Presentations of rings with a chain of semidualizing modules

Authors

  • Ensiyeh Amanzadeh
  • Mohammad T. Dibaei

DOI:

https://doi.org/10.7146/math.scand.a-96668

Abstract

Inspired by Jorgensen et al., it is proved that if a Cohen-Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$-modules of length $n$, then $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

References

Avramov, L. L. and Martsinkovsky, A., Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393–440. https://doi.org/10.1112/S0024611502013527

Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839–1883. https://doi.org/10.1090/S0002-9947-01-02627-7

Chrstensen, L. W. and Sather-Wagstaff, S., A Cohen-Macaulay algebra has only finitely many semidualizing modules, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 3, 601–603. https://doi.org/10.1017/S0305004108001552

Foxby, H.-B., Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284. https://doi.org/10.7146/math.scand.a-11434

Frankild, A. and Sather-Wagstaff, S., Reflexivity and ring homomorphisms of finite flat dimension, Comm. Algebra 35 (2007), no. 2, 461–500. https://doi.org/10.1080/00927870601052489

Frankild, A. and Sather-Wagstaff, S., The set of semidualizing complexes is a nontrivial metric space, J. Algebra 308 (2007), no. 1, 124–143. https://doi.org/10.1016/j.jalgebra.2006.06.017

Gerko, A., On the structure of the set of semidualizing complexes, Illinois J. Math. 48 (2004), no. 3, 965–976.

Golod, E. S., $G$-dimension and generalized perfect ideals, Trudy Mat. Inst. Steklov. 165 (1984), 62–66.

Hartshorne, R., Residues and duality, Lecture Notes in Mathematics, no. 20, Springer-Verlag, Berlin-New York, 1966.

Holm, H. and Jørgensen, P., Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205 (2006), no. 2, 423–445. https://doi.org/10.1016/j.jpaa.2005.07.010

Jorgensen, D. A., Leuschke, G. J., and Sather-Wagstaff, S., Presentations of rings with non-trivial semidualizing modules, Collect. Math. 63 (2012), no. 2, 165–180. https://doi.org/10.1007/s13348-010-0024-6

Nasseh, S. and Sather-Wagstaff, S., A local ring has only finitely many semidualizing complexes up to shift-isomorphism, preprint arXiv:1201.0037v2 [math.AC], 2012.

Reiten, I., The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420. https://doi.org/10.2307/2037829

Sather-Wagstaff, S., Semidualizing modules, http://www.ndsu.edu/pubweb/~ssatherw/DOCS/sdm.pdf, 2010.

Sather-Wagstaff, S., Lower bounds for the number of semidualizing complexes over a local ring, Math. Scand. 110 (2012), no. 1, 5–17. https://doi.org/10.7146/math.scand.a-15192

Sharp, R. Y., Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings, Proc. London Math. Soc. (3) 25 (1972), 303–328. https://doi.org/10.1112/plms/s3-25.2.303

Vasconcelos, W. V., Divisor theory in module categories, North-Holland Mathematics Studies, no. 14, Notas de Matemática no. 53, North-Holland Publishing Co., Amsterdam, 1974.

Published

2017-10-22

How to Cite

Amanzadeh, E., & Dibaei, M. T. (2017). Presentations of rings with a chain of semidualizing modules. MATHEMATICA SCANDINAVICA, 121(2), 161–185. https://doi.org/10.7146/math.scand.a-96668

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Articles