Intersections of Leray complexes and regularity of monomial ideals

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 k ⩾ 0 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 ; K ) = 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 [ x 1 , … , x n ] , then reg ( I + J ) ⩽ reg ( I ) + reg ( J ) − 1 , where reg ( I ) denotes the Castelnuovo–Mumford regularity of I.