# COHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9

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1 COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We want to show that m = 0. Then, let ϕ : U M and ψ : M N be the sequence maps. Note that by properties of exact sequences that ψ is surjective and ϕ is injective. Since xm = 0, ψ(xm) = xψ(m) = 0 = ψ(m) = 0, since x is regular in N. Thus, m Ker ψ = ϕ(u) so that m = ϕ(u) for some u U. Then, certainly xm = ϕ(xu) = 0, and by injectivity, we know that xu = 0. Since x is regular, we conclude that u = 0 so that m = ϕ(0) = 0, and x is regular in M. Suppose now that x = x 1,..., x n is weak U and N regular for all i < n. Then, merely apply the argument of the base case to the exact sequence induced: 0 U/(x 1,..., x n 1 )U M/(x 1,..., x n 1 )M N/(x 1,..., x n 1 )N 0 To show that x must be a weak M regular sequence as well. Now, suppose that x is weak U-regular and N-regular. Using Proposition 1.1.4, we have an induced exact sequence Date: September 3,

2 2 KELLER VANDEBOGERT 0 U/xU M/xM N/xN 0 Suppose for contradiction that M/xM = 0. Then, by properties of exactness, the induced map ψ is surjective and we conclude N/xN = 0, contradicting the fact that x is N-regular. Thus, x is also M-regular. 2. Problem (a). Suppose that x and x are both M-regular elements. Suppose (xx )m = 0 for some m M. Then, since x is regular, x m = 0, and since x is regular, we conclude m = 0. Thus, xx is M-regular, and the general case for weak M-regular sequences follows by induction. Now, assume that M/xM 0 and M/x M 0, and suppose for contradiction that M/xx M = 0. Then, since M = xx M and x M M, we multiply by x to see that M xm so that M/xM = 0 (and by commutativity we also see M/x M = 0), a clear contradiction. Thus, M/xx M 0 so that the general case follows by induction. (b). This result follows from part (a) almost immediately, where we note that we can apply this to any M-regular sequence and itself, merely identifying the ith element each time. More precisely, suppose x = x 1,..., x n is regular. Then, by induction and the result of (a), x e 1 1, x 2,..., x n is regular as well, with e 1 some integer 1. Now successively apply this for every other index 1 < i n to see that x e 1 1, x e 2 2,..., x e n 1 n 1, x en n is also M-regular. 3. Problem We want to prove the following: Suppose x is a weak-m R N sequence, and N is faithfully flat over R. Then x is a weak M-sequence.

3 COHEN-MACAULAY RINGS SELECTED EXERCISES 3 Let x be a regular M N element (as usual, the general case follows from induction). We have the homothety ψ : M N M N, m n x(m n), which is injective since x is regular. By definition, x(m n) = (xm) n so that ψ(m n) = (xm) n. By faithful flatness of N, the map x id N is injective iff x is injective (x : M M, m xm). But this is equivalent to saying that the element x is M-regular, so we are done. 4. Exercise Since M has a rank, say r, we have that M Q = Q r, Q denotes our field of fractions of R. We can choose a basis {e 1,..., e r } for the above vector space. Since M is finitely generated, we can also choose a generating set {x 1,..., x n }. Consider the inclusion M Q r. We can find a ij, b ij R such that x i = j can then consider the inclusion a ij b ij e j. Take b := i,j b ij. We M Re 1 b Re 2 b Re r b = R r The above is obviously a free module, and the inclusion is injective since M is torsion free. Therefore, M is isomorphic to its image, which is a submodule of the free module R r. Note also that R r clearly has the same rank, as R r Q = Q r, so the problem is solved. 5. Problem Since R is Noetherian and M, N are finitely generated, M is finitely presented. Choose a presentation R m R n M 0

