COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS

RYO TAKAHASHI

Contents

Introduction 1 1. Associated primes 1 2. Depth 3 3. 6 4. Cohen–Macaulay rings and Gorenstein rings 9 Appendix A. Proof of Theorem 1.12 11 Appendix B. Proof of Theorem 4.8 13 References 17

Introduction

Goal. The structure theorem of (maximal) Cohen–Macaulay modules over commutative Gorenstein local rings Throughout. • {R : a commutative with 1 A = k[[X,Y ]]/(XY ) • k : a field B = k[[X,Y ]]/(X2,XY ) • x := X, y := Y

1. Associated primes Def 1.1. Let I ⊊ R be an ideal of R (1) I is a maximal ideal if there exists no ideal J of R with I ⊊ J ⊊ R (2) I is a if (ab ∈ I ⇒ a ∈ I or b ∈ I) (3) Spec R := {Prime ideals of R} Ex 1.2. Spec A = {(x), (y), (x, y)}, Spec B = {(x), (x, y)} Prop 1.3. I ⊆ R an ideal (1) I is prime iff R/I is an (2) I is maximal iff R/I is a field 2010 Mathematics Subject Classification. 13C14, 13C15, 13D07, 13H10. Key words and phrases. Associated prime, Cohen–Macaulay module, Cohen–Macaulay ring, Ext module, Gorenstein ring, Krull dimension. 1 2 RYO TAKAHASHI

(3) Every maximal ideal is prime Throughout the rest of this section, let M be an R-module. Def 1.4. p ∈ Spec R is an associated prime of M if ∃ x ∈ M s.t p = ann(x)

AssR M := {Associated primes of M} Ex 1.5.

AssA A = {(x), (y)}, AssB B = {(x), (x, y)} Prop 1.6. TFAE for p ∈ Spec R

(1) p ∈ AssR M (2) ∃ f : R/p → M an injective homomorphism Proof. (1) ⇒ (2) ∃ x ∈ M s.t p = ann(x) Define f : R/p → M by f(a) = ax (2) ⇒ (1) x := f(1) ∈ M p = ann(x) ■ Prop 1.7.

max{ann(x) | 0 ≠ x ∈ M} ⊆ AssR M Proof. p := ann(x) ETS p ∈ Spec R x ≠ 0 ⇒ 1 ∈/ p ⇒ p ⊊ R Suppose a, b ∈ R, ab ∈ p, a∈ / p. Then abx = 0 ∴ p = ann(x) ⊊ ann(bx)(a ∈ ann(bx) − p) The maximality of p shows ann(bx) = R ∴ bx = 0 ∴ x ∈ p ■ Cor 1.8.

M ≠ 0 ⇐⇒ AssR M ≠ ∅ Proof. (⇐) Trivial (⇒) As R is Noetherian, LHS in Prop 1.7 is nonempty. ■ Def 1.9. x ∈ R, M an R-module (1) x is a zerodivisor (ZD) on M if 0 ≠ ∃ m ∈ M s.t xm = 0 (2) x is a nonzerodivisor (NZD) on M if x is not a ZD on M

Cor 1.10. ∪ {ZDs on M} = p p∈Ass M COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 3

Proof. ∪ ⊇ ∈ ( ) Let a p∈Ass M p ⇒ a ∈ ∃ p ∈ Ass M ⇒ ∃ x ∈ M s.t p = ann(x) ∴ ax = 0, x ≠ 0 ∴ a is a ZD on M (⊆) Let a ∈ R be a ZD on M ⇒ 0 ≠ ∃ z ∈ M s.t az = 0 ⇒ a ∈ ann(z) ann(z) ⊆ ∃ q ∈ max{ann(x) | 0 ≠ x ∈ M} ⊆ Ass M by Prop 1.7 ∪ ∴ ∈ ∈ ∴ ∈ ■ a q Ass M a p∈Ass M p Ex 1.11. (1) {ZDs on A} = (x) ∪ (y) ∴ x − y is a NZD on A (2) {ZDs on B} = (x, y) ∴ All nonunits of B are ZDs on B Thm 1.12.

If M is finitely generated, then # AssR M < ∞ Sketch. (Details: Appendix A)

(1) ∃ 0 = M0 ⊊ M1 ⊊ ··· ⊊ Mn = M submodules s.t. ∼ Mi/Mi−1 = R/pi, pi ∈ Spec R (2) If 0 → X → Y → Z → 0 is an exact seq of R-modules, then Ass Y ⊆ Ass X ∪ Ass Z

(3) AssR(R/p) = {p} ∀ p ∈ Spec R

0 → Mi−1 → Mi → R/pi → 0 ⇝ Ass Mi ⊆ Ass Mi−1 ∪ Ass R/pi = Ass Mi−1 ∪ {pi} ⇝ Ass M ⊆ {p1,..., pn} ■ 2. Depth Def 2.1. R is local if R has a unique maximal ideal m k := R/m is the residue field of R We say (R, m, k) is a Ex 2.2. (1) (A, (x, y)A, k) is local (2) (B, (x, y)B, k) is local

