A Term of
Commutative Algebra
By Allen ALTMAN and Steven KLEIMAN
c 2013, Worldwide Center of Mathematics, LLC
Version of September 1, 2013: 13Ed.tex
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. v. edition number for publishing purposes ISBN 978-0-9885572-1-5
iv
Contents Preface .
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1. 2. 3. 4. 5.
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v
Rings and Ideals . . . . . . Prime Ideals . . . . . . . Radicals . . . . . . . . . Modules . . . . . . . . . Exact Sequences . . . . . . Appendix: Fitting Ideals . . . 6. Direct Limits . . . . . . . 7. Filtered Direct Limits . . . . 8. Tensor Products . . . . . . 9. Flatness . . . . . . . . . 10. Cayley–Hamilton Theorem . . 11. Localization of Rings . . . . 12. Localization of Modules . . . 13. Support . . . . . . . . 14. Krull–Cohen–Seidenberg Theory 15. Noether Normalization . . . Appendix: Jacobson Rings . . 16. Chain Conditions . . . . . 17. Associated Primes . . . . . 18. Primary Decomposition . . . 19. Length . . . . . . . . . 20. Hilbert Functions . . . . . Appendix: Homogeneity . . . 21. Dimension . . . . . . . 22. Completion . . . . . . . 23. Discrete Valuation Rings . . Appendix: Cohen–Macaulayness 24. Dedekind Domains . . . . 25. Fractional Ideals . . . . . 26. Arbitrary Valuation Rings . .
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1 7 11 17 24 30 35 42 48 54 60 66 72 77 84 88 93 96 101 106 112 116 122 124 130 138 143 148 152 157
Solutions . . . . . . . . . 1. Rings and Ideals . . . . 2. Prime Ideals . . . . . 3. Radicals . . . . . . . 4. Modules . . . . . . . 5. Exact Sequences . . . . 6. Direct Limits . . . . . 7. Filtered direct limits . . . 8. Tensor Products . . . . 9. Flatness . . . . . . . 10. Cayley–Hamilton Theorem 11. Localization of Rings . .
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162 162 164 166 173 175 179 182 185 188 191 194
iii
Contents 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Lo calization of Modules . . . Support . . . . . . . . . Krull–Cohen–Seidenberg Theory Noether Normalization . . . Chain Conditions . . . . . Associated Primes . . . . . Primary Decomposition . . . Length . . . . . . . . . Hilbert Functions . . . . . Dimension . . . . . . . . Completion . . . . . . . Discrete Valuation Rings . . . Dedekind Domains . . . . . Fractional Ideals . . . . . . Arbitrary Valuation Rings . .
Bibliography
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198 201 211 214 218 220 221 224 226 229 232 236 241 243 245
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249
Disposition of the Exercises in [3] . . . . . . . . . . . . . . .
250
Index . .
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iv
Contents Preface .
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1. 2. 3. 4. 5.
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v
Rings and Ideals . . . . . . Prime Ideals . . . . . . . Radicals . . . . . . . . . Modules . . . . . . . . . Exact Sequences . . . . . . Appendix: Fitting Ideals . . . 6. Direct Limits . . . . . . . 7. Filtered Direct Limits . . . . 8. Tensor Products . . . . . . 9. Flatness . . . . . . . . . 10. Cayley–Hamilton Theorem . . 11. Localization of Rings . . . . 12. Localization of Modules . . . 13. Support . . . . . . . . 14. Krull–Cohen–Seidenberg Theory 15. Noether Normalization . . . Appendix: Jacobson Rings . . 16. Chain Conditions . . . . . 17. Associated Primes . . . . . 18. Primary Decomposition . . . 19. Length . . . . . . . . . 20. Hilbert Functions . . . . . Appendix: Homogeneity . . . 21. Dimension . . . . . . . 22. Completion . . . . . . . 23. Discrete Valuation Rings . . Appendix: Cohen–Macaulayness 24. Dedekind Domains . . . . 25. Fractional Ideals . . . . . 26. Arbitrary Valuation Rings . .