5 COHEN-MACAULAY RINGS SELECTED EXERCISES 5 Now we can proceed to the problem. Assume M is torsionless and suppose that rm = 0 for 0 r R and m M. We want to prove that m = 0. Consider the map m (φ(m)) φ M. By definition of torsionless, the image of m is 0 iff m = 0 identically. Consider rm (φ(rm)) φ M = r(φ(m)) φ M. Since rm = 0, this point maps to 0. However, the above shows that r(φ(m)) φ M = 0. Since M is torsion free, we conclude that m = 0, so that M is torsion free as asserted. (b). First note that if M is a submodule of a finite free module R n, then it is obviously torsionless since (R n ) = R n, so that any free module is torsionless (and hence any submodule of a free module. Note that this improves the claim of part (a), as we now see that M is also torsionless. Now, suppose that M is torsionless. By the claim proven in part (a), M is finitely generated and we can choose a generating set {φ 1,..., φ n }. Then, we can consider the following map Θ : M R n, which takes Θ(m) = (φ i (m)) n i=1. It suffices to show that this map is injective, so that M is isomorphic to its image as a submodule of a free module. Suppose then that Θ(m) = 0. Then, φ i (m) = 0 for each generator. But then we see that φ(m) = 0 for all φ M. Since M is torsionless, m = 0, so that Θ is injective, as desired. (c). Note that since M is finitely generated, the claim of part (a) along with the remark in (b) gives that M is finitely generated and torsionless. Since M is reflexive, M = M, so M is torsionless and hence isomorphic to a submodule of a free module (by (b)). Taking F 0

6 6 KELLER VANDEBOGERT as the cokernel of the map M R n := F 1, we have the exact sequence 0 M F 1 F 0 0 So that M is a second syzygy. 7. Problem We have an exact sequence F G M 0 Dualizing yields 0 M G F D(M) 0 We have that M is finitely presented, so choose a presentation F 1 F 0 M 0, leading to a partial free resolution of D(M): F 1 F 0 G F D(M) 0 Dualizing our presentation of M, we get the exact sequence Leading to the sequence 0 M F 0 F 1 G M h M F 0 Which can be extended to the sequence F G F0 F1 Now, let us consider computing the cohomology in the above sequence. The cohomology at G is going to be the kernel of the map taking g g +Im φ h(g +Im φ) F0 modulothe image of φ, which is M. That is, g + Im φ Ker h, so that Ker(G F0 ) = {g : g + Im φ Ker h}. Hence, taking the quotient Ker(G F0 )/ Im φ gives precisely Ker h.

7 COHEN-MACAULAY RINGS SELECTED EXERCISES 7 Now consider the cohomology at F0. Im(G F0 ) is isomorphic to Im h viewed as a submodule of F0 (under the inclusion M F0 ). Also, by the exact sequence 0 M F0 F1, we have that Ker(F0 F1 ) = M. Now, to solve the problem, dualize our partial resolution for D(M). We then see that the above cohomology groups are precisely isomorphic to Ext 1 R(D(M), R) and Ext 2 R(D(M), R), respectively. 8. Exercise Exercise Exercise Problem Since R is given as a PID, it suffices to show the quotient K/R of the field of fractions by R is divisible. Note: K/R = {r/s K/R s 1} Hence, given r/s K/R, we can write this as r(1/s), r R, hence every element is divisible so that K/R is injective (and hence 0 K K/R 0 is an injective resolution). 12. Problem We proceed by proving that Hom R (R, k) is the injective hull of the residue field, which is indecomposable by Theorem Take S := Soc(Hom R (R, k)). since First note that injectivity is clear,

8 8 KELLER VANDEBOGERT Hom R (, Hom k (R, k)) = Hom R ( k R, k) = HomR (, k) Since k is a field, it is obvious that Hom R (, k) is exact, so that Hom R (R, k) is an injective module over R. Continuing, since R has finite k-dimension, Hom k (R, k) is an Artinian module, so that S is the intersection of all essential extensions of Hom R (R, k), however is also characterized as S = Hom R (R/m, Hom k (R, k) = Hom R (k k R, k) = Homk (k, k) = k From the set of all simple submodules of Hom k (R, k), we can choose a minimal element M. By definition of essential extension, M S 0 and hence we see that Hom k (R, k) is an injective essential extension of S, so that Hom k (R, k) = E R (S) = E R (k), which is nondecomposable, so we are done. 13. Problem Suppose there exists a nonzero finitely generated injective module E. We have that dim R (E) id R (E) = 0, so that E is an Artinian module (since E is also finitely generated) with Supp(E) = {m}. By 3.2.8, E = E(R/m), implying that E(R/m) must also be finitely generated. If T (E) denotes our Matlis dual (so that T ( ) = Hom(, E(k))), we see that T (E(k)) must be Artinian as an R-module. T (R) = E(k), and hence: However,

