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

• Autores: Lê Tuân Hoa, Tran Nam Trung
• Localización: Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, Vol. 149, Nº 2, 2010, págs. 229-246
• Idioma: inglés
• DOI: 10.1017/s0305004110000071
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• Resumen

  • Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+njIn1j + · · · + njIpjn. 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?8ad(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 = e = d = 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.