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1 7 11 17 24 30 35 42 48 54 60 66 72 77 84 88 93 96 101 106 112 116 122 124 130 138 143 148 152 157
Solutions . . . . . . . . . 1. Rings and Ideals . . . . 2. Prime Ideals . . . . . 3. Radicals . . . . . . . 4. Modules . . . . . . . 5. Exact Sequences . . . . 6. Direct Limits . . . . . 7. Filtered direct limits . . . 8. Tensor Products . . . . 9. Flatness . . . . . . . 10. Cayley–Hamilton Theorem 11. Localization of Rings . .
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162 162 164 166 173 175 179 182 185 188 191 194
iii
Contents 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Lo calization of Modules . . . Support . . . . . . . . . Krull–Cohen–Seidenberg Theory Noether Normalization . . . Chain Conditions . . . . . Associated Primes . . . . . Primary Decomposition . . . Length . . . . . . . . . Hilbert Functions . . . . . Dimension . . . . . . . . Completion . . . . . . . Discrete Valuation Rings . . . Dedekind Domains . . . . . Fractional Ideals . . . . . . Arbitrary Valuation Rings . .
Bibliography
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198 201 211 214 218 220 221 224 226 229 232 236 241 243 245
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249
Disposition of the Exercises in [3] . . . . . . . . . . . . . . .
250
Index . .
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Preface
vii
preserve other direct limits. Here the theory briefly climbs to a higher level of abstraction. The discussion is completely elementary, but by far the most abstract in the book. The extra abstraction can be difficult, especially for beginners. Below, filtered direct limits are treated too. They are closer to the kind of limits treated by Atiyah and Macdonald. In particular, filtered direct limits preserve exactness and flatness. Further, they appear in the following lovely form of Lazard’s Theorem: in a canonical way, every module is the direct limit of free modules of finite rank; moreover, the module is flat if and only if that direct limit is filtered. Atiyah and Macdonald treat primary decomposition in a somewhat dated fashion. First, they study primary decompositions of ideals in rings. Then, in the exercises, they indicate how to translate the theory to modules. The decompositions need not exist, as the rings and modules need not be Noetherian. Associated primes play a secondary role: they are defined as the radicals of the primary components, and then characterized as the primes that are the radicals of annihilators of elements. Finally, they prove that, when the rings and modules are Noetherian, decompositions exist and the associated primes are annihilators. To prove existence, they use irreducible modules. Nowadays, associated primes are normally defined as prime annihilators of elements, and studied on their own at first; sometimes, as below, irreducible modules are not considered at all in the main development. There are several other significant differences between Atiyah and Macdonald’s treatment and the one below. First, the Noether Normalization Lemma is proved below in a stronger form for nested sequences of ideals; consequently, for algebras that are finitely generated over a field, dimension theory can be developed directly without treating Noetherian local rings first. Second, in a number of results below, the modules are assumed to be finitely presented over an arbitrary ring, rather than finitely generated over a Noetherian ring. Third, there is an elementary treatment of regular sequences below and a proof of Serre’s Criterion for Normality. Fourth, below, the Adjoint-Associativity Formula is proved over a pair of base rings; hence, it yields both a left and a right adjoint to the functor of restriction of scalars. The present book is a second beta edition. Please do the community a service by sending the authors comments and corrections. Thanks! Allen B. Altman and Steven L. Kleiman 31 August 2013
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Rings and Ideals (1.17)
5
Exercise (1.10). — Let R be ring, and P := R[X 1 , . . . , Xn ] the polynomial ring. Let m n and a1 , . . . , am R. Set p := X 1 a1 , . . . , Xm am . Prove that P/ p = R[X m+1 , . . . , Xn ].
≤
∈
−
−
∈
(1.11) (Idempotents ). — Let R be a ring. Let e R be an idempotent; that is, e2 = e. Then Re is a ring with e as 1, because (xe)e = xe. But Re is not a subring of R unless e = 1, although Re is an ideal. Set e ′ := 1 e. Then e ′ is idempotent and e e′ = 0. We call e and e ′ complementary idempotents. Conversely, if two elements e 1 , e2 R satisfy e 1 + e2 = 1 and e 1 e2 = 0, then they are complementary idempotents, as for each i,
−
·
∈
ei = e i 1 = e i (e1 + e2 ) = e 2i .