9 COHEN-MACAULAY RINGS SELECTED EXERCISES 9 T (E(k)) = T (T (R)) = R So that R is Artinian as an R-module, and hence as a ring. 14. Problem We proceed by induction on depth R M. Assume M 0 (else the statement holds trivially) and suppose first that depth R M = 0. Then, m Ass(M) so that we have an exact sequence 0 R/m x M L 0 Set r := id R N, and apply Hom R (, N) to the above. induced exact sequence: We have an Ext r R(M, N) Ext r R(R/m, N) Ext r+1 R (L, N) Obviously Ext r R(R/m, N) 0 and Ext r+1 R (L, N) = 0 by definition of injective dimension. Hence, Ext r R(M, N) 0, and Ext i R(M, N) = 0 for all i > r. More precisely, id R N = r = sup{i Ext i R(M, N) 0}. Since depth R M = 0, this completes the base case. Inductive Step: Suppose depth R M > 0, so we can choose some M- regular element x R, giving an exact sequence 0 M x M M/xM 0 Apply Hom R (, N) to the above: Ext i R(M/xM, N) Ext i R(M, N) x Ext i R(M, N) Ext i+1 R (M/xM, N)

10 10 KELLER VANDEBOGERT By the inductive hypothesis, we have that id R N = depth R (M/xM) + sup{i Ext i R(M/xM, N) 0}. Set t := sup{i Ext i R(M/xM, N) 0}, so that for i t the above exact sequence implies Ext t R(M/xM, N) Ext t R(M, N) x Ext t R(M, N) 0 is exact, so that Ext t R(M, N) = x Ext i R(M, N). By Nakayama s Lemma, we have that Ext t R(M, N) = 0, and likewise Ext i R(M, N) = 0 for all i t. We also have that Ext t 1 R (M, N) 0, since the above shows we have an exact sequence Ext t 1 R (M, N) Ext t R(M/xM, N) 0 so that the vanishing of Ext t 1 R (M, N) implies the vanishing of Extt R(M/xM, N), in contradiction with the definition of t. Thus, we deduce sup{i Ext R (M, N) 0} = t 1. Using this and the standard equality depth R M/xM = depth R M 1, we see: id R N = depth R (M/xM) + t = depth R M 1 + sup{i Ext R (M, N) 0} + 1 = depth R M + sup{i Ext R (M, N) 0} Using that id R N = depth R, the above implies depth R depth R M = sup{i Ext R (M, N) 0} which completes the proof. 15. Problem We have that R is Gorenstein, M is finitely generated. Assume first that pd M = n <. We can choose a minimal free resolution

11 COHEN-MACAULAY RINGS SELECTED EXERCISES 11 0 F n F 0 M 0 Since R is Gorenstein and each F i is free, id F i = id R <. By the above resolution it is also clear that id M id F i + i <, and hence id M <, as desired. Conversely, suppose that id M <. We proceed by induction on the depth of R. When depth R = 0, we have that R is injective (id R = depth R). Since M is finitely generated with dimension 0, it is finitely presented so that there exists an exact sequence 0 R n R m M 0 for positive integer m, n. Since R is injective, so is R n, and hence the above sequence splits so that R m = R n M. Then M is the direct summand of a free module, hence projective with pd M = 0 <. Inductive Step: Suppose depth R = n, and choose some R-regular element x. As M is finitely generated, we have an exact sequence 0 K F M 0 where K denotes the kernel of the surjective map F M, and F is free. It suffices to show that pd K <, since any projective resolution for K can be extended to a resolution for M. Since F is free, x is also F regular and hence K-regular. We have the equality: id R/xR K/xK = id R K 1 Since R/xR is also Gorenstein, we can employ the inductive hypothesis to deduce pd R/xR K/xK <. However, pd R/xR K/xK = pd R K,