Def 2.3. (R, m, k): local M: a f.g R-mod The depth of M is: { ⩾ | i ̸ } depthR M = inf i 0 ExtR(k, M) = 0 Prop 2.4. (R, m, k): local 0 → L → M → N → 0 an exact seq of f.g R-modules Then (1) depth L ⩾ inf{depth M, depth N + 1} 4 RYO TAKAHASHI

(2) depth M ⩾ inf{depth L, depth N} (3) depth N ⩾ inf{depth M, depth L − 1} Proof. (1) n := inf{depth M, depth N + 1} ⇒ depth M ⩾ n, depth N ⩾ n − 1 Extn−2(k, N) −−−→ Extn−1(k, L) −−−→ Extn−1(k, M) −−−→ Extn−1(k, N) −−−→ Extn(k, L) −−−→ Extn(k, M) −−−→ Extn(k, N) ∴ depth L ⩾ n (2)(3) Similar to (1) ■ Cor 2.5 (Depth Lemma). (R, m, k): local 0 → L → M → N → 0 an exact seq of f.g R-modules depth M > depth N =⇒ depth L = depth N + 1 Proof. depth M ⩾ depth N + 1 By Prop 2.4(1), depth L ⩾ inf{depth M, depth N + 1} = depth N + 1 Assume depth L > depth N + 1. Then by Prop 2.4(3) depth N ⩾ inf{depth M, depth L − 1} > depth N This contradiction shows depth L = depth N + 1 ■ Prop 2.6. (R, m, k): local M: a f.g R-mod ∈ ⇐⇒ m AssR M depthR M = 0 Proof.

(⇒) Prop 1.6 ⇒ k = R/m ,→ M ⇒ HomR(k, M) ≠ 0 (⇐) HomR(k, M) ≠ 0 ⇒ ∃ f : k → M, f(1) = x ≠ 0 a ≠ 0 ⇒ a ∈/ m ⇒ a ∈ R× ⇒ ax ≠ 0 ∴ f is injective Prop 1.6 implies m ∈ Ass M ■ ∈ Ex 2.7. (1) depthA A > 0 since (x, y) / AssA A ∈ (2) depthB B = 0 since (x, y) AssB B Lem 2.8 (Prime Avoidance).

I ⊆ R an ideal p1,..., pn ∈ Spec R

I ⊆ p1 ∪ · · · ∪ pn =⇒ I ⊆ pℓ (∃ ℓ) Proof. Induction on n n = 1 Trivial

n ⩾ 2 May assume ∄ inclusion relation among p1,..., pn Suppose I ⊈ pi (∀ i) Ind hyp ⇒ I ⊈ p1 ∪ · · · ∪ pn−1 ⇒ ∃ x ∈ I − (p1 ∪ · · · ∪ pn−1) COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 5

As I ⊆ (p1 ∪ · · · ∪ pn−1) ∪ pn, we have x ∈ pn

∃ y ∈ I − pn,

∃ zi ∈ pi − pn (1 ⩽ ∀ i ⩽ n − 1),

w := x + yz1 ··· zn−1

(1) x, y ∈ I ⇒ w ∈ I ⊆ (p1 ∪ · · · ∪ pn−1) ∪ pn (2) x∈ / p1 ∪ · · · ∪ pn−1, z1 ∈ p1 ⇒ w∈ / p1 ∪ · · · ∪ pn−1 (3) x ∈ pn, y, z1, . . . , zn−1 ∈/ pn ⇒ w∈ / pn (1), (2), (3) yield a contradiction ■ Prop 2.9. (R, m, k) local M a f.g R-mod TFAE

(1) depthR M > 0 (2) ∃ x ∈ m a NZD on M Proof. By Prop 2.6, depth M > 0 ⇔ m ∈/ Ass M (2) ⇒ (1) ∪ ⇒ ∈ ⇒ ∈ ∀ ∈ Cor 1.10 x / p∈Ass M p x / p Ass M As x ∈ m, we must have m ∈/ Ass M (1) ⇒ (2) m ∈/ Ass M ⇒ m ⊈ ∀ p ∈ Ass M Thm 1.12 implies # Ass M < ∞ Lem 2.8 and Cor 1.10 yield ∪ m ⊈ p = {ZDs on M} p∈Ass M ∴ ∃ x ∈ m a NZD on M ■ Ex{ 2.10. depth A > 0 (1) − {x y is a NZD on A depth B = 0 (2) ∄ NZD on B in the maximal ideal of B Lem 2.11. M,N: R-modules x ∈ ann N: a NZD on R,M Then i+1 ∼ i ∀ ∈ Z ExtR (N,M) = ExtR/(x)(N, M/xM)( i ) i i+1 − Proof. T := ExtR ( ,M) ETS: (1) ∀ 0 → X → Y → Z → 0 exact seq of R/(x)-modules ∃ 0 → T 0(Z) → T 0(Y ) → T 0(X) → T 1(Z) → · · · exact 0 ∼ (2) T = HomR/(x)(−, M/xM) (3) T i(P ) = 0 (∀ P projective R-mod, ∀ i > 0) by Axioms for (contravariant) Ext [5, Theorem 6.64] ■ Prop 2.12. (R, m, k) local M a f.g R-mod x ∈ m a NZD on M 6 RYO TAKAHASHI − (1) depthR M/xM = depthR M 1 − (2) If x a NZD on R, then depthR/(x) M/xM = depthR M 1

