Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T13:21:46.087Z Has data issue: false hasContentIssue false
  • English
  • Français

REFLEXIVITY AND CONNECTEDNESS

Published online by Cambridge University Press:  22 December 2014

SEAN SATHER-WAGSTAFF
SEAN SATHER-WAGSTAFF*
Affiliation:
North Dakota State University, Department of Mathematics # 2750, P.O. Box 6050, Fargo, ND 58108-6050, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given a finitely generated module over a commutative noetherian ring that satisfies certain reflexivity conditions, we show how failure of the semidualizing property for the module manifests in a disconnection of the prime spectrum of the ring.

Keywords

13D0213D0713D0913G05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 231 - 240
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra (Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969). MR 0242802 (39#4129)Google Scholar
2.Auslander, M., Anneaux de Gorenstein, et torsion en algèbre commutative, Séminaire d'Algèbre Commutative dirigé par Pierre Samuel, vol. 1966/67, (Secrétariat mathématique, Paris, 1967). MR 37 #1435Google Scholar
3.Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, vol. 94 (American Mathematical Society, Providence, R.I., 1969). MR 42 #4580Google Scholar
4.Avramov, L. L. and Foxby, H.-B., Ring homomorphisms and finite Gorenstein dimension, Proc. London Math. Soc. 75 (3) (1997), 241–270. MR 98d:13014Google Scholar
5.Avramov, L. L., Iyengar, S. B. and Lipman, J., Reflexivity and rigidity for complexes. I. Commutative rings, Algebra Number Theory 4 (1) (2010), 47–86. MR 2592013CrossRefGoogle Scholar
6.Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Am. Math. Soc. 353 (5) (2001), 1839–1883. MR 2002a:13017Google Scholar
7.Enochs, E. E., Jenda, O. M. G. and Xu, J. Z., Foxby duality and Gorenstein injective and projective modules, Trans. Amer. Math. Soc. 348 (8) (1996), 3223–3234. MR 1355071 (96k:13010)CrossRefGoogle Scholar
8.Foxby, H.-B., Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284. MR 48 #6094Google Scholar
9.Foxby, H.-B., Gorenstein dimensions over Cohen-Macaulay rings, in Proceedings of the international conference on commutative algebra (Osnabrück, Germany) (Bruns, W., Editor) (Universität Osnabrück, 1994), 59–63.Google Scholar
10.Frankild, A. and Sather-Wagstaff, S., Reflexivity and ring homomorphisms of finite flat dimension, Commun. Algebra 35 (2) (2007), 461–500. MR 2294611CrossRefGoogle Scholar
11.Frankild, A. and Sather-Wagstaff, S., The set of semidualizing complexes is a nontrivial metric space, J. Algebra 308 (1) (2007), 124–143. MR 2290914Google Scholar
12.Frankild, A. J., Sather-Wagstaff, S. and Taylor, A., Relations between semidualizing complexes, J. Commut. Algebra 1 (3) (2009), 393–436. MR 2524860CrossRefGoogle Scholar
13.Golod, E. S., G-dimension and generalized perfect ideals, Algebraic geometry and its applications, Trudy Mat. Inst. Steklov. 165 (1984), 62–66. MR 85m:13011Google Scholar
14.Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, vol. 20 (Springer-Verlag, Berlin, 1966). MR 36 #5145Google Scholar
15.Hartshorne, R., Local cohomology, A seminar given by A. Grothendieck, Harvard University, Fall, vol. 1961 (Springer-Verlag, Berlin, 1967). MR 0224620 (37 #219)Google Scholar
16.Sather-Wagstaff, S., Complete intersection dimensions and Foxby classes, J. Pure Appl. Algebra 212 (12) (2008), 2594–2611. MR 2452313 (2009h:13015)CrossRefGoogle Scholar
17.Sather-Wagstaff, S., Bass numbers and semidualizing complexes, Commutative Algebra and its Applications (Walter de Gruyter, Berlin, 2009), 349–381. MR 2640315Google Scholar
18.Vasconcelos, W. V., Divisor theory in module categories, North-Holland Mathematics Studies, No. 14, Notas de Matemática No. 53. [Notes on Mathematics, No. 53] (North-Holland Publishing Co., Amsterdam, 1974). MR 0498530 (58 #16637)Google Scholar
19.Yassemi, S., G-dimension, Math. Scand. 77 (2) (1995), 161–174. MR 97d:13017Google Scholar