·
We denote the set of all idempotents by Idem(R). Let ϕ : R R′ be a ring map. Then ϕ(e) is idempotent. So the restriction of ϕ to Idem(R) is a map
→
Idem(ϕ): Idem(R)
→ Idem(R′).
Example (1.12). — Let R := R ′ R′′ be a product of two rings: its operations are performed componentwise. The additive identity is (0, 0); the multiplicative identity is (1, 1). Set e := (1, 0) and e ′ := (0, 1). Then e and e ′ are complementary idempotents. The next proposition shows this example is the only one possible.
×
Proposition (1.13). — Let R be a ring with complementary idempotents e and e′ . Set R′ := Re and R′′ := Re′, and form the map ϕ: R R ′ R′′ defined by ϕ(x) := (xe,xe′ ). Then ϕ is a ring isomorphism.
→ ×
Proof: Define a map ϕ ′ : R R ′ by ϕ ′ (x) := xe. Then ϕ ′ is a ring map since xye = xye 2 = (xe)(ye). Similarly, define ϕ′′ : R R′′ by ϕ′′ (x) := xe ′ ; then ϕ′′ is a ring map. So ϕ is a ring map. Further, ϕ is surjective, since (xe,x′ e′ ) = ϕ(xe+x′ e′ ). Also, ϕ is injective, since if xe = 0 and xe ′ = 0, then x = xe + xe′ = 0. Thus ϕ is an isomorphism.
→
→
Exercise (1.14) (Chinese Remainder Theorem ). — Let R be a ring. (1) Let a and b be comaximal ideals; that is, a + b = R. Prove (a) ab = a
∩ b and
(b) R/ab = (R/a)
× (R/b).
(2) Let a be comaximal to both b and b ′ . Prove a is also comaximal to bb ′ . (3) Let a , b be comaximal, and m, n 1. Prove a m and b n are comaximal. (4) Let a 1 , . . . , an be pairwise comaximal. Prove
≥
··· ∩···∩ ·· · ··· −→
an are comaximal; (a) a1 and a 2 (b) a1 an = a 1 an ; an ) ∼ (c) R/(a1 (R/ai ).
≥
Exercise (1.15). — First, given a prime number p and a k 1, find the idempotents in Z / pk . Second, find the idempotents in Z / 12 . Third, find the number N of idempotents in Z / n where n = i=1 pni i with p i distinct prime numbers.
Exercise (1.16). — Let R := R′ R′′ be a product of rings, a R an ideal. Show a = a ′ a′′ with a ′ R′ and a ′′ R′′ ideals. Show R/ a = (R′ /a′) (R′′ /a′′ ).
×
⊂
× ⊂
⊂
×
6
Rings and Ideals (1.17)
Exercise (1.17). — Let R be a ring, and e, e′ idempotents. (See (10.7) also.) (1) Set a := e . Show a is idempotent; that is, a 2 = a . (2) Let a be a principal idempotent ideal. Show a f with f idempotent. (3) Set e ′′ := e + e′ ee′. Show e, e′ = e′′ and e ′′ is idempotent. (4) Let e 1 , . . . , er be idempotents. Show e1 , . . . , er = f with f idempotent. (5) Assume R is Boolean. Show every finitely generated ideal is principal.
−
Radi Radica cals ls (3.4 (3.40) 0)
15
√ say xn ∈ a and y m ∈ a. Then Proof: Take x, Take x, y ∈ a; say x n+m−1
(x + y )
=
n+m−1 i+j=m+n−1 j
i j
xy .
This sum belongs to a as, in each summand, either x i or y or y j does, since, if i i n 1 and j and j m 1, then i then i + j m + n 2. Thus x Thus x + y a. So clearly a is an ideal. Alternatively, given any collection of ideals aλ , note that aλ is also an ideal. So a is an ideal owing to (3.29). (3.29).
≤ − √
≤
∈ √
−
√
≤ −
√ a is finitely
Exercise (3.32). (3.32). — Let R be a ring, and a an an ideal. ideal. Assume Assume n generated. Show large n.. a a for all large n
√ ⊂
Exercise (3.33). (3.33). — Let R be a ring, q an ideal, p a finitely generated prime. Prove that p = q if and only if there is n 1 such that p q pn.
