| Intersections of Leray Complexes and Regularity of Monomial Ideals (2005) | |||||||||||||||
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| For a simplicial complex X and a field K, let ˜ hi(X) = dim ˜ Hi(X; K). It is shown that if X, Y are complexes on the same vertex set, then for k ≥ 0 ˜hk−1(X ∩ Y) ≤ � � ˜hi−1(X[σ]) · ˜ hj−1(lk(Y, σ)). σ∈Y i+j=k A simplicial complex X is d-Leray over K, if ˜ Hi(Y; K) = 0 for all induced subcomplexes Y ⊂ X and i ≥ d. Let LK(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 LK(X ∩ Y) ≤ LK(X) + LK(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,..., xn], then reg(I + J) ≤ reg(I) + reg(J) − 1 where reg(I) denotes the Castelnuovo-Mumford regularity of I. | |||||||||||||||
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