University of Paderborn Department of Mathematics Diploma Thesis ...

3.2. EXPANSIONS AND CONTRACTIONS OF MONOMIALS 49Definition 3.9. Let I ⊂ R be a saturated stable ideal, and let x B = x b 00 · . . . · x bss ∈ M,b s > 0, s ≤ n − 1 be a monomial with x B /∈ I and x B · x s ∈ I g . Furthermore, let L(x B ) becontained in the ideal I. Then we call x B contractible in I (or simply contractible, if theideal is clear, we are considering) and the contraction of x B in I is defined to be the idealI con , generated by the setI cong := I g ∪ { x B} \ { x B · x s , x B · x s+1 , . . . , x B · x n−1}.For consistency (which we will see later), we need a special definition of the contraction ofx B = 1. We define Igcon to be the setI cong := I g ∪ {1}\ {x s } ,where x s is the variable, such that s is the maximal index with x B · x s = x s ∈ I g .Remark 3.10. The definition of a contraction of a monomial above differs from the originaldefinition of a contraction in the thesis by Alyson Reeves (see [16]). There are good reasonsfor changing the original definitions. These reasons will become apparent to the reader,who is familiar with the results of [16], later on in Example 3.20.Note that by the definition of the contraction of a monomial x B in a stable ideal I, themonomial x B itself is not contained in I but in the ideal I con . Although this might sounda bit strange, this notation also appears in [16] and has been adapted to prevent a toostrong deviation from the work of Alyson Reeves.Remark 3.11. Let again I ⊂ R be a stable saturated ideal and let x B = x b 00 · . . . · x bssbe contractible in I.(i) We assert that I con again is a stable ideal. Since I is stable and x B is the onlymonomial included into the set Igcon , it suffices to show that x B fulfills the conditionof Theorem 2.7. We know that L(x B ) is a subset of I. Since we have{ }xB · x s , x B · x s+1 · . . . · x B · x n−1 ∩ L(x B ) = ∅,the set L(x B ) is still contained in the set of minimal generators of I con . Hence, allmonomials of the formx b 00 · . . . · x bs+1s · x b s+1−1s+1 · . . . · x brrfor 0 ≤ s ≤ r − 1 must be contained in I con and therefore fulfill the condition ofTheorem 2.7. We have to show that each of the monomialsx b 00 · . . . · x b j−1j−1 · xb j+1j · x b j+1j+1 · . . . · xb k−1k−1 · xb k−1k· x b k+1k+1 · . . . · xbs sis contained in I con for all 0 ≤ j < k ≤ s. We know that the monomialx b 00 · . . . · x b j−1j−1 · xb jj · x b j+1j+1 · . . . · xb k−1+1k−1· x b k−1k· x b k+1k+1 · . . . · xbs sis contained in I con . Thus, we can divide the monomial by the variable x k−1 andmultiply it by x j , which proves the assertion./∈ I