University of Paderborn Department of Mathematics Diploma Thesis ...

24 CHAPTER 2. STABLE IDEALSm ∈ M of I the term U c (m) is contained in I (if U c is an upper triangular matrix). SinceI is a monomial ideal (i.e. I is homogeneous, especially), it follows I = U c (I). Since the setof all upper triangular matrices of the form U c and all diagonal matrices in G forms a basisof all upper triangular matrices of G, I is fixed under the action of all upper triangularmatrices of G.This characterization of Borel-fixed ideals will be used in the following sections of thisthesis and we will call an ideal, which fulfills the condition of the preceding Theorem 2.7,a stable ideal.Definition 2.8. Let K be a field of characteristic zero and let R := K[x 0 , . . . , x n ]. Amonomial ideal I ⊂ R is called a stable ideal, if for every monomial m = x a 00 · x a 11 · . . . · x annof I we havefor i < j and s ≤ a j .x a 00 · . . . · x a i−1i−1 · xa i+si · x a i+1i+1 · . . . · xa j−1j−1 · xa j−sj · x a j+1j+1 · . . . · xan n ∈ IRemark 2.9. Indeed, it is easy to see that the condition of Definition 2.8 is equivalentto the one in Theorem 2.7: Any stable ideal fulfills the condition of Theorem 2.7. On theother hand, let I ⊂ R be a monomial ideal fulfilling the condition of Theorem 2.7. We haveto show that I is stable. Let m ∈ I be a monomial. Then there is a monomial generatorm ′ ∈ I and a monomial m ′′ ∈ M, such that m = m ′ · m ′′ . Let x k , k > 0, be a variabledividing m. It suffices to show that x l · m ∈ I for 0 ≤ l < k. If x k divides m ′′ , we havex kx l · mx k= m ′ ·hence, x l · mx k=) (x l · m′′x k ) (x l · m′· m ′′ ∈ I, which proves the assertion.x k∈ I, since m ′ ∈ I. If x k divides m ′ , we know x l · m′x k∈ I and,Example 2.10. To come back to Example 2.2 of the preceding section, we can easily seethat the ideal I = (x 3 0, x 2 0x 1 , x 2 0x 2 ) is stable, sincex 2 0x 1x 1· x 0 = x 3 0 ∈ I,x 2 0x 2x 2· x 1 = x 2 0x 1 ∈ I and x2 0x 2x 2· x 0 = x 3 0 ∈ I.In the following, we will present an easy way to compute the saturation of a stable ideal.2.3 Saturation of stable idealsIn order to decide, whether a given stable ideal is saturated and, if this is not the case,to compute its saturation, we have to examine the associated prime ideals in the primarydecomposition of the stable ideal.We will prove that all associated prime ideals of a given stable ideal are of the form(x 0 , . . . , x i ) for an i ≥ 0. Within the proof of this assertion, we will make use of a theoremby Bayer and Stillman (see [6], Proposition 15.24 ).