University of Paderborn Department of Mathematics Diploma Thesis ...

2.8. AN APPLICATION TO GOTZMANN’S REGULARITY THEOREM 43Proof. Let p R/I (z) be the Hilbert polynomial of R/I for some saturated ideal I ⊂ R. ByTheorem 2.25, there is a unique lexicographic ideal L p , such that the Hilbert polynomialof R/I equals that of R/L p , i.e. p R/I (z) = p R/Lp (z). Let d denote the maximal degree ofthe minimal generators of L p . Since L p is lexicographic, we know by Remark 1.20 thath R/Lp (d + 1) = h R/Lp (d) 〈d〉 . Hence, leth R/Lp (d) =( ) ( )kd kd−1+ + . . . +d d − 1( )kssdenote the d-th Macaulay representation of h R/Lp (d). By Gotzmann’s Persistence Theorem,we know( ) ( ) ( )z + (kd − d) z + (kd−1 − d)z + (ks − d)p R/I (z) = p R/Lp (z) =++ . . . +.(k d − d) (k d−1 − d) + 1(k s − d) + d − sThus, it follows by Gotzmann’s Regularity Theorem that d−s+1 is the upper bound to thedegrees of the generators of any ideal associated to p R/I (z) = p R/Lp (z). Since d ≤ d − s + 1by this upper bound, we must have d = d − s + 1, i.e. s = 1. Thus, the upper bound forthe degree of the generators is d and the ideal L p contains at least one minimal generatorof degree d, which proves the assertion of the theorem.The result of Theorem 2.34 is a special case of the regularity results presented in [1],Chapter 2.Remark 2.35. By Theorem 2.34, we conclude that to a given Hilbert polynomial p(z),the lexicographic ideal of Theorem 2.25 is maximal with respect to the degree of its monomialgenerators. In particular, it has a minimal monomial generator of maximal degreeallowed by Gotzmann’s Regularity Theorem. This degree bounds the degrees of minimalgenerators of all homogeneous ideals I ⊂ R with Hilbert polynomial p R/I (z) = p(z).This is the first remarkable property of the ideal L p – as we will see later at the endof Chapter 3 in Corollary 3.40, the ideal L p has some more interesting properties e.g. concerningits Hilbert function.We consider one example, where we explicitly compute the lexicographic ideal L p to agiven Hilbert polynomial p(z), such that the degree of one of the minimal generators of L preaches Gotzmann’s upper bound.We start with a Hilbert polynomial p R/I (z), where I ⊂ R is even not a stable ideal:Example 2.36. Let R := K[x 0 , x 1 , x 2 , x 3 , x 4 ] and I := (x 2 0, x 2 1), which provides p R/I (z) =2z 2 + 2z + 1. We have to write the Hilbert polynomial in the form of Theorem 2.32: