Definition 10.103.1. Let R be a Noetherian local ring. Let M be a finite R-module. We say M is Cohen-Macaulay if \dim (\text{Supp}(M)) = \text{depth}(M).
10.103 Cohen-Macaulay modules
Here we show that Cohen-Macaulay modules have good properties. We postpone using Ext groups to establish the connection with duality and so on.
A first goal will be to establish Proposition 10.103.4. We do this by a (perhaps nonstandard) sequence of elementary lemmas involving almost none of the earlier results on depth. Let us introduce some notation.
Let R be a local Noetherian ring. Let M be a Cohen-Macaulay module, and let f_1, \ldots , f_ d be an M-regular sequence with d = \dim (\text{Supp}(M)). We say that g \in \mathfrak m is good with respect to (M, f_1, \ldots , f_ d) if for all i = 0, 1, \ldots , d-1 we have \dim (\text{Supp}(M) \cap V(g, f_1, \ldots , f_ i)) = d - i - 1. This is equivalent to the condition that \dim (\text{Supp}(M/(f_1, \ldots , f_ i)M) \cap V(g)) = d - i - 1 for i = 0, 1, \ldots , d - 1.
Lemma 10.103.2. Notation and assumptions as above. If g is good with respect to (M, f_1, \ldots , f_ d), then (a) g is a nonzerodivisor on M, and (b) M/gM is Cohen-Macaulay with maximal regular sequence f_1, \ldots , f_{d - 1}.
Proof. We prove the lemma by induction on d. If d = 0, then M is finite and there is no case to which the lemma applies. If d = 1, then we have to show that g : M \to M is injective. The kernel K has support \{ \mathfrak m\} because by assumption \dim \text{Supp}(M) \cap V(g) = 0. Hence K has finite length. Hence f_1 : K \to K injective implies the length of the image is the length of K, and hence f_1 K = K, which by Nakayama's Lemma 10.20.1 implies K = 0. Also, \dim \text{Supp}(M/gM) = 0 and so M/gM is Cohen-Macaulay of depth 0.
Assume d > 1. Observe that g is good for (M/f_1M, f_2, \ldots , f_ d), as is easily seen from the definition. By induction, we have that (a) g is a nonzerodivisor on M/f_1M and (b) M/(g, f_1)M is Cohen-Macaulay with maximal regular sequence f_2, \ldots , f_{d - 1}. By Lemma 10.68.4 we see that g, f_1 is an M-regular sequence. Hence g is a nonzerodivisor on M and f_1, \ldots , f_{d - 1} is an M/gM-regular sequence. \square
Lemma 10.103.3. Let R be a Noetherian local ring. Let M be a Cohen-Macaulay module over R. Suppose g \in \mathfrak m is such that \dim (\text{Supp}(M) \cap V(g)) = \dim (\text{Supp}(M)) - 1. Then (a) g is a nonzerodivisor on M, and (b) M/gM is Cohen-Macaulay of depth one less.
Proof. Choose a M-regular sequence f_1, \ldots , f_ d with d = \dim (\text{Supp}(M)). If g is good with respect to (M, f_1, \ldots , f_ d) we win by Lemma 10.103.2. In particular the lemma holds if d = 1. (The case d = 0 does not occur.) Assume d > 1. Choose an element h \in R such that (i) h is good with respect to (M, f_1, \ldots , f_ d), and (ii) \dim (\text{Supp}(M) \cap V(h, g)) = d - 2. To see h exists, let \{ \mathfrak q_ j\} be the (finite) set of minimal primes of the closed sets \text{Supp}(M), \text{Supp}(M)\cap V(f_1, \ldots , f_ i), i = 1, \ldots , d - 1, and \text{Supp}(M) \cap V(g). None of these \mathfrak q_ j is equal to \mathfrak m and hence we may find h \in \mathfrak m, h \not\in \mathfrak q_ j by Lemma 10.15.2. It is clear that h satisfies (i) and (ii). From Lemma 10.103.2 we conclude that M/hM is Cohen-Macaulay. By (ii) we see that the pair (M/hM, g) satisfies the induction hypothesis. Hence M/(h, g)M is Cohen-Macaulay and g : M/hM \to M/hM is injective. By Lemma 10.68.4 we see that g : M \to M and h : M/gM \to M/gM are injective. Combined with the fact that M/(g, h)M is Cohen-Macaulay this finishes the proof. \square
Proposition 10.103.4. Let R be a Noetherian local ring, with maximal ideal \mathfrak m. Let M be a Cohen-Macaulay module over R whose support has dimension d. Suppose that g_1, \ldots , g_ c are elements of \mathfrak m such that \dim (\text{Supp}(M/(g_1, \ldots , g_ c)M)) = d - c. Then g_1, \ldots , g_ c is an M-regular sequence, and can be extended to a maximal M-regular sequence.
