Presentations of rings with a chain of semidualizing modules


  • Ensiyeh Amanzadeh
  • Mohammad T. Dibaei



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.


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.

Christensen, L. W., Semi-dualizing complexes and their Auslander categories, Trans. Amer. Math. Soc. 353 (2001), no. 5, 1839–1883.

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.

Foxby, H.-B., Gorenstein modules and related modules, Math. Scand. 31 (1972), 267–284.

Frankild, A. and Sather-Wagstaff, S., Reflexivity and ring homomorphisms of finite flat dimension, Comm. Algebra 35 (2007), no. 2, 461–500.

Frankild, A. and Sather-Wagstaff, S., The set of semidualizing complexes is a nontrivial metric space, J. Algebra 308 (2007), no. 1, 124–143.

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.

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.

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.

Sather-Wagstaff, S., Semidualizing modules,, 2010.

Sather-Wagstaff, S., Lower bounds for the number of semidualizing complexes over a local ring, Math. Scand. 110 (2012), no. 1, 5–17.

Sharp, R. Y., Finitely generated modules of finite injective dimension over certain Cohen-Macaulay rings, Proc. London Math. Soc. (3) 25 (1972), 303–328.

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.



How to Cite

Amanzadeh, E., & Dibaei, M. T. (2017). Presentations of rings with a chain of semidualizing modules. MATHEMATICA SCANDINAVICA, 121(2), 161–185.