$C$-CANONICAL MODULES

Abstract

Let $C$ be a semidualizing module over a commutative Noetherian local ring $R$. In this paper we introduce a new class of modules, namely $C$-canonical modules which are a generalization of canonical modules. It is shown that if the canonical module exists then the $C$-canonical module exists and the converse holds under special conditions. Also, a new characterization of Gorenstein local rings is given via $C$-canonical modules.

Keywords

• Semidualizing modules
• dualizing modules
• local cohomology
• Auslander class
• Bass class

References

1. Y. Aoyama, On the depth and the projective dimension of the canonical module, Japan. J. Math., 6 (1980), 61-66.
2. Y. Aoyama, Some basic results on canonical modules, J. Math. Kyoto Univ., 23 (1983), 85-94.
3. M. P. Brodmann, R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Second edition, Cambridge University Press, 2013.
4. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised edition, Cambridge University Press, Cambridge, 1993.
5. L. W. Christensen, Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc., 353(5) (2001), 1839-1883.
6. L. W. Christensen and S. Sather-Wagstaff, A Cohen-Macaulay algebra has only finitely many semidualizing modules, Math. Proc. Cambridge Philos. Soc., 145(3) (2008), 601-603.
7. Mohammad T. Dibaei and Arash Sadeghi, Linkage of modules and the Serre conditions, J. Pure Appl. Algebra, 219 (2015), 4458-4478.
8. H.-B. Foxby, Gorenstein modules and related modules, Math. Scand., 31 (1972), 267-284.

9. A. J. Frankild, S. Sather-Wagstaff, and A. Taylor, Relations between semidualizing complexes, J. Commut. Algebra, 1(3) (2009), 393-436.
10. E. S. Golod, $G$-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov., 165 (1984), 62-66.
11. R. Hartshorne, Local Cohomology, A seminar given by A. Grothendieck, Harvard University, Fall 1961, Springer-Verlag, Berlin-New York, 1967.
12. J. Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings, Lect. Notes in Math., Vol. 238, Springer-Verlag, Berlin-New York, 1971.
13. M. Hochster and C. Huneke, Indecomposable canonical modules and connectedness, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 197-208.
14. H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1986.
15. S. Nasseh and S. Sather-Wagstaff, Geometric aspects of representation theory for DG algebras: answering a question of Vasconcelos, J. Lond. Math. Soc. (2), 96(1) (2017), 271-292.
16. I. Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc., 32 (1972), 417-420.
17. R. Y. Sharp, Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings, Proc. Lond. Math. Soc. (3), 25 (1972), 303-328.
18. R. Takahashi and D. White, Homological aspects of semidualizing modules, Math. Scand., 106(1) (2010), 5-22.
19. W. V. Vasconcelos, Divisor Theory in Module Categories, North-Holland Mathematics Studies, No. 14., North-Holland Publishing Co., Amsterdam-Oxford, 1974.
20. S. Sather-Wagstaff, Semidualizing modules and the divisor class group, Illinois J. Math., 51(1) (2007), 255-285.
21. S. Sather-Wagstaff, Semidualizing Modules, in preperation, URL: https://www.ndsu.edu/pubweb/ ssatherw/DOCS/sdm.pdf.