University of Paderborn Department of Mathematics Diploma Thesis ...

64 CHAPTER 3. OPERATIONS ON STABLE IDEALSRemark 3.33. The theorem above appears in [16] – but similar to the definition of thecontraction of a monomial (see Definition 3.9) it had to be modified. In [16] the theoremis stated without the assumption that the ideals mentioned in the theorem must have thesame Hilbert polynomial.Note that the theorem is wrong without this assumption: Consider the saturated stableideals I := (x 2 0, x 0 x 1 , x 0 x 2 2) and J := (x 2 0, x 0 x 1 , x 0 x 3 2) in K[x 0 , x 1 , x 2 , x 3 ]. They are bothstable and saturated and do have the same double saturation given by sat x2 ,x 3(I) = (x 0 ) =sat x2 ,x 3(J), but they do not have the same Hilbert polynomial. A contraction of themonomial x 0 x 2 2 in J provides the ideal I, which shows that there is no sequence of pairedexpansions and contractions taking I to J or vice versa as in the theorem above.One more example shall illustrate the algorithm proposed by Theorem 3.32 to obtain asaturated ideal I ′ ⊂ R from another saturated stable ideal I ⊂ R with the same doublesaturation and the same Hilbert polynomial.Example 3.34. We consider the polynomial ring R := K[x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] and theidealsI := (x 0 , x 1 , x 3 2, x 2 2x 2 3, x 2 2x 3 x 4 , x 2 2x 2 4)andI ′ := (x 2 0, x 0 x 1 , x 0 x 2 , x 0 x 3 , x 0 x 4 , x 2 1, x 1 x 2 , x 1 x 3 , x 1 x 4 , x 3 2, x 2 2x 3 , x 2 2x 4 ).I and I ′ are both stable and saturated. By Algorithm 2.29, we compute the Hilbert polynomialsp R/I (z) and p R/I ′(z), which provides p R/I (z) = z 2 +2z+4 = p R/I ′(z). Furthermore,we obtain sat x4 ,x 5(I) = (x 0 , x 1 , x 2 2) = sat x4 ,x 5(I ′ ) by setting x 4 := 1 in all generators of bothideals. Hence, I and I ′ fulfill the conditions of Theorem 3.32.In the first step, we compute the ideal J from I, which contains only one monomial generatorwith the last variable x 4 . Therefore, we contract the monomial the x 2 2x 3 , whichprovidesI (1) = (x 0 , x 1 , x 3 2, x 2 2x 3 , x 2 2x 2 4).Afterwards we expand the monomial x 2 2x 2 4, such thatJ := I (2) = (x 0 , x 1 , x 3 2, x 2 2x 3 , x 2 2x 3 4).Note that J has the same double saturation as I and I ′ and, again by Algorithm 2.29, thesame Hilbert polynomial, since p R/J (z) = z 2 + 2z + 4.To take I to sat x4 ,x 5(I), we have to perform the contractions of x 2 2x 3 , x 2 2x 4 and x 2 2. To takeJ to sat x4 ,x 5(J) = sat x4 ,x 5(I), we have to contract x 2 2x 2 4, x 2 2x 4 and x 2 2.Since the last two monomials are equal in both sequences of contractions, we can firstcontract x 2 2x 3 in I and then expand x 2 2x 2 4 in I exp (expansions via x 2 2x 3 ), which obviouslyprovides the ideal J.In the next step we consider the sequence of monomials, which has to be contracted totake I ′ to sat x4 ,x 5(I ′ ). We have to contract x 0 , x 1 and x 2 2 to obtain the double saturation.Since the last monomial x 2 2 in the sequence taking J to its double saturation equals the lastmonomial in the sequence of contractions taking I ′ to sat x4 ,x 5(I ′ ), we only have to consider