University of Paderborn Department of Mathematics Diploma Thesis ...

26 CHAPTER 2. STABLE IDEALSRemark and Definition 2.14. Let I ⊂ R be a stable ideal. The double saturation ofthe ideal I is defined to be the saturation of sat xn (I) in ¯R := K[x 0 , . . . , x n−1 ]. It will bedenoted by sat xn−1 ,x n(I) and viewed as an ideal in R = K[x 0 , . . . , x n ].Note that the above definition makes sense, since none of the generators of the ideal sat xn (I)contains the last variable x n . Thus, we may view sat xn (I) as a possibly non-saturated stableideal in ¯R and compute its saturation in ¯R by setting x n−1 := 1 in all of its generators.Hence, the double saturation of I is obtained by first setting x n := 1 and then x n−1 := 1.Example 2.15. Let I again be the ideal (x 3 0, x 2 0x 1 , x 2 0x 2 ). We know that I is stable. Nowit depends on the polynomial ring R if I is saturated or not. If R = K[x 0 , x 1 , x 2 ], theideal I is not saturated and we obtain sat x2 (I) from I by setting x 2 := 1 in the set of itsmonomial generators. This provides sat x2 (I) = (x 2 0). The double saturation sat x1 ,x 2(I) of Iis obtained from sat x2 (I) by setting x 1 := 1 in the set of generators of sat x2 (I). Hence, weobtain sat x1 ,x 2(I) = (x 2 0) = sat x2 (I). If R = K[x 0 , . . . , x n ], n ≥ 3, the ideal I is saturated.Furthermore, if we have n ≥ 4, we obtain I = sat xn (I) = sat xn−1 ,x n(I), i.e. the processes ofsaturation and double saturation do not affect the monomial generators of I.In the next section of this chapter about stable ideals, we will present a simple formula tocompute the Hilbert series of a stable ideal.2.4 Hilbert series of stable idealsWe start with a relationship between the Hilbert series of R/(f 1 , . . . , f r , f), R/(f 1 , . . . , f r )and R/((f 1 , . . . , f r ) : (f)) for homogeneous polynomials f 1 , . . . , f r , f ∈ R.Lemma 2.16. Let I = (f 1 , . . . , f r ) ⊂ R be a homogeneous ideal and f ∈ R a homogeneouspolynomial of degree d > 0.Then the Hilbert series of R/(f 1 , . . . , f r , f) is given bywhere I : f denotes the ideal I : (f).Proof. Consider the exact sequence0 −−−→ (R/(I : f))(−d)The sequence yieldsH R/(f1 ,...,f r,f)(t) = H R/I (t) − t d · H R/(I:f) ,f−−−→ R/I −−−→ R/(f 1 , . . . , f r , f) −−−→ 0.h R/(f1 ,...,f r,f)(i) = h R/I (i) − h (R/(I:f))(−d) (i),(∗)for all i ∈ Z, since the vector space dimension is additive on exact sequences. For theHilbert series of (R/(I : f))(−d), we getH (R/(I:f))(−d) (t) = ∑ i∈Zh (R/(I:f))(−d) (i) · t i = ∑ i∈Zh R/(I:f) (i − d) · t i