University of Paderborn Department of Mathematics Diploma Thesis ...

56 CHAPTER 3. OPERATIONS ON STABLE IDEALSProposition 3.23. Let p R/I (z) be the Hilbert polynomial of R/I for some saturated stableideal I sat xn−1 ,x n(I). Let m and ˜m be minimal monomial generators of I, such that m is˜mexpandable in I, ˜m contains the variable x n−1 and is contractible in I exp (expansionvia the monomial m). Then the ideal J = (I exp ) con , which can be obtained from I after the˜mexpansion of m in I and the contraction of in I exp has the same Hilbert polynomialas I, i.e. p R/J (z) = p R/I (z).x n−1Proof. Note that since I is saturated, the monomial m cannot contain the variable x n . Theexpansion of m in I provides the ideal I exp , where the minimal monomial generator m of Iis removed from the set of generators of I and the monomials m · x r , m · x r+1 , . . . , m · x n−1are included into the set of generators. Hence, the ideal I exp contains exactly one monomialof degree deg m + 1 less than the ideal I, namely the monomial m · x n , which is containedin I, but not in I exp . Since all other monomial generators of I are still contained in I exp ,similar observations for m · x i n, i > 0, showfor all j ≥ deg m + 1.x n−1h I exp(j) = h I (j) − 1Contraction of the monomial˜mx n−1in I exp provides the idealJ = (I exp ) con , where the monomial generator ˜m of I exp ˜m(and of I) is replaced by .x n−1Consequently, the ideal J contains exactly one more monomial of degree deg ˜m than I exp ,i.e. the monomiali > 0, we obtain˜mx n−1· x n , which is contained in J, but not in I exp . Consideringh J (j) = h I exp(j) + 1˜mx n−1· x i n,for all j ≥ deg ˜m. It follows h I (j) = h J (j) for all j ≥ max{deg m + 1, deg ˜m} andp I (z) = p J (z), which shows p R/I (z) = p R/J (z).Corollary 3.24. With the notation in Proposition 3.23, let m be a monomial generatormof I containing the variable x n−1 , such that is contractible in I. Let ˜m be expandablex n−1in I con m, contraction via the monomial . Then the ideal J := (I con ) exp , expansion viax n−1the monomial ˜m, has the same Hilbert polynomial as I.Proof. The assertion follows from the proof of Proposition 3.23, since we can again usemthe arguments on the growth of the Hilbert function: Contraction of gives h I con(j) =x n−1h I (j)+1. Expansion of ˜m in I con provides the ideal J with h J (j) = h I con(j)−1. Thus, as inthe proof of the proposition, it follows h I (j) = h j (j) for all j ≥ max{deg m−1, deg ˜m}.An example will demonstrate the preceding results, before we come to another corollaryof Proposition 3.23.