Intersections of Leray complexes and regularity of monomial ideals

For a simplicial complex X and a field K, let over(h, ̃)_i (X) = dim over(H, ̃)_i (X ; K). It is shown that if X, Y are complexes on the same vertex set, then for k ≥ 0over(h, ̃)_{k - 1} (X ∩ Y) ≤ under(∑, σ ∈ Y) under(∑, i + j = k) over(h, ̃)_{i - 1} (X [σ]) ṡ over(h, ̃)_{j - 1} (lk (Y, σ)) . A simplicial complex X is d-Leray over K, if over(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 thenL_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], thenreg (I + J) ≤ reg (I) + reg (J) - 1, where reg (I) denotes the Castelnuovo-Mumford regularity of I.