√
≥
⊃ ⊃
Proposition Proposition (3.34). (3.34). — A ring R is reduced and has only one minimal prime q if and only if R R is a domain.
Proof: Suppose R Suppose R is reduced, or 0 = 0 . Then 0 is equal to the intersection of all the prime ideals p by (3.29) by (3.29).. By (3.14) By (3.14),, every p contains q . So 0 = q . Thus R Thus R is a domain. The converse is obvious.
Exercise (3.35). (3.35). — Let R be R be a ring. Assume R Assume R is reduced and has finitely many minimal prime ideals p1, . . . , pn . Prove ϕ Prove ϕ : R (R/pi) is injective, and for each i, there is some (x ( x1 , . . . , xn ) Im(ϕ Im(ϕ) with x with x i = 0 but x but x j = 0 for j for j = i. i .
→
∈
Exercise (3.36). (3.36). — Let R be a ring, X a X a variable, f variable, f := a := a 0 + a1 X + + + an X n m and g := b0 + b + b 1 X + + + bm X polynomials with an = 0 and bm = 0. Call Call f primitive if primitive if a0 , . . . , an = R. R . Prove the following statements:
···
···
(1) Then f Then f is is nilpotent if and only if a a 0 , . . . , an are nilpotent. (2) Then f Then f is is a unit if and only if a if a 0 is a unit and a and a 1 , . . . , an are nilpotent. (3) If f is f is a zerodivisor, then there is a nonzero b R with bf = bf = 0; in fact, if f g = 0 with m with m minimal, then f then f bm = 0 (or m (or m = = 0). (4) Then f Then f g is primitive if and only if f if f and g and g are primitive.
∈
Exercise (3.37) Exercise (3.37).. — Generalize — Generalize (3.36) to the polynomial ring P := R := R[[X 1 , . . . , Xr ]. For (3), reduce to the case of one variable Y variable Y via via this standard device: take d suitably d suitably i large, and define ϕ : P R[Y ] Y ] by ϕ by ϕ((X i ) := Y := Y d .
→
Exercise (3.38). (3.38). — Let R be a ring, X ring, X a variable. Show that rad(R rad(R[X ]) ]) = nil(R nil(R[X ]) ]) = nil(R nil(R)R[X ]. Exercise (3.39). (3.39). — Let R be a ring, a an ideal, X ideal, X a a variable, R variable, R[[ [[X X ]] ]] the formal power series ring, M R[[X [[X ]] ]] be a maximal ideal, and f and f := an X n R[[X [[X ]]. ]]. Set bn X n bn a . Prove the following statements: m := M R and A := (1) (2) (3) (4) (5)
⊂
∩
|
∈
∈
If f is f is nilpotent, then a n is nilpotent for all n all n.. The converse is false. Then f Then f rad(R rad(R[[X [[X ]]) ]]) if and only if a a 0 rad(R rad(R). Assume Assume X M. Then X Then X and m generate M . Assume Assume M is maximal. Then X Then X M and m is maximal. a is finitely generated, then a R[[X If a [[X ]] ]] = A . The converse may fail.
∈ ∈
∈ ∈
∈ ∈
∈
16
Radi Radica cals ls (3.4 (3.40) 0)
Example (3.40) Example (3.40).. — Let R be a ring, R[[ R [[X X ]] ]] the formal power series series ring. Then every prime p of R is R is the contraction of a prime of R of R[[ [[X X ]]. ]]. Indeed, p R[[X [[X ]] ]] R = p . So by (3.13) by (3.13),, there is a prime q of R of R[[ [[X X ]] ]] with q R = p . In fact, a specific choice for q is the set of series an X n with a with a n p. Indeed, the canonical map R map R R/p induces a surjection R[[X [[X ]] ]] R/p with kernel q; hence, R[[X [[X ]]/ ]]/q = (R/p)[[X )[[X ]]. ]]. Plainly (R/ (R/p)[[X )[[X ]] ]] is a domain. domain. But (3.39) But (3.39)(5) (5) shows q may not be equal to pR[[X [[X ]]. ]].