Proof. Let Z = \text{Supp}(M) \subset \mathop{\mathrm{Spec}}(R). By Lemma 10.60.13 in the chain Z \supset Z \cap V(g_1) \supset \ldots \supset Z \cap V(g_1, \ldots , g_ c) each step decreases the dimension at most by 1. Hence by assumption each step decreases the dimension by exactly 1 each time. Thus we may successively apply Lemma 10.103.3 to the modules M/(g_1, \ldots , g_ i) and the element g_{i + 1}.
To extend g_1, \ldots , g_ c by one element if c < d we simply choose an element g_{c + 1} \in \mathfrak m which is not in any of the finitely many minimal primes of Z \cap V(g_1, \ldots , g_ c), using Lemma 10.15.2. \square
Having proved Proposition 10.103.4 we continue the development of standard theory.
Lemma 10.103.5. Let R be a Noetherian local ring with maximal ideal \mathfrak m. Let M be a finite R-module. Let x \in \mathfrak m be a nonzerodivisor on M. Then M is Cohen-Macaulay if and only if M/xM is Cohen-Macaulay.
Proof. By Lemma 10.72.7 we have \text{depth}(M/xM) = \text{depth}(M)-1. By Lemma 10.63.10 we have \dim (\text{Supp}(M/xM)) = \dim (\text{Supp}(M)) - 1. \square
Lemma 10.103.6. Let R \to S be a surjective homomorphism of Noetherian local rings. Let N be a finite S-module. Then N is Cohen-Macaulay as an S-module if and only if N is Cohen-Macaulay as an R-module.
Proof. Omitted. \square
Lemma 10.103.7.reference Let R be a Noetherian local ring. Let M be a finite Cohen-Macaulay R-module. If \mathfrak p \in \text{Ass}(M), then \dim (R/\mathfrak p) = \dim (\text{Supp}(M)) and \mathfrak p is a minimal prime in the support of M. In particular, M has no embedded associated primes.
Proof. By Lemma 10.72.9 we have \text{depth}(M) \leq \dim (R/\mathfrak p). Of course \dim (R/\mathfrak p) \leq \dim (\text{Supp}(M)) as \mathfrak p \in \text{Supp}(M) (Lemma 10.63.2). Thus we have equality in both inequalities as M is Cohen-Macaulay. Then \mathfrak p must be minimal in \text{Supp}(M) otherwise we would have \dim (R/\mathfrak p) < \dim (\text{Supp}(M)). Finally, minimal primes in the support of M are equal to the minimal elements of \text{Ass}(M) (Proposition 10.63.6) hence M has no embedded associated primes (Definition 10.67.1). \square
Definition 10.103.8. Let R be a Noetherian local ring. A finite module M over R is called a maximal Cohen-Macaulay module if \text{depth}(M) = \dim (R).
In other words, a maximal Cohen-Macaulay module over a Noetherian local ring is a finite module with the largest possible depth over that ring. Equivalently, a maximal Cohen-Macaulay module over a Noetherian local ring R is a Cohen-Macaulay module of dimension equal to the dimension of the ring. In particular, if M is a Cohen-Macaulay R-module with \mathop{\mathrm{Spec}}(R) = \text{Supp}(M), then M is maximal Cohen-Macaulay. Thus the following two lemmas are on maximal Cohen-Macaulay modules.
Lemma 10.103.9.slogan Let R be a Noetherian local ring. Assume there exists a Cohen-Macaulay module M with \mathop{\mathrm{Spec}}(R) = \text{Supp}(M). Then any maximal chain of prime ideals \mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_ n has length n = \dim (R).