Proof. (1) t := depthR M From the exact seq 0 → M −→x M → M/xM → 0 we get an exact seq −−−→ t−2 −−−→x t−2 −−−→ t−2 ExtR (k, M) ExtR (k, M) ExtR (k, M/xM) 0 −−−→ t−1 −−−→x t−1 −−−→ t−1 ExtR (k, M) ExtR (k, M) ExtR (k, M/xM) 0 −−−→ t −−−→x t −−−→ t ExtR(k, M) ExtR(k, M) ExtR(k, M/xM) { 0 − Ext<(t 1)(k, M/xM) = 0, ∴ R t−1 ∼ t ̸ ExtR (k, M/xM) = ExtR(k, M) = 0 ∴ − depthR M/xM = t 1 (2) Lem 2.11 implies i ∼ i+1 ExtR/(x)(k, M/xM) = ExtR (k, M) for i ∈ Z ■ Ex 2.13. x − y is a NZD on A By Prop 2.12, − − depthA/(x−y) A/(x y) = depthA A 1 ∼ A/(x − y) = k[X]/(X2) k ,→ k[X]/(X2), 1 7→ x ∴ − ∴ depthA/(x−y) A/(x y) = 0 depthA A = 1

3. Krull dimension Lem 3.1. (R, m, k) local (1) k is a simple R-module ∼ (2) If M is a simple R-module, then M = k Prop 3.2 (Nakayama’s Lemma (NAK)). (R, m, k) local M a f.g R-mod I ⊊ R an ideal IM = M =⇒ M = 0 Proof. May assume M ≠ 0 As M is Noetherian, ∃ L ⊊ M a maximal submod The maximality shows M/L is a simple R-mod ∼ Lemma 3.1 ⇒ M/L = k ⇒ m(M/L) = 0 ⇒ mM ⊆ L M = IM ⊆ mM ⊆ L ⊊ M This contradiction shows the proposition ■ Def 3.3. The (Krull) dimension of R is:

dim R = sup{n ⩾ 0 | ∃ p0 ⊊ p1 ⊊ ··· ⊊ pn in Spec R} COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 7

Fact 3.4. [1, Theorem 11.14] If (R, m, k) is local, then { } ∃ x , . . . , x ∈ m s.t dim R = inf n ⩾ 0 1 n R/(x1, . . . , xn) is Artinian Ex 3.5. dim A = dim B = 1 A/(x − y),B/(y) are Artinian Cor 3.6. (R, m, k) local TFAE (1) R is Artinian (2) mn = 0 (∃ n ⩾ 0) (3) Spec R = {m} (4) dim R = 0 Proof. (1) ⇒ (2) R ⊇ m ⊇ m2 ⊇ · · · ⇒ ∃ n ⩾ 0 s.t mn = mn+1 By Prop 3.2, mn = 0 (2) ⇒ (3) p ∈ Spec R ⇒ n ⩾ 0 s.t mn = 0 ⊆ p ⇒ m ⊆ p ⇒ p = m (3) ⇒ (4) By definition (4) ⇒ (1) By Fact 3.4 ■ Prop 3.7. (R, m, k) local M a f.g R-module ⩽ ∀ ∈ depthR M dim R/p ( p AssR M)

Proof. Induction on t := depthR M t = 0 NTS t ⩾ 1 By Prop 2.9, ∃ x ∈ m a NZD on M. By Prop 2.12(1), − − depthR M/xM = depthR M 1 = t 1 From the exact seq 0 → M −→x M → M/xM → 0 we get an exact seq x f 0 → HomR(R/p,M) −→ HomR(R/p,M) −→ HomR(R/p, M/xM)

As p ∈ Ass M, HomR(R/p,M) ≠ 0. By Prop 3.2, ∼ Im f = HomR(R/p,M)/x HomR(R/p,M) ≠ 0

∴ HomR(R/p, M/xM) ≠ 0. Take 0 ≠ f ∈ HomR(R/p, M/xM) f : R/p → M/xM, f(1) =: z ≠ 0 a ∈ p ⇒ a = 0 in R/p ⇒ a · z = az = 0 in M/xM

