Geodesic
Ideals Generated by Reverse Lexicographic Segments
Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 53-69.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $k$ be a field, and let $S=k[x_1,\dots,x_n]$ be the polynomial ring in $x_1,\dots,x_n$ with coefficients in the field $k$. We study ideals of $S$ which are generated by reverse lexicographic segments of monomials of $S$. An ideal generated by a reverse lexicographic segment is called a completely reverse lexicographic segment ideal if all iterated shadows of the set of generators are reverse lexicographic segments. We characterize all completely reverse lexicographic segment ideals of $S$ and determine conditions under which they have a linear resolution.
Keywords: polynomial ring, ideal, reverse lexicographic segment, iterated shadow, linear resolution. 
@article{MZM_2011_89_1_a5,
     author = {M. Crupi and M. La Barbiera},
     title = {Ideals {Generated} by {Reverse} {Lexicographic} {Segments}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {53--69},
     publisher = {mathdoc},
     volume = {89},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a5/}
}
TY  - JOUR
AU  - M. Crupi
AU  - M. La Barbiera
TI  - Ideals Generated by Reverse Lexicographic Segments
JO  - Matematičeskie zametki
PY  - 2011
SP  - 53
EP  - 69
VL  - 89
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a5/
LA  - ru
ID  - MZM_2011_89_1_a5
ER  - 
%0 Journal Article
%A M. Crupi
%A M. La Barbiera
%T Ideals Generated by Reverse Lexicographic Segments
%J Matematičeskie zametki
%D 2011
%P 53-69
%V 89
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a5/
%G ru
%F MZM_2011_89_1_a5
M. Crupi; M. La Barbiera. Ideals Generated by Reverse Lexicographic Segments. Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 53-69. http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a5/

[1] H. Hulett, H. M. Martin, “Betti numbers of lex-segment ideals”, J. Algebra, 275:2 (2004), 629–638 | DOI | MR | Zbl

[2] A. Aramova, E. De Negri, J. Herzog, “Lexsegment ideals with linear resolution”, Illinois J. Math., 42:3 (1998), 509–523 | MR | Zbl

[3] E. De Negri, J. Herzog, “Completely lexsegment ideals”, Proc. Amer. Math. Soc., 126:12 (1998), 3467–3473 | DOI | MR | Zbl

[4] F. S. Macaulay, “Some properties of enumerations in the theory of modular systems”, Proc. London Math. Soc. (2), 26:1 (1927), 531–555 | DOI | Zbl

[5] D. Bayer, M. Stillman, “A theorem on refining division orders by the reverse lexicographic order”, Duke Math. J., 55:2 (1987), 321–328 | DOI | MR | Zbl

[6] D. Bayer, M. Stillman, “A criterion for detecting $m$-regularity”, Invent. Math., 87:1 (1987), 1–11 | DOI | MR | Zbl

[7] T. Deery, “Rev-lex segment ideals and minimal Betti numbers”, The Curves Seminar at Queen's (Kingston, ON, 1995), v. X, Queen's Papers in Pure and Appl. Math., 102, Queen's Univ., Kingston, ON, 1996, 193–219 | MR | Zbl

[8] A. Aramova, J. Herzog, “Koszul cycles and Eliahou–Kervaire type resolutions”, J. Algebra, 181:2 (1996), 347–370 | DOI | MR | Zbl

[9] S. Eliahou, M. Kervaire, “Minimal resolutions of some monomial ideals”, J. Algebra, 129:1 (1990), 1–25 | DOI | MR | Zbl

[10] D. Eisenbud, Commutative Algebra. With a View Toward Algebraic Geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995 | MR | Zbl

[11] M. Crupi, M. La Barbiera, Minimal Graded Resolutions of Reverse Lexsegment Ideals, Preprint, 2009