Proof. We will prove this by induction on \dim (R). If \dim (R) = 0, then the statement is clear. Assume \dim (R) > 0. Then n > 0. Choose an element x \in \mathfrak p_1, with x not in any of the minimal primes of R, and in particular x \not\in \mathfrak p_0. (See Lemma 10.15.2.) Then \dim (R/xR) = \dim (R) - 1 by Lemma 10.60.13. The module M/xM is Cohen-Macaulay over R/xR by Proposition 10.103.4 and Lemma 10.103.6. The support of M/xM is \mathop{\mathrm{Spec}}(R/xR) by Lemma 10.40.9. After replacing x by x^ n for some n, we may assume that \mathfrak p_1 is an associated prime of M/xM, see Lemma 10.72.8. By Lemma 10.103.7 we conclude that \mathfrak p_1/(x) is a minimal prime of R/xR. It follows that the chain \mathfrak p_1/(x) \subset \ldots \subset \mathfrak p_ n/(x) is a maximal chain of primes in R/xR. By induction we find that this chain has length \dim (R/xR) = \dim (R) - 1 as desired. \square
Lemma 10.103.10. Suppose R is a Noetherian local ring. Assume there exists a Cohen-Macaulay module M with \mathop{\mathrm{Spec}}(R) = \text{Supp}(M). Then for a prime \mathfrak p \subset R we have
Proof. Follows immediately from Lemma 10.103.9. \square
Lemma 10.103.11. Suppose R is a Noetherian local ring. Let M be a Cohen-Macaulay module over R. For any prime \mathfrak p \subset R the module M_{\mathfrak p} is Cohen-Macaulay over R_\mathfrak p.
Proof. We may and do assume \mathfrak p \not= \mathfrak m and M not zero. Choose a maximal chain of primes \mathfrak p = \mathfrak p_ c \subset \mathfrak p_{c - 1} \subset \ldots \subset \mathfrak p_1 \subset \mathfrak m. If we prove the result for M_{\mathfrak p_1} over R_{\mathfrak p_1}, then the lemma will follow by induction on c. Thus we may assume that there is no prime strictly between \mathfrak p and \mathfrak m. Note that \dim (\text{Supp}(M_\mathfrak p)) \leq \dim (\text{Supp}(M)) - 1 because any chain of primes in the support of M_\mathfrak p can be extended by one more prime (namely \mathfrak m) in the support of M. On the other hand, we have \text{depth}(M_\mathfrak p) \geq \text{depth}(M) - \dim (R/\mathfrak p) = \text{depth}(M) - 1 by Lemma 10.72.10 and our choice of \mathfrak p. Thus \text{depth}(M_\mathfrak p) \geq \dim (\text{Supp}(M_\mathfrak p)) as desired (the other inequality is Lemma 10.72.3). \square
Definition 10.103.12. Let R be a Noetherian ring. Let M be a finite R-module. We say M is Cohen-Macaulay if M_\mathfrak p is a Cohen-Macaulay module over R_\mathfrak p for all primes \mathfrak p of R.
By Lemma 10.103.11 it suffices to check this in the maximal ideals of R.
Lemma 10.103.13. Let R be a Noetherian ring. Let M be a Cohen-Macaulay module over R. Then M \otimes _ R R[x_1, \ldots , x_ n] is a Cohen-Macaulay module over R[x_1, \ldots , x_ n].
Proof. By induction on the number of variables it suffices to prove this for M[x] = M \otimes _ R R[x] over R[x]. Let \mathfrak m \subset R[x] be a maximal ideal, and let \mathfrak p = R \cap \mathfrak m. Let f_1, \ldots , f_ d be a M_\mathfrak p-regular sequence in the maximal ideal of R_{\mathfrak p} of length d = \dim (\text{Supp}(M_{\mathfrak p})). Note that since R[x] is flat over R the localization R[x]_{\mathfrak m} is flat over R_{\mathfrak p}. Hence, by Lemma 10.68.5, the sequence f_1, \ldots , f_ d is a M[x]_{\mathfrak m}-regular sequence of length d in R[x]_{\mathfrak m}. The quotient
has support equal to the primes lying over \mathfrak p because R_\mathfrak p \to R[x]_\mathfrak m is flat and the support of M_{\mathfrak p}/(f_1, \ldots , f_ d)M_{\mathfrak p} is equal to \{ \mathfrak p\} (details omitted; hint: follows from Lemmas 10.40.4 and 10.40.5). Hence the dimension is 1. To finish the proof it suffices to find an f \in \mathfrak m which is a nonzerodivisor on Q. Since \mathfrak m is a maximal ideal, the field extension \kappa (\mathfrak m)/\kappa (\mathfrak p) is finite (Theorem 10.34.1). Hence we can find f \in \mathfrak m which viewed as a polynomial in x has leading coefficient not in \mathfrak p. Such an f acts as a nonzerodivisor on
and hence acts as a nonzerodivisor on Q. \square
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