COHEN{MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS RYO TAKAHASHI Contents Introduction 1 1. Associated primes 1 2. Depth 3 3. Krull dimension 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 Noetherian ring 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 prime ideal if (ab 2 I ) a 2 I or b 2 I) (3) Spec R := fPrime ideals of Rg Ex 1.2. Spec A = f(x); (y); (x; y)g; Spec B = f(x); (x; y)g Prop 1.3. I ⊆ R an ideal (1) I is prime iff R=I is an integral domain (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 2 Spec R is an associated prime of M if 9 x 2 M s.t p = ann(x) AssR M := fAssociated primes of Mg Ex 1.5. AssA A = f(x); (y)g; AssB B = f(x); (x; y)g Prop 1.6. TFAE for p 2 Spec R (1) p 2 AssR M (2) 9 f : R=p ! M an injective homomorphism Proof. (1) ) (2) 9 x 2 M s.t p = ann(x) Define f : R=p ! M by f(a) = ax (2) ) (1) x := f(1) 2 M p = ann(x) ■ Prop 1.7. maxfann(x) j 0 =6 x 2 Mg ⊆ AssR M Proof. p := ann(x) ETS p 2 Spec R x =6 0 ) 1 2= p ) p ( R Suppose a; b 2 R, ab 2 p, a2 = p. Then abx = 0 ) p = ann(x) ( ann(bx)(a 2 ann(bx) − p) The maximality of p shows ann(bx) = R ) bx = 0 ) x 2 p ■ Cor 1.8. M =6 0 () AssR M =6 ; Proof. (() Trivial ()) As R is Noetherian, LHS in Prop 1.7 is nonempty. ■ Def 1.9. x 2 R, M an R-module (1) x is a zerodivisor (ZD) on M if 0 =6 9 m 2 M s.t xm = 0 (2) x is a nonzerodivisor (NZD) on M if x is not a ZD on M Cor 1.10. [ fZDs on Mg = p p2Ass M COHEN{MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 3 Proof. S ⊇ 2 ( ) Let a p2Ass M p ) a 2 9 p 2 Ass M ) 9 x 2 M s.t p = ann(x) ) ax = 0, x =6 0 ) a is a ZD on M (⊆) Let a 2 R be a ZD on M ) 0 =6 9 z 2 M s.t az = 0 ) a 2 ann(z) ann(z) ⊆ 9 q 2 maxfann(x) j 0 =6 x 2 Mg ⊆ Ass M by Prop 1.7 S ) 2 2 ) 2 ■ a q Ass M a p2Ass M p Ex 1.11. (1) fZDs on Ag = (x) [ (y) ) x − y is a NZD on A (2) fZDs on Bg = (x; y) ) All nonunits of B are ZDs on B Thm 1.12. If M is finitely generated, then # AssR M < 1 Sketch. (Details: Appendix A) (1) 9 0 = M0 ( M1 ( ··· ( Mn = M submodules s.t. ∼ Mi=Mi−1 = R=pi; pi 2 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) = fpg 8 p 2 Spec R 0 ! Mi−1 ! Mi ! R=pi ! 0 Ass Mi ⊆ Ass Mi−1 [ Ass R=pi = Ass Mi−1 [ fpig Ass M ⊆ fp1;:::; png ■ 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 local ring 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: f > j i 6 g 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 > inffdepth M; depth N + 1g 4 RYO TAKAHASHI (2) depth M > inffdepth L; depth Ng (3) depth N > inffdepth M; depth L − 1g Proof. (1) n := inffdepth M; depth N + 1g ) 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 > inffdepth M; depth N + 1g = depth N + 1 Assume depth L > depth N + 1. Then by Prop 2.4(3) depth N > inffdepth M; depth L − 1g > depth N This contradiction shows depth L = depth N + 1 ■ Prop 2.6. (R; m; k): local M: a f.g R-mod 2 () m AssR M depthR M = 0 Proof. ()) Prop 1.6 ) k = R=m ,! M ) HomR(k; M) =6 0 (() HomR(k; M) =6 0 ) 9 f : k ! M, f(1) = x =6 0 a =6 0 ) a 2= m ) a 2 R× ) ax =6 0 ) f is injective Prop 1.6 implies m 2 Ass M ■ 2 Ex 2.7. (1) depthA A > 0 since (x; y) = AssA A 2 (2) depthB B = 0 since (x; y) AssB B Lem 2.8 (Prime Avoidance). I ⊆ R an ideal p1;:::; pn 2 Spec R I ⊆ p1 [···[ pn =) I ⊆ p` (9 `) Proof. Induction on n n = 1 Trivial n > 2 May assume @ inclusion relation among p1;:::; pn Suppose I * pi (8 i) Ind hyp ) I * p1 [···[ pn−1 ) 9 x 2 I − (p1 [···[ pn−1) COHEN{MACAULAY MODULES OVER GORENSTEIN LOCAL RINGS 5 As I ⊆ (p1 [···[ pn−1) [ pn, we have x 2 pn 9 y 2 I − pn; 9 zi 2 pi − pn (1 6 8 i 6 n − 1); w := x + yz1 ··· zn−1 (1) x; y 2 I ) w 2 I ⊆ (p1 [···[ pn−1) [ pn (2) x2 = p1 [···[ pn−1; z1 2 p1 ) w2 = p1 [···[ pn−1 (3) x 2 pn; y; z1; : : : ; zn−1 2= pn ) w2 = 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) 9 x 2 m a NZD on M Proof. By Prop 2.6, depth M > 0 , m 2= Ass M (2) ) (1) S ) 2 ) 2 8 2 Cor 1.10 x = p2Ass M p x = p Ass M As x 2 m, we must have m 2= Ass M (1) ) (2) m 2= Ass M ) m * 8 p 2 Ass M Thm 1.12 implies # Ass M < 1 Lem 2.8 and Cor 1.10 yield [ m * p = fZDs on Mg p2Ass M ) 9 x 2 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 2 ann N: a NZD on R; M Then i+1 ∼ i 8 2 Z ExtR (N; M) = ExtR=(x)(N; M=xM)( i ) i i+1 − Proof. T := ExtR ( ;M) ETS: (1) 8 0 ! X ! Y ! Z ! 0 exact seq of R=(x)-modules 9 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 (8 P projective R-mod, 8 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 2 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 6 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 2 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.