∴ p ⊆ annR(z) p ⊊ annR(z) ⊆ ∃ q ∈ AssR(M/xM), where

(⊊) holds as x ∈ annR(z) − p by Cor 1.10, and 8 RYO TAKAHASHI

(⊆) follows from Prop 1.7. In particular, p ⊊ q ∴ ⩾ − dim R/p > dim R/q depthR(M/xM) = t 1, where (⩾) is by ind hyp. Thus t ⩽ dim R/p ■ Ex 3.8. depth A = dim A/(x) = dim A/(y) = 1 depth B = dim B/(x, y) = 0 < 1 = dim B/(x) Cor 3.9. (R, m, k) local M ≠ 0 a f.g R-mod ⩽ depthR M dim R Proof. As M ≠ 0, we find p ∈ Ass M by Cor 1.8 ⩽ ⩽ ■ By Prop 3.7, depthR M dim R/p dim R Def 3.10. p ∈ Spec R is a minimal prime if there is no q ∈ Spec R with q ⊊ p. Set Min R = {Minimal primes of R} Cor 3.11. Min R ⊆ Ass R Proof. Let p ∈ Spec R. Define { a | ∈ ∈ − } Rp := s a R, s R p , where a b ⇐⇒ ∃ ∈ − − s = t u R p s.t u(ta sb) = 0 ⇝ Rp is a commutative Noetherian ring by a b ta+sb a · b ab s + t := st , s t := st { a | ∈ ∈ − } ⊆ ⇝ Set pRp := s a p, s R p Rp (Rp, pRp) local

p ∈ Min R ⇒ ∄ P ∈ Spec R s.t P ⊆ pRp

⇒ Spec Rp = {pRp}

⇒ dim Rp = 0 by Cor 3.6 By Cor 20, ⩽ ⩽ ∴ 0 depthRp Rp dim Rp = 0 depthRp Rp = 0 ∈ ∴ ∃ a ∈ a By Prop 2.6, pRp AssRp Rp s Rp s.t pRp = annRp ( s ) Write p = (b1, . . . , bn) bi ∈ a ⇒ bia 0 1 pRp = annRp ( s ) s = 1

⇒ ∃ ti ∈ R − p s.t tibia = 0

t := t1 ··· tn ∈ R − p Claim p = annR(ta)(∴ p ∈ AssR R) (⊆) bi(ta) = t1 ··· ti−1ti+1 ··· tn(tibia) = 0 (⊇) c ∈ ann(ta) ⇒ 0 = c(ta) = t(ca) ⇒ ca 0 ⇒ c ∈ a ⇒ ∈ ■ s = 1 1 ann( s ) = pRp c p COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 9

Ex 3.12. Min A = {(x), (y)} = Ass A Min B = {(x)} ⊊ {(x), (x, y)} = Ass B By Cor 3.11 and Thm 1.12 : Cor 3.13. # Min R < ∞ Prop 3.14. (R, m, k) local x ∈ m (1) dim R/(x) ⩾ dim R − 1 (2) “=” holds if x is a NZD on R Proof. n := dim R/(x)

(1) By Fact 3.4, ∃ y1,..., yn ∈ m/(x) s.t

(R/(x))/(y1,..., yn) = R/(x, y1, . . . , yn) is Artinian. By Fact 3.4 again, dim R ⩽ n + 1

(2) ∃ p0,..., pn ∈ Spec R s.t

(x) ⊆ p0 ⊊ p1 ⊊ ··· ⊊ pn

Suppose p0 ∈ Min R. Then p0 ∈ Ass R by Cor 3.11 Cor 1.10 says all elements in p0 are ZDs on R, contradicting that x is a NZD on R ∴ p0 ∈/ Min R ∴ ∃ q ∈ Spec R s.t q ⊊ p0 The chain

q ⊊ p0 ⊊ p1 ⊊ ··· ⊊ pn in Spec R implies dim R ⩾ n + 1 ■

Ex{ 3.15. x − y a NZD on A (1) − 2 − {dim A/(x y) = dim k[[X]]/(X ) = 0 = dim A 1 x a ZD on A (2) dim A/(x) = dim k[[Y ]] = 1 > 0 = dim A − 1

4. Cohen–Macaulay rings and Gorenstein rings Throughout this section, let (R, m, k) be local. Def 4.1. (1) A f.g R-mod M is (maximal) Cohen–Macaulay (CM) if (M = 0 or) depth M = dim R (2) R is Cohen–Macaulay (CM) if it is CM as an R-module, i.e., depth R = dim R Rem 4.2. Any f.g free R-mod is CM if R is CM Ex 4.3. (1) A is a CM local ring (2) B is not a CM local ring (3) A/(x), A/(y) are CM A-modules 10 RYO TAKAHASHI

Def 4.4. R is Gorenstein if {

i ∼ k (i = dim R), Ext (k, R) = R 0 (i ≠ dim R). Prop 4.5. A Gor local ring is CM Prop 4.6. Let x ∈ m be a NZD on R (1) R is CM ⇐⇒ R/(x) is CM (2) R is Gor ⇐⇒ R/(x) is Gor Proof. (1) By Props 2.12(2) and 3.14(2), − depthR/(x) R/(x) = depthR R 1 dim R/(x) = dim R − 1 ∴ ⇔ depthR R = dim R depthR/(x) R/(x) = dim R/(x) (2) By Lem 2.11 and Prop 3.14(2), i ∼ i+1 ExtR/(x)(k, R/(x)) = ExtR (k, R) dim R/(x) = dim R − 1 { {

i ∼ k (i = dim R) i ∼ k (i = dim R/(x)) ∴ Ext (k, R) = ⇔ Ext (k, R/(x)) = R 0 (i ≠ dim R) R/(x) 0 (i ≠ dim R/(x)) ■