→
∈
∩
∩ →
Modules (4.20)
23
Exercise (4.19). — Let L be a module, Λ a nonempty set, M λ a module for λ Λ. Prove that the injections ι κ : M κ M λ induce an injection
∈
→ Hom(L, M λ ) → Hom(L,
5. Exact Sequences
M λ ),
and that it is an isomorphism if L is finitely generated.
∈
Exercise (4.20). — Let a be an ideal, Λ a nonempty set, M λ a module for λ Λ. aM λ . Prove a ( M λ ) = aM λ if a is finitely generated. Prove a M λ =
In the study of modules, the exact sequence plays a central role. We relate it to the kernel and image, the direct sum and direct product. We introduce diagram chasing, and prove the Snake Lemma, which is a fundamental result in homological algebra. We define projective modules, and characterize them in four ways. Finally, we prove Schanuel’s Lemma, which relates two arbitrary presentations of a module. In an appendix, we use deteminants to study free modules. Definition (5.1). — A (finite or infinite) sequence of module homomorphisms α α · · · → M i−1 −−−→ M i −→ M i+1 → · · · i−1
i
is said to be exact at M i if Ker(αi ) = Im(αi−1 ). The sequence is said to be exact if it is exact at every M i , except an initial source or final target. α −→
→
Example (5.2). — (1) A sequence 0 L M is exact if and only if α is injective. If so, then we often identify L with its image α(L). Dually —that is, in the analogous situation with all arrows reversed—a seβ quence M N 0 is exact if and only if β is surjective. β α (2) A sequence 0 L M N is exact if and only if L = Ker(β ), where ‘=’
−→ →
→ −→ −→
means “canonically isomorphic.” Dually, a sequence L and only if N = Coker(α) owing to (1) and (4.6.1).
β α −→ → M − N → 0 is exact if β −→ →
α → −→
(5.3) (Short exact sequences ). — A sequence 0 L M N 0 is exact if and only if α is injective and N = Coker(α), or dually, if and only if β is surjective and L = Ker(β ). If so, then the sequence is called short exact, and often we regard L as a submodule of M , and N as the quotient M /L. For example, the following sequence is clearly short exact: 0
ι π → L −→ L ⊕ N −−→ N → 0 L
N
ιL (l) := (l, 0) and
where
πN (l, n) := n.
Often, we identify L with ι L L and N with ι N N . Proposition (5.4). — For λ Λ, let M λ′ M λ M λ′′ be a sequence of module homomorphisms. If every sequence is exact, then so are the two induced sequences
∈
→ → M λ′
M λ
→ →
M λ′′
and
→ → M λ′
M λ
M λ′′ .
Conversely, if either induced sequence is exact then so is every original one. Proof: The assertions are immediate from (5.1) and (4.15). M ′
M ′′
⊂
M ′
Exercise (5.5). — Let and be modules, N a submodule. Set M := M ′ M ′′ . Using (5.2)(1) and (5.3) and (5.4), prove M/N = M ′ /N M ′′ .
⊕
→
M ′
→
→
M ′′
⊕
→
Exercise (5.6). — Let 0 M 0 be a short exact sequence. Prove that, if M ′ and M ′′ are finitely generated, then so is M . β α −→ −→
Proposition (5.7). — Let 0 M ′ M M ′′ 0 be a short exact sequence, 1 ′ − and N M a submodule. Set N := α (N ) and N ′′ := β (N ). Then the induced sequence 0 N ′ N N ′′ 0 is short exact.
⊂
→ → →
→ →
24
→
Solutions: (26.20)
247
Ideal Theorem (21.10). Then R[y]p is Noetherian of dimension 1. But L/K is a finite field extension, so L/ Frac(R[y]) is one too. Hence the integral closure R ′ of R[y]p in L is a Dedekind domain by (26.18). So by the Going-up Theorem (14.3), there’s a prime q of R ′ lying over p R[y]p . Then as R ′ is Dedekind, R q′ is a DVR of L by (24.7). Further, y qR′q . Thus x / Rq′ , as desired.
∈
∈
