Lectures on Commutative Algebra Weizhe Zheng Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, China University of the Chinese Academy of Sciences, Beijing 100049, China Email: [email protected] Contents Convention v 7 UFDs 1 8 Primary decomposition 3 9 DVRs and Dedekind domains 5 10 Completions 11 11 Dimension theory 17 Summary of properties of rings 23 iii iv CONTENTS Convention Rings are assumed to be commutative. v vi CONVENTION Chapter 7 UFDs Theorem 7.1 (Fundamental theorem of arithmetic, cf. Euclid’s Elements 9.14). Every nonzero integer n ∈ Z can be factorized uniquely (up to permutation of factors) as a product of primes n = ±p1 ··· pm. Definition 7.2. Let R be a domain. (1) x ∈ R is irreducible if x 6= 0, x 6∈ R×, and x = yz implies y ∈ R× or z×. (2) x, y ∈ R are associates if there exists u ∈ R× such that x = uy. (3) R is a unique factorization domain (UFD, or factorial ring) if every x ∈ R satisfying x 6= 0 and x 6∈ R admits a factorization x = a1 ··· am with ai irreducible, and if x = b1 ··· bn with bj irreducible, then m = n and there exists a bijection σ : {1, . , n} → {1, . , n} such that ai and bσ(i) are associates for every i). We say that x ∈ R is prime if xR is a prime ideal. Prime elements are irreducible. The converse holds in a UFD. Theorem 7.3. Let R be a domain. Then R is a UFD if and only if the following conditions are satisfied: (1) The ascending chain condition for principle ideals of R. (2) Irreducible elements in R are prime. Lemma 7.4. Assume that the ascending chain condition holds for principle ideals of a ring R. Then every x ∈ R satisfying x 6= 0 and x 6∈ R admits a factorization x = a1 ··· am with ai irreducible. Corollary 7.5. A PID is a UFD. √ Example√7.6. (1) Z, Z[ −1] are PIDs, hence UFDs.√ √ (2) Z[ −5] is not a UFD. Indeed, 2 · 3 = 6 = (1 + −5)(1 − −5). (3) If R is a UFD, R[X] is a UFD. Note however that neither Z[X] nor k[X, Y ] (where k is a field) is a PID. (4) If R is a nonzero UFD, R[X1,X2,... ] is a UFD, but not a Noetherian ring. S∞ 1/n (5) In R = n=1 k[X ], factorization does not exist in general. In particular, R does not satisfy the ascending chain condition for principal ideals. 1 2 CHAPTER 7. UFDS Chapter 8 Primary decomposition We largely followed [AM, Chapters 4 and 7]. Most other sources define primes associated to an ideal I of a ring R differently, as prime ideals of the form (I : x). The two definitions coincide when R is Noetherian. 3 4 CHAPTER 8. PRIMARY DECOMPOSITION Chapter 9 DVRs and Dedekind domains We studied Artinian rings, which are Noetherian rings of dimension 0. In this chapter, we study the next simplest case, Noetherian domains of dimension 1. We start by applying primary decomposition to such domains. Proposition 9.1. Let R be a Noetherian domain of dimension 1. Every nonzero ideal I ⊆ R can be uniquely written as a product of primary ideals whose radicals are distinct. We would like to further decompose primary ideals into prime powers. We first look at the local case. Discrete valuation rings Recall that the value group of a valuation ring R is K×/R×, where K = Frac(R). Definition 9.2. A discrete valuation ring (DVR) is a valuation ring whose value group is isomorphic to Z. Proposition 9.3. Let R be a valuation ring. The following are equivalent: (1) R is a DVR. (2) R is a PID. (3) R is Noetherian. For (2) ⇒ (1), we use the following. Lemma 9.4. Let R be a local ring of dimension > 0. Assume that the maximal T∞ n ideal m of R is principal and n=1 m = 0. Then R is a DVR. The assumption T∞ n n=1 m = 0 holds if R is a Noetherian domain. T∞ n We will see later that the assumption n=1 m = 0 holds for R Noetherian. A generator of the maximal ideal of a DVR is called a uniformizer. Example 9.5. (1) Let K = Q and let p be a rational prime. The p-adic valuation × a u vp : K → Z is defined by vp(p v ) = a, where a, u, v ∈ Z, (u, p) = (v, p) = 1. × Every valuation of K is equivalent to vp (Ostrowski’s theorem). 