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Lectures on Unique Factorization Domains

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Lectures on Unique Factorization Domains Lectures On Unique Factorization Domains By P. Samuel Tata Institute Of Fundamental Research, Bombay 1964 Lectures On Unique Factorization Domains By P. Samuel Notes by M. Pavman Murthy No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 5 Tata Institute of Fundamental Research Bombay 1964 Contents 1 Krull rings and factorial rings 1 1 Divisorial ideals . 1 2 Divisors.......................... 2 3 Krullrings......................... 4 4 Stability properties . 8 5 Two classes of Krull rings . 11 6 Divisor class groups . 14 7 Applications of the theorem of Nagata . 19 8 Examples of factorial rings . 25 9 Power series over factorial rings . 32 2 Regular rings 37 1 Regular local rings . 37 2 Regular factorial rings . 41 3 The ring of restricted power series . 42 3 Descent methods 45 1 Galoisian descent . 45 2 The Purely inseparable case . 50 3 Formulae concerning derivations . 51 4 Examples: Polynomial rings . 53 5 Examples: Power series rings . 56 i Chapter 1 Krull rings and factorial rings In this chapter we shall study some elementary properties of Krull rings 1 and factorial rings. 1 Divisorial ideals Let A be an integral domain (or a domain) and K its quotient field. A fractionary ideal U is an A-sub-module of K for which there exists an element d ∈ A(d , 0) such that dU ⊂ A i.e. U has “common denominator” d). A fractionary ideal is called a principal ideal if it is generated by one element. U is said to be integral if U ⊂ A. We say that U is divisorial if U , (0) and if U is an intersection of principal ideals. Let U be a fractionary ideal and V non-zero A-submodule of K. Then the set U : V = x ∈ K xV ⊂ U is a fractionary ideal. The following formulae are easy to verify. (1) ( Ui : ( V j) = (Ui : V j). Ti Pj Ti, j (2) U : VV ′ = (U : V ) : V ′. (3) If x ∈ K, x , 0, then V : Ax = x−1V . 1 2 1. Krull rings and factorial rings Lemma 1.1. Let U be a fractionary ideal , (0) and V a divisorial ideal. Then V : U is divisorial. Axi Proof. Let V = Axi. Then V : U = (Axi : U ) = ( ) Ti Ti Ti aT∈U a 2 Lemma 1.2. (1) Let U , (0) be a fractionary ideal. Then the smallest divisorial ideal containing U , denoted by U¯,isA : (A : U ). (2) If U , V , (0), then U¯ = V¯ ⇔ A : U = A : V . Proof. (1) By Lemma 1.1, A : (A : U ) is divisorial. Obviously U ⊂ A : (A : U ). Suppose now that U ⊂ Ax, x , 0, x ∈ K. Then A : U ⊃ A : Ax = Ax−1. Thus A : (A : U ) ⊂ A : Ax−1 = Ax and (1) is proved. (2) is a trivial consequence of (1) and the proof of Lemma 1.2 is com- plete. 2 Divisors Let I(A) denote the set of non-zero fractionary ideals of the integral domain A. In I(A), we introduce an equivalence relation ∼, called Artin equivalence (or quasi Gleichheit) as follows: U ∼ V ⇔ U¯ = V¯ ⇔ A : U = A : V . The quotient set I(A)/ ∼ of I(A) by the equivalence relation ∼ is called the set of divisors of A. Thus there is an l-1 correspondence between the set D(A) of divisors and the set of divisorial ideals. Let d denote the canonical mapping d : I(A) → I(A)/ ∼. Now, I(A) is partially ordered by inclusion and we have U ⊂ V ⇒ U¯ ⊂ V¯. Thus this partial order goes down to the quotient set I(A)/ ∼ by d. If U ⊂ V , we write d(V ) ≤ d(U ). On I(A) we have the structure of an ordered commutative monoid given by the composition law (U , V ) UV -with A acting as the unit element. Let U , U ′, V ∈ I(A) and U ∼ U ′i.e. A : U = A : U ′. We have A : UV = (A : U ) : V = (A : U ′) : V = A : U ′V . Hence UV ∼ 3 U ′V . Thus D(A) acquires the structure of a commutative monoid, with 2. Divisors 3 the composition law (U¯, V¯) → UV . We write the composition law in D(A) additively. Thus d(UV ) = d(U ) + d(V ) for U , V ∈ I(A). Since the order in D(A) is compatible with the compositional law in D(A), D(A) is a commutative ordered monoid with unit. We note that d(U ∩ V ) ≥ sup(d(U ), d(V ) and d(U + V ) = inf (d(U ), d(V ). Let K∗ be the set of non-zero elements of K. For x ∈ K∗, we write