## Commutative rings |

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Page 49

Auslander-Buchsbaum and simultaneously by Serre). ' The result can also be

proved using Ext. If R is commutative Noetherian,

exact sequence x x giving rise to Extn+1 ( A* ,B) -«— Extn ( A ,B ) «-^- Extn ( A ,B )

.

Auslander-Buchsbaum and simultaneously by Serre). ' The result can also be

proved using Ext. If R is commutative Noetherian,

**x i z(A**), x t J(R), consider theexact sequence x x giving rise to Extn+1 ( A* ,B) -«— Extn ( A ,B ) «-^- Extn ( A ,B )

.

Page 61

If A is stable and

sequences. But we don't need all of that). Proof. Write B = A/xA and take P 2 Ann (

B) O Ann (A). Consider the exact sequence 0+A+A+B+O Localize: 0 + A $ A + ...

If A is stable and

**x i z(A**) , then A/xA is stable. (This is really part of the theory of R-sequences. But we don't need all of that). Proof. Write B = A/xA and take P 2 Ann (

B) O Ann (A). Consider the exact sequence 0+A+A+B+O Localize: 0 + A $ A + ...

Page 102

By induction, d^K/xK) = G(M*) = G(M) - 1 = d^K) = dp (A) - 1. If G(M,A) > 0, pick x £

z(R),

Hence the result. (The hypothesis

By induction, d^K/xK) = G(M*) = G(M) - 1 = d^K) = dp (A) - 1. If G(M,A) > 0, pick x £

z(R),

**x i z(A**) and consider A/xA: G(M,A/xA) = G(M,A) - 1 = d(A^sA) □ d(A) + 1.Hence the result. (The hypothesis

**x i z**(R) in the proof is unnecessary 102.### What people are saying - Write a review

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### Contents

Prime ideals and the Nullstellensatz | 1 |

The ascending chain condition | 13 |

Zerodivisors | 19 |

7 other sections not shown

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### Common terms and phrases

A/xA algebraic Ann(A annihilator assume Auslander-Buchsbaum chains of prime clearly co-dimension coefficients commutative ring complete the proof contains a non-zero contradiction Corollary Dedekind ring defined Definition dimension direct summand discrete valuation ring dR(A elements Exercise exists finite free resolution finite number finitely generated R-module follows free module G-ideal Given grade hence Hilbert ring hypothesis ideal Q induction integral domain integrally closed intersection invertible ideal Krull Lemma little rank Macaulay ring maximal chain maximal ideal maximal R-sequence minimal prime ideal modules with FFR Noetherian ring non-zero prime ideal Note polynomials principal ideal theorem principal prime projective prove PrUfer ring quotient field R-projective R*-free rank Q regular local n-dimensional regular local ring result ring with quotient short exact sequences submodule subring take x e torsion modules unmixedness theorem write x i z(A zero zero-divisors