University of Paderborn Department of Mathematics Diploma Thesis ...

70 CHAPTER 3. OPERATIONS ON STABLE IDEALSAfterwards we expand some monomial ˜m > m (with respect to the lexicographic order) ofdegree ˜d < d in L conp . Hence, we obtain h (L conp ) exp(i) = h L con (i)−1 for all i ≥ ˜d. By combiningpthe two equations it follows h R/(L conp) exp(i) = h R/L p(i) for 0 ≤ i < ˜d, h R/(L conp ) exp(i) ≥ h R/L p(i)for ˜d ≤ i ≤ d − 1 and h R/(L conp ) exp(i) = h R/L p(i) for i ≥ d. Since L H is obtained from L pby such pairs of expansions and contractions, we conclude by induction that h R/LH (i) ≥h R/Lp (i) for all i ≥ 0.Later, in Chapter 4, we give an algorithm to compute the values of Hilbert functions.There we consider an example, where we can see that the values of the Hilbert function ofR/L p are minimal among all values of the Hilbert functions associated to the same Hilbertpolynomial p(z) in the sense of Corollary 3.40.3.4 Stable ideals with the same Hilbert polynomialIn the preceding section we provided the theoretical basis for the construction of all saturatedstable ideals with the same double saturation and the same Hilbert polynomial:We saw that we can use the unique lexicographic ideal of Theorem 2.25 associated to thegiven Hilbert polynomial to construct all saturated stable ideals with the same doublesaturation and the same Hilbert polynomial. Since our aim is to construct all saturatedstable ideals to a given Hilbert polynomial, we need to find a way to compute even thosesaturated stable ideals, whose double saturation differs from the double saturation of theunique lexicographic ideal associated to the Hilbert polynomial.In a special situation, i.e. in the case that the Hilbert polynomial is constant, we can findall saturated stable ideals with the given Hilbert polynomial by the following lemma:Lemma 3.41. Let p(z) = c, c ∈ N, be the Hilbert polynomial of R/I for I ⊂ R asaturated stable ideal. Then sat xn−1 ,x n(I) = R and all saturated stable ideals J ⊂ R suchthat p R/J (z) = p(z) can be computed from I by pairs of contractions and expansions as inProposition 3.28 and Theorem 3.32.Proof. First, we claim that if I ⊂ R is a saturated stable ideal with p(z) = p R/I (z) = c,c ∈ N, there are integers a 0 , . . . , a n−1 ∈ N, such that x a 00 , . . . , x a n−1n−1 ∈ I.Assume x j n−1 /∈ I for all j ∈ N. Since I is stable, all monomials of the form x j−in−1 · x i n,0 ≤ i ≤ j, are not contained in I either. Hence, for j ∈ N there are at least j+1 monomials,which are not contained in I. But then we obtain h R/I (j) ≥ j +1, which is a contradiction,since c = p(j) = h R/I (j) ≥ j + 1 is not satisfied for all j ≥ c.Since I contains x a n−1n−1 for some integer a n−1 , it follows that x a 00 , . . . , x a n−2n−2 are contained inI for some integers a 0 , . . . , a n−2 , since I is stable. The double saturation sat xn−1 ,x n(I) of I isobtained from I by setting x n−1 := 1 in all of its generators. This shows sat xn−1 ,x n(I) = R.It follows that the double saturation of all saturated stable ideals with a constant Hilbertpolynomial equals the whole ring R. Hence, by Proposition 3.28 and Theorem 3.32, weconclude that all such ideals are linked by finite sequences of expansions and contractions,which proves the assertion.