12 12 KELLER VANDEBOGERT hence K has finite projective dimension, so that pd M < as well, completing the proof. 16. Problem Suppose that id k <. Then, note: id k = sup{i Ext i R(k, k) 0} = pd k Hence pd k < as well. By the Auslander-Buchsbaum-Serre Theorem, we conclude that R is regular. 17. Problem (a). Let E R (k) := E (E R (k) denotes the injective hull of the residue field k). Note that the natural homomorphism ϕ : E E R R explicitly takes x x 1, where 1 = {1 + m t } t. We first show surjectivity. Let x {a t + m t } t E R R. For some positive integer n, we have that m n x = 0. We then deduce that m n R annihilates x 1, since m n R(x 1) = (m n x) ( R) = 0 R = 0 Consider then {a n + m t } t, where a n is of the residue class of m n in {a t + m t } t. Claim: x {a n + m t } t = x {a t + m t } t The above is equivalent to showing {a n a t + m t } t (x 1) = 0. We have two cases: t > n and t n. If t n, then by definition of the inverse limit we have that a n a t + m t = 0. Now, if t > n, then a n a t m n. Since m n annihilates x, the above claim follows.

13 COHEN-MACAULAY RINGS SELECTED EXERCISES 13 Finally, using the above, we have: ϕ(a n x) = (a n x) {1 + m t } t = x {a n + m t } t = x {a t + m t } t (by claim) So that ϕ is surjective. It remains to show injectivity. This is much easier, however, as R is faithfully flat over R. Since the identity is clearly injective, we have that 1 1 is injective by flatness. Then ϕ is an isomorphism, so that E = E R R. (b). Let F denote the injective hull of E over R. Then, we want to show that in fact F = E, giving that F = E R(k) = E. Note first that F is also the injective hull (over R) of the residue field k. We then have that given x F there exists a positive integer n such that m n Rx = 0. Since k = R/m = R/m R, we have the following commutative diagram (rows exact): 0 k = k E = E R R E R( k) Since F = E R(k) is an essential extension, φ is a monomorphism, giving us an exact sequence: 0 E F F/ Ker φ 0 By injectivity of E, we have that F = E E/ Ker φ. However, E and E/ Ker φ both have an R-module structure by part (a), hence we conclude that E/ Ker φ = 0 by indecomposability of F. Thus E = E R(k), as desired. φ

14 14 KELLER VANDEBOGERT (c). Note first that since N is finitely generated, N = N R R. We have the following: Hom R( N, E) = Hom R(N R R, E) = Hom R (N, Hom R( R, E)) = Hom R (N, E) Where we ve employed Hom-Tensor adjointness and that fact that since E = E R(k) by part (b), Matlis Duality yields Hom R( R, E) = T ( R) = E. 18. Problem Assume that (R, m, k) is an Artinian local ring. We proceed with the proof: (a) = (b): Suppose R is Gorenstein. R is Artinian, hence id R = dim R = 0, so that R is injective. Let M be a finitely generated R- module. M must also be injective, and as R is Artinian, we have an exact sequence 0 R m R n M 0 Applying Hom R (, R) twice, we note that our sequence remains exact by injectivity and also that R n and R m are reflexive since they are free. Thus the resulting exact sequence becomes 0 R m R n Hom R (Hom R (M, R), R) 0 and we immediately see that M must be reflexive. (b) = (c): Suppose that every finitely generated R module is reflexive. Note that Ann I = Hom R (R/I, R), so that Hom R (Ann I, R) = R/I by reflexivity. Since Ann R/I = I, we then see that I = Ann(Hom R (Ann I, R)).

15 COHEN-MACAULAY RINGS SELECTED EXERCISES 15 Hence, let x I. Then x Hom R (Ann I, R) = 0. Thus, if we merely choose the inclusion map i : Ann I R, the above shows that xi 0 so that x Ann Ann I. Therefore we see I Ann Ann I. To show the reverse inclusion, let x Ann Ann I. x Ann I = 0 giving: By definition, x Ann I = 0 = x Hom(Ann I, R) = 0 = x Ann(Hom R (Ann I, R) = x I Thus we conclude Ann Ann I = I. (c) = (d): Assume we are given nonzero ideal I, J R. We see: I J = Ann Ann I Ann Ann J = Ann(Ann I + Ann J) Hence I J = 0 if and only if Ann I + Ann J = R. Since I, J 0, however, Ann I and Ann J are contained in the maximal ideal m. This immediately gives that Ann I + Ann J m, and hence by the above I J 0. (d) = (a): We only need show that R is Cohen-Macaulay and of type r(r) = 1. Since R is Artinian, we trivially have that dim R = depth R = 0, so it remains to prove that r(r) = 1. Recall that type is defined as the vector space dimension of Ext depth R R (k, R). As depth R = 0, we want to find dim k Hom R (k, R). We recognize Hom R (k, R) as the socle (0 : m) R. Using the fact that R is Artinian gives that m n = 0 for some integer n, and hence (0 : m) R is nontrivial = dim k Hom R (k, R) 1. It remains to prove the reverse inequality.

