Intersections of Leray complexes and regularity of monomial ideals

2006/01/30 by Kalai, Gil, Meshulam, Roy · 1 citation
#13F55 #55U10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0601745

Abstract

For a simplicial complex X and a field K, let h_i(X)=dim H_i(X;K). It is shown that if X,Y are complexes on the same vertex set, then for all k h_k-1(X∩ Y) ≤ ∑_{σ∈ Y} ∑_i+j=k h_i-1(X[σ])⋅ h_j-1(\lk(Y,σ)) . A simplicial complex X is d-Leray over K, if h_i(Y)=0 for all induced subcomplexes Y ⊂ X and i ≥ d. Let L_K(X) denote the minimal d such that X is d-Leray over K. The above theorem implies that if X,Y are simplicial complexes on the same vertex set then L_K(X ∩ Y) ≤ L_K(X) +L_K(Y). Reformulating this inequality in commutative algebra terms, we obtain the following result conjectured by Terai: If I,J are square-free monomial ideals in S=K[x₁,...,x_n], then reg(I+J) ≤ reg(I)+reg(J)-1 where reg(I) denotes the Castelnuovo-Mumford regularity of I.

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