University of Paderborn Department of Mathematics Diploma Thesis ...

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50 CHAPTER 3. OPERATIONS ON STABLE IDEALS(ii) Since I is saturated and the monomial x B does not contain the variable x n , the idealI con must also be saturated by Theorem 2.13 and the definition of the contraction.(iii) A contraction of the monomial x B of degree d > 0 will always include one moreminimal generator of degree d into the set of generators (i.e. the monomial x B itself)and remove a set of monomials of degree d + 1 from the set of generators.We sum up the two main results of the preceding remark in a lemma.Lemma 3.12. (Properties of the ideal I con ) Let I ⊂ R be a saturated stable ideal and letx B = x b 00 · . . . · x bss ∈ M, b s > 0, s ≤ n − 1 be a monomial with x B /∈ I, x B · x s ∈ I gand L(x B ) be contained in I. Then, the ideal I con (contraction of the monomial x B ) issaturated and stable.Proof. The assertion has been proved within Remark 3.11.Example 3.13. We consider the ideal I := (x 3 0, x 2 0x 1 , x 2 0x 2 2) ⊂ K[x 0 , x 1 , x 2 , x 3 ], which issaturated and stable. The monomial x 2 0x 2 is contractible in I, since it is not contained inI and the set L(x 2 0x 2 ) consists of the two monomials x 3 0 and x 2 0x 1 by Example 3.4. Thus,we obtain L(x 2 0x 2 ) ⊂ I, which shows that the conditions in Definition 3.9 are fulfilled. Thecontraction of x 2 0x 2 provides the setI cong = I g ∪ {x 2 0x 2 }\{x 2 0x 2 2} = {x 3 0, x 2 0x 1 , x 2 0x 2 }.As one can easily see, the ideal I con = (x 3 0, x 2 0x 1 , x 2 0x 2 ) is again stable and saturated.Example 3.14. As another example, consider J := (x 2 0, x 0 x 1 , x 0 x 2 ) in K[x 0 , x 1 , x 2 , x 3 ].The double saturation of J is the ideal sat x2 ,x 3(J) = (x 0 ). In the following, we will alsouse contractions, to compute the double saturation of saturated stable ideals. Hence, notethat we obtain the ideal sat x2 ,x 3(J) by contracting the monomial x 0 in J. This is possible,since x 0 x 2 is contained in J, x 0 itself is not a minimal generator of J and L(x 0 ) = ∅ isobviously a subset of the ideal. Contraction of x 0 provideswhich is the set, generating sat x2 ,x 3(J).J cong = J g ∪ {x 0 }\{x 2 0, x 0 x 1 , x 0 x 2 } = {x 0 },The process of the contraction of a monomial will not always reduce the number of monomialgenerators of the ideal considered. As one can see in the example stated above, thenumber of monomial generators of the new ideal after the contraction, might be the sameas the number of generators of the ideal one started with.Now we examine the effect of an expansion followed by a contraction of the same monomialon the set of generators of a given saturated and stable ideal I ⊂ R. In detail, we willanswer the questions: Given a saturated stable ideal I ⊂ R and a monomial x A ∈ M, x A
3.2. EXPANSIONS AND CONTRACTIONS OF MONOMIALS 51expandable in I g . Will x A be contractible in Igexpmonomial generators we started with?and, if yes, do we obtain the same set ofIndeed, the lemma stated below, gives a positive answer to this question.Lemma 3.15. Let I ⊂ R be a saturated stable ideal and I g its set of minimal generators.Let x A = x a 00 · . . . · x arr ∈ I g , a r > 0, be expandable in I g . Then the monomial x A iscontractible in Ig exp and (I exp ) con = I.Proof. Let x A be expandable in I g . Since I is stable, we get L(x A ) ⊂ I by Remark 3.6.Expansion via x A provides the set of monomial generatorsI expg = I g \ { x A} ∪ { x A · x r , x A · x r+1 , . . . , x A · x n−1}.Since none of the monomials of L(x A ) has been removed by the expansion, we get L(x A ) ⊂I exp . Furthermore, x A is not contained in I exp , since otherwise this contradicts the fact thatx A is a minimal generator of the ideal I, we started with. Thus, by Definition 3.9, themonomial x A is indeed contractible in Igexp . Contraction of x A in Igexp provides the setwhich shows I g = (I expg ) con .(Igexp ) con = Ig exp ∪ {x A }\{x A · x r , x A · x r+1 , . . . , x A · x A · x n−1 },Note that a contraction of a monomial followed by an expansion of the same monomial ispossible, but it will not always yield the same ideal, as one can see in the example below.Example 3.16. Let I := (x 2 0, x 0 x 1 , x 0 x 3 2) ⊂ K[x 0 , x 1 , x 2 , x 3 ]. Then x 0 is not contained inI, x 2 0 is contained in I and the left-shift of x 0 is the empty set, which is a subset of I.Hence, x 0 is contractible in I and the contraction provides the ideal generated by the setI cong = {x 2 0, x 0 x 1 , x 0 x 3 2} ∪ {x 0 }\{x 2 0, x 0 x 1 , x 0 x 2 } = {x 0 },i.e. I con = (x 0 , x 0 x 3 2) = (x 0 ). Since R(x 0 ) = {x 1 } is not contained in the set of minimalgenerators of I con , we expand the monomial x 0 in I con . This provides the set of monomialgenerators {x 2 0, x 0 x 1 , x 0 x 2 }, but the ideal generated by these monomials is not equal to theideal I, i.e. (I con ) exp ≠ I.In contrast to this, if we first expand the monomial x 0 x 1 in I, we obtain the ideal I exp =(x 2 0, x 0 x 2 1, x 0 x 1 x 2 , x 0 x 3 2). Contraction of the same monomial x 0 x 1 is possible and provides(I exp ) con = (x 2 0, x 0 x 1 , x 0 x 3 2) = I.The fact that the contraction of a monomial m in some saturated stable ideal I andafterwards the expansion of the same monomial does not yield the same ideal I again, isnot a problem at all. We will see later that in general we do not need that any contractionof a monomial followed by the expansion of the same monomial provides the same idealagain. It suffices that some special contractions fulfill this condition.
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