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Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals

Published online by Cambridge University Press:  16 March 2010

Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam. e-mail: and
Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam. e-mail: and


Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits also exist.

As a consequence, it is shown that there are integers p > 0 and 0 ≤ ed = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

Research Article
Copyright © Cambridge Philosophical Society 2010

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