Squarefree monomial ideals with maximal depth

Ahad Rahimi

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 4, page 1111-1124
  • ISSN: 0011-4642

Abstract

top
Let ( R , 𝔪 ) be a Noetherian local ring and M a finitely generated R -module. We say M has maximal depth if there is an associated prime 𝔭 of M such that depth M = dim R / 𝔭 . In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.

How to cite

top

Rahimi, Ahad. "Squarefree monomial ideals with maximal depth." Czechoslovak Mathematical Journal 70.4 (2020): 1111-1124. <http://eudml.org/doc/297184>.

@article{Rahimi2020,
abstract = {Let $(R,\mathfrak \{m\})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\mathfrak \{p\}$ of $M$ such that depth $M=\dim R/\mathfrak \{p\}$. In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.},
author = {Rahimi, Ahad},
journal = {Czechoslovak Mathematical Journal},
keywords = {maximal depth; cycle graph; line graph; whisker graph; transversal polymatroidal ideal; power of edge ideal},
language = {eng},
number = {4},
pages = {1111-1124},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Squarefree monomial ideals with maximal depth},
url = {http://eudml.org/doc/297184},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Rahimi, Ahad
TI - Squarefree monomial ideals with maximal depth
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 4
SP - 1111
EP - 1124
AB - Let $(R,\mathfrak {m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\mathfrak {p}$ of $M$ such that depth $M=\dim R/\mathfrak {p}$. In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.
LA - eng
KW - maximal depth; cycle graph; line graph; whisker graph; transversal polymatroidal ideal; power of edge ideal
UR - http://eudml.org/doc/297184
ER -

References

top
  1. Brodmann, M., 10.2307/2042097, Proc. Am. Math. Soc. 74 (1979), 16-18. (1979) ZblA0395.13008MR0521865DOI10.2307/2042097
  2. Bruns, W., Herzog, J., 10.1017/cbo9780511608681, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). (1998) Zbl0909.13005MR1251956DOI10.1017/cbo9780511608681
  3. Faridi, S., 10.1016/j.jpaa.2003.11.014, J. Pure Appl. Algebra 190 (2004), 121-136. (2004) Zbl1045.05029MR2043324DOI10.1016/j.jpaa.2003.11.014
  4. Francisco, C. A., Hà, H. T., 10.1016/j.jcta.2007.06.004, J. Comb. Theory, Ser. A 115 (2008), 304-316. (2008) Zbl1142.13021MR2382518DOI10.1016/j.jcta.2007.06.004
  5. Frühbis-Krüger, A., Terai, N., Bounds for the regularity of monomial ideals, Mathematiche, Suppl. 53 (1998), 83-97. (1998) Zbl0951.13017MR1696019
  6. Herzog, J., Hibi, T., 10.1007/978-0-85729-106-6, Graduate Texts in Mathematics 260, Springer, London (2011). (2011) Zbl1206.13001MR2724673DOI10.1007/978-0-85729-106-6
  7. Herzog, J., Rauf, A., Vladoiu, M., 10.1007/s10801-012-0367-z, J. Algebr. Comb. 37 (2013), 289-312. (2013) Zbl1258.13014MR3011344DOI10.1007/s10801-012-0367-z
  8. Jacques, S., Betti Numbers of Graph Ideals: Ph.D. Thesis, University of Sheffield, Sheffield (2004), Available at https://arxiv.org/abs/math/0410107. (2004) 
  9. Lam, H. M., Trung, N. V., 10.1090/tran/7662, Trans. Am. Math. Soc. 372 (2019), 3211-3236. (2019) Zbl1420.13023MR3988608DOI10.1090/tran/7662
  10. Martínez-Bernal, J., Morey, S., Villarreal, R. H., 10.1007/s13348-011-0045-9, Collect. Math. 63 (2012), 361-374. (2012) Zbl1360.13027MR2957976DOI10.1007/s13348-011-0045-9
  11. Miller, E., Sturmfels, B., Yanagawa, K., 10.1006/jsco.1999.0290, J. Symb. Comput. 29 (2000), 691-708. (2000) Zbl0955.13008MR1769661DOI10.1006/jsco.1999.0290
  12. Rahimi, A., 10.1142/S021949881850202X, J. Algebra Appl. 17 (2018), Article ID 1850202, 12 pages. (2018) Zbl1409.13023MR3879078DOI10.1142/S021949881850202X
  13. Villarreal, R. H., 10.1201/b18224, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton (2015). (2015) Zbl1325.13004MR3362802DOI10.1201/b18224

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.