## Download PDF by Dummit D. S: Abstract algebra

By Dummit D. S

ISBN-10: 0130047716

ISBN-13: 9780130047717

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Example text

The defect of D relative to deg is then deﬁned to be def(B), and is denoted by def(D). The reason for deﬁning the defect of a derivation is that, if B = ∪i∈Z Bi is the ﬁltration of B induced by the degree function deg, then D (nonzero) respects this ﬁltration if and only if def(D) is ﬁnite. The defect has the following basic properties. 13. Let a, b ∈ B, and let S be a non-empty subset of B. (a) def(S) = −∞ if and only if S ⊂ ker D. (b) def(D) = −∞ if and only if D = 0. 4 The Defect of a Derivation (c) (d) (e) (f ) (g) (h) 41 D is homogeneous relative to deg if and only if def is constant on B − 0.

Suppose D ∈ LND(B) is irreducible, and set A = ker D. By Prop. 4, D has a minimal local slice y. Suppose Dy ∈ B ∗ . Then there exists irreducible x ∈ B dividing Dy. Since A is factorially closed, x ∈ A. ¯ = D (mod x) on B ¯ = B (mod x). Since D is irreducible, D ¯ = 0. In Let D ¯ = 1. By Cor. 24, it follows that B ¯ = k [1] and ker D ¯ = k. k B ¯ y = 0, we have that y ∈ xB + k. Write y = xz + λ for some z ∈ B Since D¯ and λ ∈ k. Then y − λ = xz is irreducible, by Prop. 5. But this implies z ∈ B ∗ ⊂ A, and thus y = xz + λ ∈ A, a contradiction.

Suppose D ∈ LND(B) is irreducible, and set A = ker D. By Prop. 4, D has a minimal local slice y. Suppose Dy ∈ B ∗ . Then there exists irreducible x ∈ B dividing Dy. Since A is factorially closed, x ∈ A. ¯ = D (mod x) on B ¯ = B (mod x). Since D is irreducible, D ¯ = 0. In Let D ¯ = 1. By Cor. 24, it follows that B ¯ = k [1] and ker D ¯ = k. k B ¯ y = 0, we have that y ∈ xB + k. Write y = xz + λ for some z ∈ B Since D¯ and λ ∈ k. Then y − λ = xz is irreducible, by Prop. 5. But this implies z ∈ B ∗ ⊂ A, and thus y = xz + λ ∈ A, a contradiction.