University of Paderborn Department of Mathematics Diploma Thesis ...

66 CHAPTER 3. OPERATIONS ON STABLE IDEALSProof of Claim 1. By the algorithm described in Remark 3.31, we successively contract thelargest monomial generators of L H (with respect to the lexicographic order) with the lastvariable x n−1 , until there is only one monomial generator left, which contains x n−1 . Let mbe the first monomial in L H with respect to the lexicographic order, such that m containsmlast variable x n−1 . Then we contract in L H , which provides some ideal L conHx .n−1Assume L conH is not a lexicographic ideal. Since the only effect of the contraction of min L H is removing m from the set of minimal generators and includingx n−1mx n−1into the set ofminimal generators, L conH is not a lexicographic ideal, if [Lconsubset of [R] deg m−1 . Hence, there is at least one monomial ˜m ∈ [R] deg m−1 with ˜m >H ] deg m−1 is not a lexicographicmand ˜m /∈ [L conH ] deg m−1. It follows ˜m · x n−1 > m. Since m ∈ [L conH ] deg m and [L conH ] deg mis a lexicographic subset of [R] deg m , we obtain ˜m · x n−1 ∈ [L conH ] deg m, and in particular˜m·x n−1 ∈ L H . Therefore, there is a minimal monomial generator ¯m·x n−1 in L H , such that˜m · x n−1 is divisible by ¯m · x n−1 . Note that the monomial generator contains x n−1 , sinceotherwise it would divide ˜m, which is a contradiction to the fact that ˜m /∈ [L conH ] deg m−1.Furthermore, we have deg( ¯m · x n−1 ) ≤ deg( ˜m · x n−1 ) = deg m. But this contradicts thefact that we have chosen m to be the first monomial generator in L H with the last variablex n−1 .It follows that L conHx n−1is a lexicographic ideal. Iteration of the process proves that any ofthe contractions performed successively due to Remark 3.31 provides a lexicographic ideal.Since any of the expansions described in Remark 3.31 to take L H to J only increases theexponent of the last and only monomial generator containing the last variable x n−1 , theseexpansions will always give a lexicographic ideal. It follows that J is a lexicographic ideal,which proves Claim 1.Since we know that J is a lexicographic ideal with only one minimal monomial generatorcontaining x n−1 , set m := x arn−r−1 · x a r−1n−r · x a r−2n−r+1 · . . . · x a 1n−2 · x a 0n−1 to be this monomialwith suitable r, a 0 , a 1 , . . . , a r−2 , a r−1 , a r .Claim 2. The set of minimal generators of J is given by{x 0 , x 1 , . . . , x n−r−2 ,x ar+1n−r−1,x arn−r−1 · x a r−1+1n−r ,x arn−r−1 · x a r−1n−r · x a r−2+1n−r+1 , . . . ,x arn−r−1 · x a r−1n−r · x a r−2n−r+1 · . . . · x a 2n−3 · x a 1+1n−2 ,x arn−r−1 · x a r−1n−r · x a r−2n−r+1 · . . . · x a 1n−2 · x a 0n−1 = m}.Proof of Claim 2. First we show that each of the monomials in the above set is containedin J. Then, since J is stable and lexicographic, we will be able to conclude that all