16 16 KELLER VANDEBOGERT Choose nonzero x, y (0 : m) R. We have that Rx 0 Ry, so that by assumption Rx Ry 0. Hence we can find elements r, s R such that rx = sy 0. Note that r, s / m, since else rx = sy = 0 by definition of (0 : m) R. This says that r and s are units and hence x = r 1 sy. Taking the above modulo m, we find that x and y are linearly dependent in the residue field. Since x and y are arbitrary, this says that any two elements are linearly dependent, so that dim k (0 : m) R 1. We conclude that dim k (0 : m) R = dim k Hom R (k, R) = 1 identically, so that R is Cohen-Macaulay of type 1, therefore R is Gorenstein. 19. Problem This problem is merely definition checking. (a). This is obviously a group as it behaves the same as a direct sum. For the ring properties, ((r, m)(s, n))(t, p) = (rs, sm + rn)(t, p) = (rst, tsm + trn + rsp) = (r, m)(st, sp + tn) = (r, m)((s, n)(t, p)) proves associativity and our identity is obviously (1, 0). (b). Given r R, send r (r, 0). This is clearly bijective, so it remains to show it is a homomorphism. Note that φ(rs) = (rs, 0) = (r, 0)(s, 0) = φ(r)φ(s), so R = R 0.

17 COHEN-MACAULAY RINGS SELECTED EXERCISES 17 (c). Given (0, m) 0 M, we see that (r, n)(0, m) = (0, rm) 0 M, so this is an ideal. Given arbitrary (0, m) and (0, n) 0 M, we see that (0, n)(0, m) = (0, 0) so that (0 M) 2 = 0. (d). Suppose we have an ideal (I, N) R M, where I R and N M. We want to deduce properties of I and N in order for the pair to be an ideal. We see: (R, M)(I, N) = (RI, RN + IM) In order for the above to remain a subset of (I, N), we require that RI I, so that I is an ideal, and that N = M. Hence, all ideals of R M are of the form I M for some ideal I R. If R is local, we deduce immediately that R M is local with maximal ideal m M. (e). By the characterization given in part (d), assume there exists a chain of ideals I 0 M I 1 M.... This corresponds directly to a chain of ideals I 0 I 1... in R. If R is Noetherian, this chain stabilizes, and hence R M is Noetherian. Choosing a maximal descending chain of prime ideals, this also corresponds to a maximal chain of prime ideals in R M. Immediately we deduce dim R = dim R M. 20. Problem Suppose that C/xC is a canonical module for R/x), where x denotes a regular sequence of length n. Since C is assumed maximal Cohen- Macaulay, we only need prove that id R C < and r(c) = 1.

18 18 KELLER VANDEBOGERT R is Cohen-Macaulay, so that dim R = depth R = id R C. We also have, id R/(x) C/xC = depth R/(x) C/xC = depth R C n = id R C n Since id R/(x) C/xC <, we conclude that id R C < as well. Finally, to show r(c) = 1, we have the isomorphism (where d := depth R) Ext d R(k, C) = Ext d n (k, C/xC) R We know that r(c/xc) = 1, that is, rank k Ext d n R (k, C/xC) = 1, where we ve used depth R/x R/x = depth R n. Then the above isomorphism gives that r(c) = 1 as well, completing the proof. 21. Problem We proceed by induction on the projective dimension of the R- module M. For the base case, set pd M = 0. As R is local, M is free, and also Tor R i (k, M) = 0 for all i > 0. Also, R is Gorenstein so that in particular it is Cohen-Macaulay, giving d := dim R = depth R. Hence, by definition of depth, Ext d i R (k, M) = 0 whenever i > 0, giving the stated equality when i > 0. Now, when i = 0, Tor R i (k, M) = k M. To prove this case, we will show Tor R 0 (k, M) = k µ(m) = Ext 0 R(k, M). The first equality is trivial, as k M = k R oplusµ(m) = k µ(m)

