  For a simplicial complex X and a field K, let h_i(X)=\dim \tilde{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\cap Y) \leq \sum_{\sigma \in Y} \sum_{i+j=k} h_{i-1}(X[\sigma])\cdot
h_{j-1}(\lk(Y,\sigma)) . A simplicial complex X is d-Leray over K, if h_i(Y)=0
for all induced subcomplexes Y \subset X and i \geq 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 \cap Y) \leq 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) \leq reg(I)+reg(J)-1 where reg(I)
denotes the Castelnuovo-Mumford regularity of I.
