Hilbert polynomials and powers of ideals

• Autores: Jürgen Herzog , Tony J. Puthenpurakal, Jugal Verma
• Localización: Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, Vol. 145, Nº 3, 2008, págs. 623-642
• Idioma: inglés
• DOI: 10.1017/s0305004108001540
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• Resumen

  • The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k=0 and (Jk)k=0 with Jk Ik for all k with the property that JkJl Jk+l and IkIl Ik+l for all k and l, and such that the algebras and are finitely generated, we show the function k e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg-1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.