Ex 4.7. x − y a NZD on A ∼ 2 A/(x − y) = k[[X]]/(X ) is Gor ∴ A is Gor by Prop 4.6 Thm 4.8 (Characterization of CM modules over Gor local rings). R: Gor M: a f.g R-mod TFAE (1) M is CM ∗ (2) The following hold, where (−) = HomR(−,R) ∼ ∗∗ (i) M = M i ∀ (ii) ExtR(M,R) = 0 ( i > 0) i ∗ ∀ (iii) ExtR(M ,R) = 0 ( i > 0) Sketch. (Details: Appendix B) Induction on d = dim R (2) ⇒ (1): d = 0 Easy d > 0 ∃ x ∈ m a NZD on R ⇝ a NZD on M,M ∗ 0 → M −→x M → M/xM → 0 ⇝ M/xM satisfies (i)–(iii) Ind hyp ⇒ M/xM a CM R/(x)-mod ⇒ M a CM R-mod (1) ⇒ (2): COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 11

d = 0 Ind on ℓ = ℓ(M) {

∼ i ∼ k (i = 0) ℓ = 1 M = k, Ext (k, R) = ⇝ done R 0 (i > 0) ℓ > 1 ∃ 0 → N → M → k → 0, ℓ(N) = ℓ − 1 Ind hyp to N ⇝ done d > 0 ∃ x ∈ m a NZD on R,M ⇝ a NZD on M ∗ M/xM a CM R/(x)-mod Ind hyp ⇒ M/xM satisfies (i)–(iii) ⇝ M satisfies (i)–(iii) ■ Ex 4.9. A/(x) is a CM A-module { ∼ ∗ ∼ ∗∗ A/(x) = (A/(x)) = (A/(x)) i ∀ ExtR(A/(x),A) = 0 ( i > 0) Cor 4.10. R: Gor M: a CM R-mod (1) M ∗ is a CM R-mod (2) ∃ exact seq of f.g R-mod 0 → M → F → N → 0 s.t F : free, N: CM Proof. (1) Immediate from Thm 4.8 (2) ∃ 0 → K → P → M ∗ → 0: exact s.t P : f.g free ⇝ 0 → M ∗∗ → P ∗ → K∗ → Ext1(M ∗,R) exact ∗∗ ∼ 1 ∗ By Thm 4.8, M = M and Ext (M ,R) = 0 ∴ 0 → M → P ∗ → K∗ → 0 exact By Prop 2.4, depth K ⩾ inf{depth P, depth M ∗ + 1} = dim R as depth P = dim R = depth M ∗ ∴ K is CM ∴ K∗ is CM by (1) F := P ∗, N := K∗ ■ Ex 4.11. • A/(x), A/(y) are CM ∗ ∗ ∼ • (A/(x)) is CM as (A/(x)) = A/(x) • ∃ 0 → A/(x) → A → A/(y) → 0

Appendix A. Proof of Theorem 1.12 To prove Theorem 1.12, we establish three lemmas. Lemma A.1. Let M be a finitely generated R-module. Then there exists a filtration

0 = M0 ⊊ M1 ⊊ M2 ⊊ ··· ⊊ Mn = M ∼ of submodules of M such that Mi/Mi−1 = R/pi for some pi ∈ Spec R. 12 RYO TAKAHASHI

Proof. We have M0 := 0 ⊆ M. If M = 0, then taking n = 0, we are done. Hence let us assume M ≠ 0. By Corollary 1.8 we find an associated prime p1 of M. Proposition 1.6 implies that there is an injective homomorphism f1 : R/p1 → M. Set M1 = Im f1 ⊆ M. If M1 = M, then taking n = 1, we are done. So we assume M1 ≠ M. Then by Corollary 1.8 we find an associated prime p2 of M/M1, and by Proposition 1.6 there is an injection f2 : R/p2 → M/M1. Writing Im f2 = M2/M1, we get M1 ⊆ M2 ⊆ M. If M2 = M, then taking n = 2, we are done. Iterating this procedure, we obtain an ascending chain

