University of Paderborn Department of Mathematics Diploma Thesis ...

More documents

Recommendations

Info

56 CHAPTER 3. OPERATIONS ON STABLE IDEALSProposition 3.23. Let p R/I (z) be the Hilbert polynomial of R/I for some saturated stableideal I sat xn−1 ,x n(I). Let m and ˜m be minimal monomial generators of I, such that m is˜mexpandable in I, ˜m contains the variable x n−1 and is contractible in I exp (expansionvia the monomial m). Then the ideal J = (I exp ) con , which can be obtained from I after the˜mexpansion of m in I and the contraction of in I exp has the same Hilbert polynomialas I, i.e. p R/J (z) = p R/I (z).x n−1Proof. Note that since I is saturated, the monomial m cannot contain the variable x n . Theexpansion of m in I provides the ideal I exp , where the minimal monomial generator m of Iis removed from the set of generators of I and the monomials m · x r , m · x r+1 , . . . , m · x n−1are included into the set of generators. Hence, the ideal I exp contains exactly one monomialof degree deg m + 1 less than the ideal I, namely the monomial m · x n , which is containedin I, but not in I exp . Since all other monomial generators of I are still contained in I exp ,similar observations for m · x i n, i > 0, showfor all j ≥ deg m + 1.x n−1h I exp(j) = h I (j) − 1Contraction of the monomial˜mx n−1in I exp provides the idealJ = (I exp ) con , where the monomial generator ˜m of I exp ˜m(and of I) is replaced by .x n−1Consequently, the ideal J contains exactly one more monomial of degree deg ˜m than I exp ,i.e. the monomiali > 0, we obtain˜mx n−1· x n , which is contained in J, but not in I exp . Consideringh J (j) = h I exp(j) + 1˜mx n−1· x i n,for all j ≥ deg ˜m. It follows h I (j) = h J (j) for all j ≥ max{deg m + 1, deg ˜m} andp I (z) = p J (z), which shows p R/I (z) = p R/J (z).Corollary 3.24. With the notation in Proposition 3.23, let m be a monomial generatormof I containing the variable x n−1 , such that is contractible in I. Let ˜m be expandablex n−1in I con m, contraction via the monomial . Then the ideal J := (I con ) exp , expansion viax n−1the monomial ˜m, has the same Hilbert polynomial as I.Proof. The assertion follows from the proof of Proposition 3.23, since we can again usemthe arguments on the growth of the Hilbert function: Contraction of gives h I con(j) =x n−1h I (j)+1. Expansion of ˜m in I con provides the ideal J with h J (j) = h I con(j)−1. Thus, as inthe proof of the proposition, it follows h I (j) = h j (j) for all j ≥ max{deg m−1, deg ˜m}.An example will demonstrate the preceding results, before we come to another corollaryof Proposition 3.23.
3.3. STABLE IDEALS WITH THE SAME DOUBLE SATURATION 57Example 3.25. Consider the stable and saturated ideal I := (x 2 0, x 0 x 1 , x 0 x 3 2) in R :=K[x 0 , x 1 , x 2 , x 3 ]. By Algorithm 2.29, we compute the Hilbert polynomial of R/I, whichyieldsp R/I (z) = 1 2 z2 + 3 2 z + 4.Contraction of the monomial x 0 x 2 2 in I is possible: We obtain the idealI (1) := I con = (x 2 0, x 0 x 1 , x 0 x 2 2),in which we perform an expansion via the monomial x 0 x 1 yielding the idealComputing p R/J (z) by Algorithm 2.29 showsJ := (I (1) ) exp = (x 2 0, x 