University of Paderborn Department of Mathematics Diploma Thesis ...

20 CHAPTER 2. STABLE IDEALSExample 2.2. Let R := K[x 0 , x 1 , x 2 ] and A = (a ij ) ∈ GL(2, K) with⎡ ⎤1 2 3A = ⎣ 0 2 3 ⎦0 0 3The action of A on the variables x 0 , x 1 , x 2 is given byA(x 0 ) = a 00 · x 0 + a 10 · x 1 + a 20 · x 2= x 0 ,A(x 1 ) = a 01 · x 0 + a 11 · x 1 + a 21 · x 2= 2x 0 + 2x 1 ,A(x 3 ) = a 02 · x 0 + a 12 · x 1 + a 22 · x 2= 3x 0 + 3x 1 + 3x 2 .For the monomial m := x 0 x 2 1x 2 ∈ R the action of A on m is given byA(m) = x 0 · (2x 0 + 2x 1 ) 2 · (3x 0 + 3x 1 + 3x 2 ).In the following, we apply the matrix A to the monomial generators of an ideal. LetI := (x 3 0, x 2 0x 1 , x 2 0x 2 ) ⊂ R. Applying A to each of the monomial generators of I, we obtainthe monomialsA(x 3 0) = x 3 0,A(x 2 0x 1 ) = x 2 0 · (2x 0 + 2x 1 )= 2x 3 0 + 2x 2 0x 1 ,A(x 2 0x 2 ) = x 2 0 · (3x 0 + 3x 1 + 3x 2 )= 3x 3 0 + 3x 2 0x 1 + 3x 2 0x 2 ,and thereby the ideal I ′ = (x 3 0, 2x 3 0 + 2x 2 0x 1 , 3x 3 0 + 3x 2 0x 1 + 3x 2 0x 2 ). One can easily see thatI ′ ⊂ I, since any monomial in the set of generators of I ′ is a linear combination withcoefficients in K of monomial generators of I. On the other hand we obtainx 3 0 ∈ I ′ ,x 2 0x 1 = 2x3 0 + 2x 2 0x 12− x 3 0 ∈ I ′ ,x 2 0x 2 = 3x3 0 + 3x 2 0x 1 + 3x 2 0x 2− x 230x 1 − x 3 0 ∈ I ′ .This shows I ⊂ I ′ and we conclude that the ideal I is fixed under the action of the matrix Aon its monomial generators. If this holds for any invertible upper triangular (3×3) matrix,such an ideal will be called a stable ideal (in the case, when K is a field of characteristiczero).Definition 2.3. Let B denote the Borel-subgroup of G, i.e. the invertible upper triangular(n + 1) × (n + 1)-matrices over K. A homogeneous ideal I ⊂ R is called Borel-fixed if it isfixed under the action of matrices in B, in the sense that the application of any matrix ofB on the generators of the ideal I reveals a set of polynomials generating the same ideal.