5 6 CHAPTER 9. DVRS AND DEDEKIND DOMAINS (2) Let k be a field and let K = k(X). Similarly to (1), for every irreduci- × ble polynomial f ∈ k[X], the f-adic valuation vf : K × → Z is defined by a u vf (f v ) = 1, where a ∈ Z, u, v ∈ k[X], (u, f) = (v, f) = 1. Every valuation of × K is equivalent to vf or v1/X . S 1/n (3) Let k be a field and let K = n k(X ). We have a non-discrete rank 1 × 1/n 1 valuation v : K → Q whose restriction to k(X ) is n vX1/n with vX1/N defined 1/n in (2). The valuation ideal m = (X )n≥1 is not finitely generated. × (4) Let F be a field and let vF : F → Γ. Let K = F (X) (or F ((X))). Then × P n K → Z × Γ carrying f = n≥N anX with aN 6= 0 to (N, v(aN )) is a valuation, of rank > 1 for vF nontrivial. Here Z × Γ is equipped with the lexicographical order. (5) Consider the particular case of (4) where F = k(Y ) and vF = vY . Let m be T n n the valuation ideal. Then m = YR is principal, but n m = (X/Y )n≥1 is not finitely generated. Definition 9.6. We say that a ring R is normal if for every prime p, Rp is an integrally closed domain. A normal domain is synonymous to an integrally closed domain. Proposition 9.7. Let R be a Noetherian local domain that is not a field. Let m be the maximal ideal and let k = m/m2. The following conditions are equivalent: (1) R is a DVR. (2) R is normal of dimension one. (3) m is principal. 2 (4) dimk(m/m ) = 1. (5) Every nonzero ideal of R is a power of m. For (2)⇒(3), we use the following. Lemma 9.8. Let R be a normal local domain and assume that the maximal ideal m is finitely generated and there exists a, b ∈ R such that m = (aR : b). Then m is principal. The proposition holds in fact for R a Noetherian local ring of dimension > 0. The proof uses the Krull intersection theorem (Corollary 10.28). Dedekind domains Theorem 9.9. Let R be a Noetherian domain of dimension one. The following are equivalent: (1) R is normal. (2) Every primary ideal of R is a power of a prime ideals. (3) Every local ring Rp (p 6= 0) is a DVR. Definition 9.10. A Dedekind domain is a Noetherian domain of dimension one satisfying the above equivalent conditions. Corollary 9.11. Every nonzero ideal of a Dedekind domain has a unique factori- zation as a product of prime ideals. 7 Remark 9.12. The following converse of Corollary 9.11 holds: An integral domain of which every nonzero ideal is a product of prime ideals is a Dedekind domain [M2, Theorem 11.6]. Example 9.13. A PID is a Dedekind domain. More examples are given by taking integral closure. Proposition 9.14. Let A be a normal domain. Let L be a finite separable extension of K = Frac(A) and let B be the integral closure of A in L. Then B is contained in a finitely generated A-submodule of L. Corollary 9.15. Let A be Dedekind domain. Let L be a finite separable extension of K = Frac(A). Then the integral closure B of A in L is a Dedekind domain. Remark 9.16. More generally, if A is a Noetherian domain of dimension 1 (not necessarily normal) and if L is a finite (not necessarily separable) extension of K = Frac(A), then the normalization B of A in L is a Dedekind domain (even when B is not finite over A). This follows from the Krull-Akizuki Theorem [M2, Theorem 11.7]. Example 9.17. Let K be a number field (namely, a finite extension of Q). Then the ring of integers OK (namely, the integral closure of Z in K) is a Dedekind domain by Corollary 9.15. √ √ Example 9.18. In particular, for√K = Q( −5), R = Z[ −5] is a Dedekind domain. 2 The prime ideal p = (2√, 1 + −5) is not principal. Indeed, since p = 2R, if 2 p = αR for α = a + b −5, a, b ∈ Z, then (NK/Qα) = NK/Q2 = 4, so that a2 + 5b2 = N α = 2, which is impossible. K/Q √ √ √ 0 0 We have√3R = qq , (1+ −5)R = pq, (1− −5)R = pq , where q = (3, 1+ −5), 0 q = (3, 1 − −5). √ The maximal ideal of the DVR Rp is (1 + −5)Rp. √ Example 9.19. Let R = [ 5]. This is a Noetherian domain of dimension one, but Z √ √ not normal. The integral closure of R in ( 5) is [ 1+ 5 ].