On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Kosar Abolfath Beigi; Kamran Divaani-Aazar; Massoud Tousi

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 989-1002
  • ISSN: 0011-4642

Abstract

top
Let R be a local ring and C a semidualizing module of R . We investigate the behavior of certain classes of generalized Cohen-Macaulay R -modules under the Foxby equivalence between the Auslander and Bass classes with respect to C . In particular, we show that generalized Cohen-Macaulay R -modules are invariant under this equivalence and if M is a finitely generated R -module in the Auslander class with respect to C such that C ⊗ R M is surjective Buchsbaum, then M is also surjective Buchsbaum.

How to cite

top

Abolfath Beigi, Kosar, Divaani-Aazar, Kamran, and Tousi, Massoud. "On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence." Czechoslovak Mathematical Journal 72.4 (2022): 989-1002. <http://eudml.org/doc/298906>.

@article{AbolfathBeigi2022,
abstract = {Let $R$ be a local ring and $C$ a semidualizing module of $R$. We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$-modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$. In particular, we show that generalized Cohen-Macaulay $R$-modules are invariant under this equivalence and if $M$ is a finitely generated $R$-module in the Auslander class with respect to $C$ such that $C\otimes _RM$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum.},
author = {Abolfath Beigi, Kosar, Divaani-Aazar, Kamran, Tousi, Massoud},
journal = {Czechoslovak Mathematical Journal},
keywords = {Auslander class; Bass class; Buchsbaum module; dualizing module; generalized Cohen-Macaulay module; local cohomology; semidualizing module; surjective Buchsbaum module},
language = {eng},
number = {4},
pages = {989-1002},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence},
url = {http://eudml.org/doc/298906},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Abolfath Beigi, Kosar
AU - Divaani-Aazar, Kamran
AU - Tousi, Massoud
TI - On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 989
EP - 1002
AB - Let $R$ be a local ring and $C$ a semidualizing module of $R$. We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$-modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$. In particular, we show that generalized Cohen-Macaulay $R$-modules are invariant under this equivalence and if $M$ is a finitely generated $R$-module in the Auslander class with respect to $C$ such that $C\otimes _RM$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum.
LA - eng
KW - Auslander class; Bass class; Buchsbaum module; dualizing module; generalized Cohen-Macaulay module; local cohomology; semidualizing module; surjective Buchsbaum module
UR - http://eudml.org/doc/298906
ER -

References

top
  1. Bourbaki, N., Elements of Mathematics. Commutative Algebra. Chapters 1-7, Springer, Berlin (1998). (1998) Zbl0902.13001MR1727221
  2. Bruns, W., Herzog, J., 10.1017/cbo9780511608681.004, Cambridge Studies in Advanced Mathematics 39. Cambridge University Press, Cambridge (1993). (1993) Zbl0788.13005MR1251956DOI10.1017/cbo9780511608681.004
  3. Christensen, L. W., 10.1090/S0002-9947-01-02627-7, Trans. Am. Math. Soc. 353 (2001), 1839-1883. (2001) Zbl0969.13006MR1813596DOI10.1090/S0002-9947-01-02627-7
  4. Christensen, L. W., Frankild, A., Holm, H., 10.1016/j.jalgebra.2005.12.007, J. Algebra 302 (2006), 231-279. (2006) Zbl1104.13008MR2236602DOI10.1016/j.jalgebra.2005.12.007
  5. Foxby, H.-B., Hyperhomological algebra & commutative rings, notes in preparation, . 
  6. Herzog, J., Komplex Auflösungen und Dualität in der lokalen Algebra: Habilitationsschrift, Universität Regensburg, Regensburg (1974), German. (1974) 
  7. Kawasaki, T., 10.1007/BF02571897, Math. Z. 218 (1995), 191-205. (1995) Zbl0814.13017MR1318153DOI10.1007/BF02571897
  8. Miyazaki, C., 10.3836/tjm/1270133544, Tokyo J. Math. 12 (1989), 1-20. (1989) Zbl0696.13016MR1001728DOI10.3836/tjm/1270133544
  9. Rotman, J. J., 10.1007/b98977, Universitext. Springer, New York (2009). (2009) Zbl1157.18001MR2455920DOI10.1007/b98977
  10. Sather-Wagstaff, S., Semidualizing modules, Available at https://www.ndsu.edu/pubweb/ {ssatherw/DOCS/sdm.pdf} (2000), 109 pages. (2000) 
  11. Schenzel, P., Trung, N. V., Cuong, N. T., 10.1002/mana.19780850106, Math. Nachr. 85 (1978), 57-73 German. (1978) Zbl0398.13014MR0517641DOI10.1002/mana.19780850106
  12. Sharp, R. Y., 10.1112/plms/s3-25.2.303, Proc. Lond. Math. Soc., III. Ser. 25 (1972), 303-328. (1972) Zbl0244.13015MR0306188DOI10.1112/plms/s3-25.2.303
  13. Stückrad, J., Vogel, W., 10.1007/978-3-662-02500-0, Springer, Berlin (1986). (1986) Zbl0606.13018MR0881220DOI10.1007/978-3-662-02500-0
  14. Takahashi, R., White, D., 10.7146/math.scand.a-15121, Math. Scand. 106 (2010), 5-22. (2010) Zbl1193.13012MR2603458DOI10.7146/math.scand.a-15121
  15. Trung, N. V., 10.1017/S0305004100060308, Math. Proc. Camb. Philos. Soc. 93 (1983), 35-47. (1983) Zbl0509.13024MR0684272DOI10.1017/S0305004100060308
  16. Yamagishi, K., 10.1017/S0305004100070341, Math. Proc. Camb. Philos. Soc. 110 (1991), 261-279. (1991) Zbl0760.13010MR1113425DOI10.1017/S0305004100070341

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.