d(x) = d(Ax); d(x) is called a principal divisor. Theorem 2.1. For D(A) to be a group it is necessary and sufficient that A be completely integrally closed. We recall that A is said to be completely integrally closed if when- ever, for x ∈ K there exist an a , 0, a ∈ As.t.axn ∈ A for every n, then x ∈ A. We remark that if A is completely integrally closed, then it is inte- grally closed. The converse also holds if A is noetherian. A valuation ring of height > 1 is an example of an integrally closed ring which is not completely integrally closed. Proof. Suppose D(A) is a group. Let x ∈ K and a be a non-zero element of A such that axn ∈ A, for every n ≥ 0. Then a ∈ Ax−n = V which is divisorial. Set d(V ) = β and α = d(x−1). nT=0 Now β = sup(nα). But β + α = Supn≥0((n + 1)α) n≥0 = sup (qα). Thus β + α ≤ β. Since D(A) is a group −β exists, 4 q≥1 and therefore α = (β + α) − β ≤ β − β = 0 i.e. d(x) ≥ 0. Hence Ax ⊂ A i.e. x ∈ A. Conversely suppose that A is completely integrally closed. Let U be a divisorial ideal. Then U = xU ′, U ′ ⊂ A. Since we already know 4 1. Krull rings and factorial rings that principal divisors are invertible, we have only to prove that integral divisorial ideals are invertible. Let V be a divisorial ideal ⊂ A. Then V .(A : V ) ⊂ A. Let V (A : V ) ⊂ Ax, for some x ∈ K. Then x−1V (A : V ) ⊂ A. Thus x−1V ⊂ A : (A : V ) = V , since V is divisorial. Thus V ⊂ V x. By induction V ⊂ V xn, for every n ≥ 0. Consider an element b , 0, b ∈ V . Then b(x−1)n ⊂ V ⊂ A for every n ≥ 0. Hence x−1 ∈ A i.e. Ax ⊃ A. Thus V (A : V ) ∼ A. Theorem 2.1 is completely proved. Notice that we have d(U ) + d(A : U ) = 0. Let us denote by F(A) the subgroup generated by the principal di- visors. If D(A) is a group, the quotient group D(A)/F(A) is called the divisor class group of A and is denoted by C(A). In this chapter we shall study certain properties of the group C(A). 3 Krull rings Let Z denote the ring of integers. Let I be a set. Consider the abelian group Z(I). We order Z(I) by means of the following relation: (I) for (αi), (βi) ∈ Z , (αi) ≥ (βi) if αi ≥ βi, for all i ∈ I. 5 The ordered group Z(I) has the following properties: (a) any two elements of Z(I) have a least upper bound and a greatest lower bound i.e. Z(I) is an ordered lattice. (b) The positive elements of Z(I) satisfy the minimum condition i.e. given a nonempty subset of positive elements of Z(I), there exists a minimal element in that set. Conversely any ordered abelian group satisfying conditions (a) and (b) is of the form Z(I) for some indexing set I (for proof see Bourbaki, Algebre, Chapter VI). Let A be an integral domain. We call A a Krull ring if D(A) ≈ Z(I), the isomorphism being order preserving. Theorem 3.1. Let A be an integral domain. Then A is Krull if and only if the following two conditions are satisfied. (a) A is completely integrally closed. (b) The divisorial ideals contained in A satisfy the maximum condition. In fact the above theorem is an immediate consequence of Theorem 2.1 and the characterization of the ordered group Z(I) mentioned above. An immediate consequence of Theorem 3.1 is: 3. Krull rings 5 Theorem 3.2. A noetherian integrally closed domain is a Krull ring. We remark that the converse of Theorem 3.2 is false. For example the ring of polynomials in an infinite number of variables over a field K is a Krull ring, but is not noetherian. In fact this ring is known to be factorial and we shall show later that any factorial ring is a Krull ring. (I) Let ei = (δi j) j∈I ∈ Z , where δi j is the usual Kronecker delta. The ei are minimal among the strictly positive ele- 6 ments. Let A be a Krull ring and let ϕ be the order preserving isomor- (I) −1 phism ϕ : D(A) → Z . Let Pi = ϕ (ei). We call the divisors Pi the prime divisors. Let P(A) denote the set of prime divisors. Then any d ∈ D(A) can be written uniquely in the form d = n P X P P∈P(A) ∗ where nP ∈ Z and nP = 0 for almost all P. Now let x ∈ K . Consider the representation d(x) = v (x)P, v (x) ∈ Z, v (x) = 0 X P P P P∈P(A) for almost all P ∈ P(A).
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