TY - JOUR
T1 - On ideals with the Rees property
AU - Migliore, Juan
AU - Miró-Roig, Rosa M.
AU - Murai, Satoshi
AU - Nagel, Uwe
AU - Watanabe, Junzo
N1 - Funding Information:
Part of the work for this paper was done while the first author was sponsored by the National Security Agency under Grant Number H98230-12-1-0204. Part of the work for this paper was done while the second author was sponsored by the Grant MTM2010-15256. Part of the work for this paper was done while the third author was sponsored by KAKENHI 22740018. Part of the work for this paper was done while the fourth author was sponsored by the National Security Agency under Grant Number H98230-12-1-0247. This work was started at the workshop “Aspects of SLP and WLP,” held at Hawaii Tokai International College (HTIC) in September 2012, which was financially supported by JSPS KAKENHI 24540050. The authors thank HTIC for their kind hospitality. Also, we would like to thank Mats Boij for valuable discussions at the workshop.
PY - 2013/11
Y1 - 2013/11
N2 - A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal I ⊂ S is said to be m-full if mI:y=I for some y ∈ m, where m is the graded maximal ideal of S. It was proved by one of the authors that m-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not m-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.
AB - A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal J ⊂ S which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal I ⊂ S is said to be m-full if mI:y=I for some y ∈ m, where m is the graded maximal ideal of S. It was proved by one of the authors that m-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not m-full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.
UR - http://www.scopus.com/inward/record.url?scp=84887477508&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84887477508&partnerID=8YFLogxK
U2 - 10.1007/s00013-013-0565-5
DO - 10.1007/s00013-013-0565-5
M3 - Article
AN - SCOPUS:84887477508
SN - 0003-889X
VL - 101
SP - 445
EP - 454
JO - Archiv der Mathematik
JF - Archiv der Mathematik
IS - 5
ER -