19 COHEN-MACAULAY RINGS SELECTED EXERCISES 19 For the second equality, we have Ext d R(k, M) = Ext d R(k, R µ(m) ) = (Ext d R(k, R)) µ(m) = k µ(m) Where the final equality follows from R being Gorenstein. Since r(r) = 1, we must have that Ext d R(k, R) = k, as they are both vector spaces of dimension 1 over k. Inductive Step: Suppose now that pd M = n. We chose a free resolution 0 F n F 0 M 0 and extract a short exact sequence 0 N F 0 M 0 Applying both k and Hom R (k, ) to the above, we have the induced sequences Tor R i (k, F 0 ) Tor R i (k, M) Tor R i+1(k, M) Tor R i+1(k, F 0 ) Ext i R(k, F 0 ) Ext i R(k, M) Ext i+1 R (k, M) Exti+1 R (k, F 0) Since F 0 is free, in particular it is flat, so that Ext i R(k, M) = Tor R i (k, M) = 0 when i > 0. Therefore the above sequences give the isomorphisms Tor R i (k, M) = Tor R i+1(k, N) Ext i R(k, M) = Ext i+1 R (k, N) Since pd N = n 1, we apply the inductive hypothesis to see Tor R i (k, M) = Tor R i+1(k, N) = Ext d i 1 R (k, N) = Ext d i R (k, M) So that Tor R i (k, M) = Ext d i R (k, M) when i > 0. For the case i = 0, first note that the exact same reasoning as in the base case gives that

20 20 KELLER VANDEBOGERT Ext d R(k, F 0 ) = k F 0, so we have the commutative diagram k N k F 0 k M Ext d i (k, N) Ext d i (k, F 0 ) Ext d i (k, M) The first vertical arrow is an isomorphism by the inductive hypothesis, and the second is an isomorphism by the above. Hence, the third vertical must also be an isomorphism, that is, Tor R i (k, M) = Ext d i R (k, M) when i = 0 as well, completing the proof. 22. Problem By the given assumptions, we see that there exists an epimorphism ω R R, inducing an exact sequence 0 K ω R R 0 As R is free, the above sequence splits, that is, ω R = R K. Since, by definition, ω R has finite injective dimension, we conclude that id R R < as well, so that R is Gorenstein. 23. Problem (a). We proceed by induction on the dimension of M (= dim R, since M is maximal Cohen-Macaulay). When dim M = 0, M is injective and dim R = dim ω R = id ω R = 0. Thus, ω R is injective as well, and by of the book, it is also of rank 1. Therefore ω R = E(R/m), E( ) denoting the injective hull (over R). However, as M is also injective, we have which completes the base case. M = E(R/m) = ω R

21 COHEN-MACAULAY RINGS SELECTED EXERCISES 21 Inductive Step: Set dim M = n > 0. Then, obviously depth R > 0, so we can choose some M-regular element x, inducing a short exact sequence 0 M M M/xM 0 As depth M/xM = n 1, we have that M/xM = ω R/(x) inductive hypothesis. Using in the book, by the ω R/(x) = ω R /xω R And, as any regular element of M is also regular in ω R, we have M/xM = ω R /xω R = ( ω R )/x( ω R ) and hence we deduce that M = ω R, which completes the proof. (b). Firstly, note that if there exists a finite ω R resolution of M, then, as each ω R has finite injective dimension, we deduce that id M < as well. To show the converse, we use the hint. Suppose id M <. Note that any maximal Cohen-Macaulay module is isomorphic to a direct sum of canonical modules ω R. That is, we can find a Cohen-Macaulay approximation: 0 Y 1 ω r 0 R M 0 Then, as Y 1 has finite injective dimension, we can use part (a) to find another Cohen-Macaulay approximation 0 Y 2 ω r 1 R Y 1

22 22 KELLER VANDEBOGERT We continue in this fashion: 0 0 Y 1 ω r 2 R ω r 1 R ω r 0 R M 0 0 Y 2 0 We continue this process, adding to the above diagram by taking successive Cohen-Macaulay approximations of each Y i. Since M has finite injective dimension, this process must terminate past some integer k, giving the desired resolution. (c).

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