Hilbert polynomials and powers of ideals

Abstract

Abstract. 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, the Hilbert coefficients ei(M/I k M ) are polynomial functions. Given two families of graded ideals (I k ) k≥0 and (J k ) k≥0 with J k ⊂ I k for all k with the property that J k J ℓ ⊂ J k+ℓ and I k I ℓ ⊂ I k+ℓ for all k and ℓ, and such that the algebras A = L k≥0 J k and B = L k≥0 I k are finitely generated, we show the function k →0 (I k /J k ) is of quasipolynomial type, say given by the polynomials P0, . . . , Pg−1.we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that lim k→∞ ℓ(Γm (S/I k ))/k n ∈ Q. We also study analogous statements in the local case.

Asymptotic growth of algebras associated to powers of ideals

“…, x n ] are polynomial functions in k for k ≫ 0. In an explicit form this statement is given in [14]. In the same paper the question is raised whether for any ideal in a Noetherian local ring (R, m, K) a similar statement is true.…”

Higher Iterated Hilbert Coefficients of the Graded Components of Bigraded Modules

“…It is shown in Corollary 3.2 that for any finitely generated graded Smodule M , the modules Tor S i (M, I k ) are finitely graded S-modules which for k ≫ 0 have constant Krull dimension, and furthermore in Corollary 3.5 it is shown that the higher iterated Hilbert coefficients (which appear as the coefficients of the higher iterated Hilbert polynomials) are all polynomials functions. A related result has been shown in [4] for the case M/I k M and in [5] for the case Tor Observe that knowing all higher iterated Hilbert coefficients of a graded module is equivalent to knowing its h-vector, and hence the Hilbert series of the module. This is the reason why we are not only interested in the ordinary Hilbert coefficients, but in all higher iterated Hilbert coefficients.…”

Multiplicities and Rees valuations

“…In [4], [3], and [7] this limit is shown to exist in many special cases. In particular, in [4] it is shown to exist for graded ideals in a finitely generated standard graded algebra of depth greater than one over a field of characteristic zero.…”