fatemeh Mohammadi Aghjeh. “Characterizing Local Rings via Complete Intersection Homological Dimensions”. Hacettepe Journal of Mathematics and Statistics 48, no. 2 (April 2019): 359-64.
EndNote
Mashhad FMA (April 1, 2019) Characterizing local rings via complete intersection homological dimensions. Hacettepe Journal of Mathematics and Statistics 48 2 359–364.
IEEE
F. M. A. Mashhad, “Characterizing local rings via complete intersection homological dimensions”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, pp. 359–364, 2019.
ISNAD
Mashhad, Fatemeh Mohammadi Aghjeh. “Characterizing Local Rings via Complete Intersection Homological Dimensions”. Hacettepe Journal of Mathematics and Statistics 48/2 (April 2019), 359-364.
JAMA
Mashhad FMA. Characterizing local rings via complete intersection homological dimensions. Hacettepe Journal of Mathematics and Statistics. 2019;48:359–364.
MLA
Mashhad, Fatemeh Mohammadi Aghjeh. “Characterizing Local Rings via Complete Intersection Homological Dimensions”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, 2019, pp. 359-64.
Vancouver
Mashhad FMA. Characterizing local rings via complete intersection homological dimensions. Hacettepe Journal of Mathematics and Statistics. 2019;48(2):359-64.