TY - JOUR
T1 - Betti numbers of lex ideals over some Macaulay-Lex rings
AU - Mermin, Jeff
AU - Murai, Satoshi
N1 - Funding Information:
The first author is supported by an NSF Postdoctoral fellowship (award No. DMS-0703625). The second author is supported by JSPS Research Fellowships for Young Scientists.
PY - 2010/3
Y1 - 2010/3
N2 - Let A = K [x1,.,xn] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R = A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Füredi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.
AB - Let A = K [x1,.,xn] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R = A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Füredi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.
KW - Colored simplicial complexes
KW - Graded Betti numbers
KW - Hilbert functions
KW - Lex ideals
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U2 - 10.1007/s10801-009-0192-1
DO - 10.1007/s10801-009-0192-1
M3 - Article
AN - SCOPUS:77952430273
SN - 0925-9899
VL - 31
SP - 299
EP - 318
JO - Journal of Algebraic Combinatorics
JF - Journal of Algebraic Combinatorics
IS - 2
ER -