0 = M0 ⊊ M1 ⊊ M2 ⊊ · · · ⊆ M ∼ of submodules of M such that for each i one has Mi/Mi−1 = Im fi = R/pi with pi ∈ Spec R. Since M is Noetherian, this procedure ends in finitely many steps. This means that there exists an integer n ⩾ 0 such that Mn = M. ■

f g Lemma A.2. Let 0 → L −→ M −→ N → 0 be an exact sequence of R-modules. Then Ass M ⊆ Ass L ∪ Ass N. Proof. Let p be an associated prime of M. Then p = ann(x) for some x ∈ M. If Rx ∩ Im f = 0, then it holds that ∼ ∼ ∼ R/p = Rx = Rx/(Rx ∩ Im f) = (Rx + Im f)/ Im f ⊆ Coker f = N, which shows that p is an associated prime of N by Proposition 1.6. Let us consider the case where Rx ∩ Im f ≠ 0. In this case, there exist elements a ∈ R and y ∈ L such that ax = f(y) ≠ 0. We claim that p = ann(y). Indeed, pick any element b ∈ p. Then bx = 0, and we have f(by) = bf(y) = b(ax) = a(bx) = a · 0 = 0. The injectivity of the map f implies by = 0, whence b ∈ ann(y). Conversely, pick any element c ∈ ann(y). Then cy = 0, and we have (ac)x = c(ax) = cf(y) = f(cy) = f(0) = 0, whence ac ∈ p. Since ax ≠ 0, the element a is not in the prime ideal p. Hence c must be in p. Now the claim follows, and p is an associated prime of L. Consequently, p belongs to the union Ass L ∪ Ass N, which shows the lemma. ■ Lemma A.3. For each prime ideal p of R one has

AssR(R/p) = {p}.

Proof. We have p = ann(1) ∈ AssR(R/p). Take any q ∈ AssR(R/p). Then q = ann(x) for some x ∈ R/p. It suffices to show q = p. Fix an element a ∈ R. If a ∈ p, then a · x = ax = 0 as ax ∈ p. Hence a ∈ q, and we get p ⊆ q. Conversely, if a ∈ q, then a · x = 0, which implies ax ∈ p. Since q is a prime ideal, q ≠ R. Hence x ≠ 0, that is, x∈ / p. Therefore a ∈ p, which shows q ⊆ p. ■ Remark A.4. (1) The filtration in Lemma A.1 can be regarded as a generalization of a composition series of a module of finite length. (2) The conclusion of Lemma A.2 can be improved as Ass L ⊆ Ass M ⊆ Ass L ∪ Ass N. COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 13

(3) Lemma A.3 especially says that the only associated prime of a Noetherian domain is the zero ideal. Now we can give a proof of Theorem 1.12. Proof of Theorem 1.12. Lemma A.1 yields a filtration

0 = M0 ⊊ M1 ⊊ M2 ⊊ ··· ⊊ Mn = M ∼ of submodules of M such that Mi/Mi−1 = R/pi for some pi ∈ Spec R. For each 1 ⩽ i ⩽ n there is an exact sequence

0 → Mi−1 → Mi → R/pi → 0. Applying Lemmas A.2 and A.3, we get

Ass Mi ⊆ Ass Mi−1 ∪ Ass R/pi = Ass Mi−1 ∪ {pi}. Thus we obtain

Ass M = Ass Mn ⊆ Ass Mn−1 ∪ {pn}

⊆ Ass Mn−2 ∪ {pn−1, pn} ⊆ · · ·

⊆ Ass M0 ∪ {p1,..., pn} = {p1,..., pn},

where the last equality holds since Ass M0 = ∅ by Corollary 1.8. Therefore Ass M is a finite set. ■

Appendix B. Proof of Theorem 4.8 To prove Theorem 4.8, we establish three lemmas. Lemma B.1. Let (R, m, k) be an Artinian local ring. For each finitely generated R-module M, there exists a filtration

0 = M0 ⊊ M1 ⊊ ··· ⊊ Mn = M ∼ of submodules of M such that Mi/Mi−1 = k for all 1 ⩽ i ⩽ n. Proof. This is a direct consequence of Lemma A.1 and Corollary 3.6. ■ Lemma B.2. Let f g 0 / L / M / N / 0

α β γ    f ′ g′ 0 / L′ / M ′ / N ′ / 0 be a commutative diagram of R-modules and R-homomorphisms with exact rows. If α and γ are isomorphisms, then so is β. Proof. This is easy to show by diagram chasing. ■ Lemma B.3. Let (R, m, k) be a local ring. Let f : M → N be a homomorphism of finitely generated R-modules. Let x ∈ m be an nonzerodivisor on N. If the induced homomorphism f : M/xM → N/xN is an isomorphism, then so is f. 14 RYO TAKAHASHI

Proof. There is an exact sequence

f 0 → K −→θ M −→ N −→π C → 0,

where K = Ker f, C = Coker f, and θ, π are natural maps. Applying − ⊗R R/(x) to this, we get an exact sequence

f M/xM −→ N/xN −→π C/xC → 0. Since f is an isomorphism, we have C/xC = 0. Proposition 3.2 implies C = 0. Thus there is a short exact sequence

f 0 → K −→θ M −→ N → 0. We claim that the induced map θ : K/xK → M/xM is injective. Indeed, let z ∈ K/xK be an element such that θ(z) = 0. Then z = 0 in M/xM. This means z ∈ xM, and z = xw for some w ∈ M. As z ∈ K = Ker f, there are equalities 0 = f(z) = f(xw) = xf(w) in N. Since x is a nonzerodivisor on N, we have f(w) = 0, which says w ∈ K. Therefore z = xw ∈ xK, and z = 0 in K/xK. The claim now follows. By this claim, the induced sequence

f 0 → K/xK −→θ M/xM −→ N/xN → 0 is exact, and we get K/xK = 0 as f is an isomorphism. Using Proposition 3.2 again, we obtain K = 0. Consequently, f is an isomorphism. ■

