Normalization of monomial ideals and Hilbert functions
نویسنده
چکیده
We study the normalization of a monomial ideal and show how to compute its Hilbert function if the ideal is zero dimensional. A positive lower bound for the second coefficient of the Hilbert polynomial is shown. 1 Normalization of monomial ideals In the sequel we use [3, 11] as references for standard terminology and notation on commutative algebra and polyhedral cones. We denote the set of non-negative real (resp. integer) numbers by R+ (resp. N). Let R = k[x1, . . . , xd] be a polynomial ring over a field k and let I be a monomial ideal of R generated by x1 , . . . , xq . If R is the Rees algebra of I, R = R[It], we call its integral closure R the normalization of I. This algebra has for components the integral closures of the powers of I, R = R⊕ It⊕ · · · ⊕ It ⊕ · · · ⊂ R⊕ It⊕ · · · ⊕ Iit ⊕ · · · = R. Two of the results below (Propositions 1.2 and 1.6) complement the following: Theorem 1.1 [12, Theorem 7.58] Ib = IIb−1 for b ≥ d. Proposition 1.2 Let r0 be the rank of the matrix (v1, . . . , vq). If v1, . . . , vq lie in a hyperplane of R not containing the origin, then Ib = IIb−1 for b ≥ r0. Proof. Assume b ≥ r0. Notice that we invariably have IIb−1 ⊂ Ib. To show the reverse inclusion take x ∈ Ib. Let R+A ′ be the cone generated by the set A = {(v1, 1), . . . , (vq, 1), e1, . . . , en}, 2000 Mathematics Subject Classification. Primary 13B22; Secondary 13D40, 13F20.
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