TY - JOUR

T1 - Linear syzygies of Stanley-Reisner ideals

AU - Reiner, V.

AU - Welker, V.

PY - 2001

Y1 - 2001

N2 - We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex Δ. The description is in terms of the homology of the canonical Alexander dual complex Δ*. As applications we are able to • prove for monomial ideals and j = 1 a conjecture of J. Herzog giving lower bounds on the number of i-syzygies in the linear strand of jth-syzygy modules. • show that the maps in the linear strand can be written using only ±1 coefficients if Δ* is a pseudomanifold, • exhibit an example where multigraded maps in the linear strand cannot be written using only ±1 coefficients. • compute the entire resolution explicitly when Δ* is the complex of independent sets of a matroid.

AB - We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex Δ. The description is in terms of the homology of the canonical Alexander dual complex Δ*. As applications we are able to • prove for monomial ideals and j = 1 a conjecture of J. Herzog giving lower bounds on the number of i-syzygies in the linear strand of jth-syzygy modules. • show that the maps in the linear strand can be written using only ±1 coefficients if Δ* is a pseudomanifold, • exhibit an example where multigraded maps in the linear strand cannot be written using only ±1 coefficients. • compute the entire resolution explicitly when Δ* is the complex of independent sets of a matroid.

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U2 - 10.7146/math.scand.a-14333

DO - 10.7146/math.scand.a-14333

M3 - Article

AN - SCOPUS:0035634921

SN - 0025-5521

VL - 89

SP - 117

EP - 132

JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

IS - 1

ER -