Remark B.4. (1) Lemma B.1 is included in the Jordan–H¨oldertheorem for modules of finite length; see [1, Chapter 6]. (2) Lemma B.2 is a special case of the five lemma; see [5, Proposition 2.72].

Now we can give a proof of Theorem 4.8.

Proof of Theorem 4.8. Put d := dim R = depth R. We may assume M ≠ 0. (2) ⇒ (1): We use induction on d. When d = 0, we have 0 ⩽ depth M ⩽ d = 0 by Corollary 3.9, whence depth M = d and M is Cohen–Macaulay. Let d ⩾ 1. We find a nonzerodivisor x ∈ m on R by Proposition 2.9. There is an exact sequence 0 → R −→x R → R/(x) → 0. ∗ ∼ ∗∗ Applying HomR(M, −) and HomR(M , −) to this and using the isomorphism M = M , we obtain exact sequences 0 → M ∗ −→x M ∗ and 0 → M −→x M, which say that x is a nonzerodivisor on M and M ∗. Hence there are exact sequences (B.4.1) 0 → M −→x M → M/xM → 0, (B.4.2) 0 → M ∗ −→x M ∗ → M ∗/xM ∗ → 0. COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 15

Applying (−)∗ to (B.4.1) makes an exact sequence: 0 −−−→ (M/xM)∗ −−−→ M ∗ −−−→x M ∗ −−−→ 1 −−−→ 1 −−−→x 1 ExtR(M/xM, R) ExtR(M,R) ExtR(M,R) −−−→ 2 −−−→ 2 −−−→x 2 ExtR(M/xM, R) ExtR(M,R) ExtR(M,R) −−−→ ··· Since Ext>0(M,R) = 0 and x is a nonzerodivisor on M ∗, we see that R { ∗ ∗ i ∼ i+1 ∼ M /xM (i = 0), Ext (M/xM, R/(x)) = Ext (M/xM, R) = R/(x) R 0 (i ≠ 0), where the first isomorphism follows from Lemma 2.11. An analogous argument using (B.4.2) shows {

i ∗ ∗ ∼ i+1 ∗ ∗ ∼ M/xM (i = 0), Ext (M /xM ,R/(x)) = Ext (M /xM ,R) = R/(x) R 0 (i ≠ 0). In particular, we have ∼ ∗ ∗ ∼ M/xM = HomR/(x)(M /xM ,R/(x)) = HomR/(x)(HomR/(x)(M/xM, R/(x)),R/(x)). Proposition 3.14(2) implies dim R/(x) = d − 1, so by the induction hypothesis M/xM is a Cohen–Macaulay R/(x)-module. Hence depthR/(x) M/xM = dim R/(x), and we have − − depthR M 1 = depthR/(x) M/xM = dim R/(x) = d 1,

where the first equality follows from Proposition 2.12(2). Therefore depthR M = d, that is, M is Cohen–Macaulay. (1) ⇒ (2): Again we use induction on d. Let us consider the case d = 0. (In this case, note that every finitely generated R-module is Cohen–Macaulay by the proof of (2) ⇒ (1) for d = 0.) As R is Gorenstein, we have {

i ∼ k (i = 0), Ext (k, R) = R 0 (i ≠ 0). Since R is Artinian by Corollary 3.6, it follows from Lemma B.1 that there is a filtration

0 = M0 ⊊ M1 ⊊ M2 ⊊ ··· ⊊ Mn = M ∼ of submodules of M such that Mi/Mi−1 = k for each 1 ⩽ i ⩽ n. Let us deduce (2) by induction on n. When n = 1, the module M is isomorphic to k. ∼ 0 ∗ Since k = ExtR(k, R) = HomR(k, R) = k , we have ∼ ∗∗ i ∗ ∼ i (B.4.3) k = k , ExtR(k ,R) = ExtR(k, R) = 0 for all i > 0.