0 x 2 1, x 0 x 1 x 2 , x 0 x 2 2).p R/J (z) = 1 2 z2 + 3 2 z + 4,i.e. the Hilbert polynomial of J indeed equals the Hilbert polynomial of I. Below we willconsider an example, where we use the corollary stated above.Corollary 3.26. Let I ⊂ R be a saturated stable ideal. Let I exp be the ideal obtained from Iby performing the expansion of some monomial and let I con be the ideal obtained from I byperforming the contraction of some monomial m in I, where m·x n−1 is a minimal generatorof I. Then the Hilbert polynomials of R/I exp and R/I con are given by p R/I exp(z) = p R/I (z)+1and p R/I con(z) = p R/I (z) − 1.Proof. The assertion immediately follows from the proof of Proposition 3.23.Example 3.27. Consider the ideals I and J of Example 3.25 in R := K[x 0 , x 1 , x 2 , x 3 ]. Weobtained the ideal I (1) from I by the contraction of x 0 x 2 2. The Hilbert polynomial of I (1)can be computed by Algorithm 2.29, which yieldsp R/I (1)(z) = 1 2 z2 + 3 2 z + 3 = p R/I(z) − 1.The ideal J was obtained from the ideal I (1) by the expansion of the monomial x 0 x 1 . Sincewe know by Example 3.25 thatp R/I (z) = p R/J (z) = 1 2 z2 + 3 2 z + 4,we see that the expansion of x 0 x 1 in I (1) does indeed increase the Hilbert polynomial ofR/J by one, i.e. p R/J (z) = p R/I (1)(z) + 1.In Corollary 3.22, we stated that all saturated stable ideals I, J ⊂ R with the same doublesaturation are linked by a finite sequence of contractions and expansions. This means thatwe can always find monomials m 1 , . . . , m l ∈ M, such that the contraction of m 1 , . . . , m l
Page 1:
University of PaderbornDepartment o
Page 5:
”Im großen Garten der Geometriek
Page 9:
AbstractThe main result of this the
Page 12 and 13:
Glossary of MuPAD procedures 129Bib
Page 14 and 15:
This thesis is organized in four ch
Page 16 and 17: 6described in Chapter 4.Additionall
Page 18 and 19: 8 CHAPTER 1. NOTATIONS AND PREREQUI
Page 20 and 21: 10 CHAPTER 1. NOTATIONS AND PREREQU
Page 30 and 31: 20 CHAPTER 2. STABLE IDEALSExample
Page 32 and 33: 22 CHAPTER 2. STABLE IDEALSset, whe
Page 34 and 35: 24 CHAPTER 2. STABLE IDEALSm ∈ M
Page 36 and 37: 26 CHAPTER 2. STABLE IDEALSRemark a
Page 38 and 39: 28 CHAPTER 2. STABLE IDEALSby Lemma
Page 40 and 41: 30 CHAPTER 2. STABLE IDEALSwith Hil
Page 42 and 43: 32 CHAPTER 2. STABLE IDEALSfor all
Page 44 and 45: 34 CHAPTER 2. STABLE IDEALS• We h
Page 46 and 47: 36 CHAPTER 2. STABLE IDEALSis stabl
Page 48 and 49: 38 CHAPTER 2. STABLE IDEALSWe will
Page 50 and 51: 40 CHAPTER 2. STABLE IDEALSAs a cor
Page 52 and 53: 42 CHAPTER 2. STABLE IDEALSTheorem
Page 54 and 55: 44 CHAPTER 2. STABLE IDEALSBinomial
Page 56 and 57: 46 CHAPTER 3. OPERATIONS ON STABLE
Page 86 and 87: 76 CHAPTER 4. ALGORITHMS FOR STABLE
Page 110 and 111: 100 CHAPTER 4. ALGORITHMS FOR STABL
Page 116 and 117:
106 CHAPTER 4. ALGORITHMS FOR STABL
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 137 and 138:
Glossary of NotationsK a field of c
Page 139:
Glossary of MuPAD procedurescompute
Page 142:
[13] F.S. Macaulay, Some properties
show all

University of Paderborn Department of Mathematics Diploma Thesis ...

Create successful ePaper yourself

Delete template?

Save as template?