Intersections of Leray complexes and regularity of monomial ideals

For a simplicial complex X   and a field KK, let h˜i(X)=dimH˜i(X;K).It is shown that if X,YX,Y are complexes on the same vertex set, then for k⩾0k⩾0h˜k−1(X∩Y)⩽∑σ∈Y∑i+j=kh˜i−1(X[σ])⋅h˜j−1(lk(Y,σ)).A simplicial complex X is d-Leray   over KK, if H˜i(Y;K)=0 for all induced subcomplexes Y⊂XY⊂X and i⩾di⩾d. Let LK(X)LK(X) denote the minimal d such that X is d  -Leray over KK. The above theorem implies that if X,YX,Y are simplicial complexes on the same vertex set thenLK(X∩Y)⩽LK(X)+LK(Y).LK(X∩Y)⩽LK(X)+LK(Y).Reformulating this inequality in commutative algebra terms, we obtain the following result conjectured by Terai: If I,JI,J are square-free monomial ideals in S=K[x1,…,xn]S=K[x1,…,xn], thenreg(I+J)⩽reg(I)+reg(J)−1,reg(I+J)⩽reg(I)+reg(J)−1, where reg(I)reg(I) denotes the Castelnuovo–Mumford regularity of I.