Therefore (2) holds when n = 1. Next, let n ⩾ 2. Setting N = Mn−1, we have an exact sequence 0 → N → M → k → 0. We can apply the hypothesis of induction (on n) to N to see that ∼ ∗∗ >0 >0 ∗ (B.4.4) N = N , ExtR (N,R) = 0 = ExtR (N ,R). 16 RYO TAKAHASHI

Applying (−)∗ to the above short exact sequence gives rise to an exact sequence: 0 −−−→ k∗ −−−→ M ∗ −−−→ N ∗ −−−→ 1 −−−→ 1 −−−→ 1 ExtR(k, R) ExtR(M,R) ExtR(N,R) −−−→ 2 −−−→ 2 −−−→ 2 ExtR(k, R) ExtR(M,R) ExtR(N,R) −−−→ · · · This exact sequence together with (B.4.3) and (B.4.4) shows that there is an exact se- quence 0 → k∗ → M ∗ → N ∗ → 0, >0 − ∗ and ExtR (M,R) = 0. Applying ( ) to this short exact sequence, we get another exact sequence: 0 −−−→ N ∗∗ −−−→ M ∗∗ −−−→ k∗∗ −−−→ 1 ∗ −−−→ 1 ∗ −−−→ 1 ∗ ExtR(N ,R) ExtR(M ,R) ExtR(k ,R) −−−→ 2 ∗ −−−→ 2 ∗ −−−→ 2 ∗ ExtR(N ,R) ExtR(M ,R) ExtR(k ,R) −−−→ · · · From this exact sequence together with (B.4.3) and (B.4.4) again, we obtain an exact sequence 0 → N ∗∗ → M ∗∗ → k∗∗ → 0, >0 ∗ and ExtR (M ,R) = 0. There is a commutative diagram with natural vertical maps: 0 / N / M / k / 0

   0 / N ∗∗ / M ∗∗ / k∗∗ / 0 By (B.4.3), (B.4.4) again, the left and right vertical maps are isomorphisms. It follows ∼ ∗∗ from Lemma B.2 that M = M . Thus the proof of the case d = 0 is completed. Next, let us consider the case d ⩾ 1. Since depth R = depth M = d ⩾ 1, the maximal ideal m is not an associated prime of R or M by Proposition 2.6. Hence m is not contained in any prime ideal belonging to Ass R ∪ Ass M, and we can choose an element x ∈ m such that ∪ x ∈/ p p∈Ass R∪Ass M by Theorem 1.12 and Lemma 2.8. Corollary 1.10 says that x is a nonzerodivisor on R and M. Thus R/(x) is Gorenstein by Proposition 4.6(2), and − − depthR/(x) M/xM = depthR M 1 = d 1 = dim R/(x) by Propositions 2.12(2) and 3.14(2). Hence M/xM is a Cohen–Macaulay R/(x)-module. The induction hypothesis implies:   ∼ M/xM = HomR/(x)(HomR/(x)(M/xM, R/(x)),R/(x)), Exti (M/xM, R/(x)) = 0 for all i ≠ 0,  R/(x) i ̸ ExtR/(x)(HomR/(x)(M/xM, R/(x)),R/(x)) = 0 for all i = 0. COHEN–MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 17

Using Lemma 2.11, we get  ∼ 1 1 M/xM = ExtR(ExtR(M/xM, R),R), Exti+1(M/xM, R) = 0 for all i ≠ 0,  R i+1 1 ̸ ExtR (ExtR(M/xM, R),R) = 0 for all i = 0. From the exact sequence 0 → M −→x M → M/xM → 0 we get an exact sequence: 0 −−−→ (M/xM)∗ −−−→ M ∗ −−−→x M ∗ −−−→ 1 −−−→ 1 −−−→x 1 ExtR(M/xM, R) ExtR(M,R) ExtR(M,R) −−−→ 2 −−−→ 2 −−−→x 2 ExtR(M/xM, R) ExtR(M,R) ExtR(M,R) −−−→ ··· ∗ >0 We see from this that x is a nonzerodivisor on M , and that ExtR (M,R) = 0 by Propo- 1 ∼ ∗ ∗ sition 3.2, whence ExtR(M/xM, R) = M /xM . Therefore ∼ 1 ∗ ∗ =1̸ ∗ ∗ M/xM = ExtR(M /xM ,R), ExtR (M /xM ,R) = 0. A similar argument for the exact sequence 0 → M ∗ −→x M ∗ → M ∗/xM ∗ → 0 shows that ∗∗ >0 ∗ 1 ∗ ∗ ∼ x is a nonzerodivisor on M , that ExtR (M ,R) = 0 and that ExtR(M /xM ,R) = ∗∗ ∗∗ ∼ ∗∗ ∗∗ M /xM . We obtain an isomorphism M/xM = M /xM , and it follows from Lemma ∼ ∗∗ B.3 that M = M . Now the proof of the theorem is completed. ■ References

[1] M. F. Atiyah; I. G. Macdonald, Introduction to , Addison-Wesley Publish- ing Co., Reading, Mass.-London-Don Mills, Ont., 1969. [2] M. P. Brodmann; R. Y. Sharp, : an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cam- bridge, 1998. [3] W. Bruns; J. Herzog, Cohen-Macaulay rings, revised edition, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998. [4] H. Matsumura, theory, Translated from the Japanese by M. Reid, Second edition, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1989. [5] J. J. Rotman, An introduction to homological algebra, Second edition, Universitext, Springer, New York, 2009.

Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan E-mail address: [email protected] URL: http://www.math.nagoya-u.ac.jp/~takahashi/