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62 CHAPTER 3. OPERATIONS ON STABLE IDEALSTo take J to its double saturation, we have to contract the monomials x 0 , x 4 1x 3 2, x 4 1x 2 2x 3 3,x 4 1x 2 2x 2 3, x 4 1x 2 2x 3 , x 4 1x 2 2. Hence, we obtain the idealsJ (0) = (x 2 0, x 0 x 1 , x 0 x 2 , x 0 x 3 , x 5 1, x 4 1x 4 2, x 4 1x 3 2x 3 , x 4 1x 2 2x 4 3)J (1) = (x 0 , x 5 1, x 4 1x 4 2, x 4 1x 3 2x 3 , x 4 1x 2 2x 4 3)J (2) = (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 4 3)J (3) = (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 3 3)J (4) = (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 2 3)J (5) = (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2x 3 )J (6) = (x 0 , x 5 1, x 4 1x 3 2, x 4 1x 2 2) = sat x3 ,x 4(J)Since only the first two contractions are different in the process taking I to sat x3 ,x 4(I) andJ to sat x3 ,x 4(J), we proceed as follows: First we contract the monomial x 4 1x 2 2x 5 3 in I, whichprovides the ideal I (1) . Then we expand the monomial x 4 1x 3 2 in I (1) , which provides theideal(x 0 , x 5 1, x 4 1x 4 2, x 4 1x 3 2x 3 , x 4 1x 2 2x 5 3).Next, we contract the monomial x 4 1x 2 2x 4 3 and obtain(x 0 , x 5 1, x 4 1x 4 2, x 4 1x 3 2x 3 , x 4 1x 2 2x 4 3).The last step is the expansion of x 0 in this ideal, which provides(x 2 0, x 0 x 1 , x 0 x 2 , x 0 x 3 , x 5 1, x 4 1x 4 2, x 4 1x 3 2x 3 , x 4 1x 2 2x 4 3) = Jand we are done.In particular, the ideals I and J are linked by the contraction-expansion pairs x 4 1x 2 2x 5 3, x 4 1x 3 2and x 4 1x 2 2x 4 3, x 0 .Remark 3.31. Our aim is to prove the result of Proposition 3.28 for all saturated stableideals I, I ′ ⊂ R with the same double saturation and the same Hilbert polynomial. Theproblem is, we cannot assume in general that one of the ideals I, I ′ contains only onemonomial generator with the last variable x n−1 . This problem can be avoided, since wecan use one of the ideals, say I, to compute an ideal J (via a sequence of special contractionsfollowed by a sequence of special expansions) with the same double saturation and the sameHilbert polynomial as I, which contains only one minimal generator with the last variablex n−1 . The ideal J can be computed as follows: LetI =: I (0) , I (1) , . . . , I (j) = sat xn−1,xn (I)be the sequence of ideals obtained when taking I to sat xn−1 ,x n(I) by performing contractionsas in Lemma 3.17. Furthermore let Ĩ := I(i) , 0 ≤ i < j, be the first ideal of the abovesequence, which contains only one minimal monomial generator m with the last variable
3.3. STABLE IDEALS WITH THE SAME DOUBLE SATURATION 63x n−1 . Let J be the ideal obtained from Ĩ by the expansion of the monomials m, m·x n−1, m·x 2 n−1, . . . , m · x i−1n−1. Thenp R/I (z) = p R/J (z),sat xn−1 ,x n(I) = sat xn−1 ,x n(J)and I and J are linked by a finite sequence of paired contractions and expansions.The contractions used to take I to Ĩ can be performed by Remark 3.18 (we always contractthe first monomial in lexicographic order containing the last variable x n−1 ). The expansionsof m, m · x n−1 , m · x 2 n−1, . . . , m · x i−1n−1 can be performed successively in Ĩ: m is expandablein Ĩ, since it is a monomial generator and R(m) is not contained in the set of minimalgenerators of Ĩ. In the ideal Ĩexp we can expand the monomial m · x n−1 , since m · x n−1 is aminimal monomial generator of Ĩexp and none of the elements of R(m · x n−1 ) is containedin Ĩexp . Finally we obtain the ideal J.It is clear that both ideals do have the same double saturation. From Corollary 3.26,we conclude p R/I (z) = p R/J (z), since the number of contractions taking I to Ĩ equalsthe number of expansions taking Ĩ to J. Finally, since I and J fulfill the conditionsof Proposition 3.28, I and J are linked by a finite sequence of paired contractions andexpansions.Now, we come to the main result of this section. We will see that the theorem statedbelow, directly follows from Proposition 3.28 and Remark 3.31. It will be of great importanceto the algorithms presented in Chapter 4 and can be viewed as an algorithmicclassification of all saturated stable ideals with the same double saturation and the sameHilbert polynomial.Theorem 3.32. All saturated stable ideals with the same double saturation and the sameHilbert polynomial are linked by a finite sequence of paired expansions and contractions.In particular, given two saturated stable ideals I and I ′ with the same double saturationand p R/I (z) = p R/I ′(z) one can find an even number of monomials m 1 , . . . , m k and idealsI (0) , I (1) , . . . , I (k) , where(i) I (0) := I and{(ii) I (l+1) (I (l) ) con if l mod 2 = 0 (contraction via the monomial m l ):=(I (l) ) exp if l mod 2 = 1 (expansion via the monomial m l )l = 0, . . . , k and I (k) = I ′ .Proof. From I we compute the ideal J with the same double saturation and the sameHilbert polynomial, which contains only one monomial generator with the last variablex n−1 , due to Remark 3.31. We know from Proposition 3.28 that I and J are linked by afinite sequence of paired expansions and contractions. It follows by the proof of Proposition3.28 that J and I ′ are also linked by a finite sequence of paired expansions and contractions.Hence, we can first take I to J and then J to I ′ by sequences of paired expansions andcontractions. This proves the assertions of the theorem.
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University of Paderborn Department of Mathematics Diploma Thesis ...

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