Geodesic
Squarefree Lexsegment Ideals with Linear Resolution
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 1, pp. 275-291.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

In this paper we determine all squarefree completely lexsegment ideals which have a linear resolution. Let $M_d$ denote the set of all squarefree monomials of degree $d$ in a polynomial ring $k[x_1, \ldots, x_n ]$ in $n$ variables over a field $k$. We order the monomials lexicographically such that $x_1 > x_2 > \ldots > x_n$, thus a lexsegment (of degree $d$) is a subset of $M_d$ of the form $L(u, v) = \{w \in M_d: u \geq w \geq v\}$ for some $u, v \in M_d$ con $u \geq v$. An ideal generated by a lexsegment is called a lexsegment ideal. We describe the procedure to determine when such an ideal has a linear resolution.
In questo articolo determiniamo tutti gli ideali completamente lexsegmento squarefree con risoluzione lineare. Sia $M_d$ l'insieme di tutti i monomi squarefree di grado $d$ in un anello di polinomi $k[x_1, \ldots, x_n ]$ in $n$ variabili su un campo $k$. Ordiniamo i monomi lessicograficamente in modo che $x_1 > x_2 > \ldots > x_n$, così un lexsegmento (di grado $d$) è un sottoinsieme di $M_{d}$ del tipo $L(u, v) = \{w \in M_d: u \geq w \geq v\}$ per qualche $u, v \in M_d$ con $u \geq v$. Un ideale generato da un lexsegmento è chiamato ideale lexsegmento. Descriviamo la procedura per determinare quando un tale ideale ha risoluzione lineare. 
@article{BUMI_2008_9_1_1_a11,
     author = {Bonanzinga, Vittoria and Sorrenti, Loredana},
     title = {Squarefree {Lexsegment} {Ideals} with {Linear} {Resolution}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {275--291},
     publisher = {mathdoc},
     volume = {Ser. 9, 1},
     number = {1},
     year = {2008},
     zbl = {1164.13009},
     mrnumber = {2424294},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_1_a11/}
}
TY  - JOUR
AU  - Bonanzinga, Vittoria
AU  - Sorrenti, Loredana
TI  - Squarefree Lexsegment Ideals with Linear Resolution
JO  - Bollettino della Unione matematica italiana
PY  - 2008
SP  - 275
EP  - 291
VL  - 1
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_1_a11/
LA  - en
ID  - BUMI_2008_9_1_1_a11
ER  - 
%0 Journal Article
%A Bonanzinga, Vittoria
%A Sorrenti, Loredana
%T Squarefree Lexsegment Ideals with Linear Resolution
%J Bollettino della Unione matematica italiana
%D 2008
%P 275-291
%V 1
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_1_a11/
%G en
%F BUMI_2008_9_1_1_a11
Bonanzinga, Vittoria; Sorrenti, Loredana. Squarefree Lexsegment Ideals with Linear Resolution. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 1, pp. 275-291. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_1_a11/

[1] A. Aramova - L. L. Avramov - J. Herzog, Resolutions of monomial ideals and cohomology over exterior algebras, Trans. AMS, 352 (2) (2000), 579-594. | DOI | MR | Zbl

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

[3] A. Aramova - J. Herzog - T. Hibi, Squarefree lexsegment ideals, Math. Z., 228 (2) (1998), 353-378. | DOI | MR | Zbl

[4] V. Bonanzinga, Lexsegment ideals in the exterior algebra, in "Geometric and Combinatorial aspects of commutative algebra", (J. Herzog and G. Restuccia Eds.), Lect. Notes in Pure and Appl. Math., 4, Dekker, New York, (1999), 43-56. | MR

[5] E. De Negri - J. Herzog, Completely lexsegment ideals, Proc. Amer. Math. Soc., 126 (12) (1998), 3467-3473. | DOI | MR | Zbl

[6] S. Eliahou - M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra, 129 (1990), 1-25. | DOI | MR | Zbl

[7] H. A. Hulett - H. M. Martin, Betti numbers of lexsegment ideals, J. Algebra, 275 (2004), 2, 629-638. | DOI | MR | Zbl