MAXIMAL COHEN–MACAULAY APPROXIMATIONSAND SERRE’S CONDITIONHIROKI MATSUI AND RYO TAKAHASHI
Dedicated to Professor Ngo Viet Trung on the occasion of his sixtieth birthday
Abstract. This paper studies the relationship between Serre’s condition (Rn ) andAuslander–Buchweitz’s maximal Cohen–Macaulay approximations. It is proved thata Gorenstein local ring satisfies (Rn ) if and only if every maximal Cohen–Macaulay
module is a direct summand of a maximal Cohen–Macaulay approximation of a (Cohen–Macaulay) module of codimension n + 1.

1. Introduction
In the 1980s, Auslander and Buchweitz [2] introduced the notion of a maximal Cohen–
Macaulay approximation of a finitely generated module over a Cohen–Macaulay local ringwith a canonical module, which has been playing a fundamental role in the representation
theory of Cohen–Macaulay rings. Several years ago Kato [6] gave the following characterization theorem of Gorenstein local rings by maximal Cohen–Macaulay approximations.
We abbreviate Cohen–Macaulay to CM and maximal Cohen–Macaulay to MCM.
Theorem 1.1 (Kato). Let R be a d-dimensional Gorenstein local ring.
(1) The following are equivalent for d ≥ 1.
(a) R is a domain.
(b) Every MCM R-module is a MCM approximation of a (CM) R-module of codimension 1.
(2) The following are equivalent for d ≥ 2.
(a) R is a unique factorization domain.
(b) Every MCM R-module is a MCM approximation of a (CM) R-module of codimension 2.
It is natural to ask what happens if in the statements (b) of the above theorem weweaken the condition of being a MCM approximation to that of being a direct summandof a MCM approximation. The main purpose of this paper is to answer this question inmore general settings. Our main results yield the following theorem.
Theorem 1.2. Let R be a d-dimensional Gorenstein local ring. The following are equivalent for each 0 ≤ c ≤ d.
Date: September 4, 2014.
2010 Mathematics Subject Classification. 13C14, 13H10.
Key words and phrases. Cohen–Macaulay ring, Gorenstein ring, maximal Cohen–Macaulay module,maximal Cohen–Macaulay approximation, Serre’s condition, nonfree locus, singular locus, syzygy, transpose.
RT was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400038.
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HIROKI MATSUI AND RYO TAKAHASHI(1) R satisfies Serre’s condition (Rc−1 ).
(2) Every MCM R-module is a direct summand of a c-th syzygy of a (CM) R-module ofcodimension c.
(3) Every MCM R-module is a direct summand of a MCM approximation of a (CM)
R-module of codimension c.
Letting c = 1, 2 in the above theorem, we obtain the following result which is analogous
to Kato’s theorem. This gives the answer to the question raised above.
Corollary 1.3. Let R be a d-dimensional Gorenstein local ring.
(1) The following are equivalent for d ≥ 1.
(a) R is reduced.
(b) Every MCM R-module is a direct summand of a MCM approximation of a (CM)
R-module of codimension 1.
(2) The following are equivalent for d ≥ 2.
(a) R is normal.
(b) Every MCM R-module is a direct summand of a MCM approximation of a (CM)
R-module of codimension 2.
This paper is organized as follows. In Section 2, we consider over a CM local ring thecondition that all MCM modules are direct summands of syzygies of certain modules. InSection 3, we study over a Gorenstein local ring the condition that all MCM modules aredirect summands of MCM approximations of certain modules. The proof of Theorem 1.2
is given at the end of this section.
2. MCM modules that are direct summands of syzygies
Throughout this paper, let R be a commutative Cohen–Macaulay local ring of Krull
dimension d. All R-modules are assumed to be finitely generated.
Let us begin with recalling some basic definitions.
Definition 2.1. (1) For an integer n ≥ 0 we denote by Ωn M an n-th syzygy of M , thatis, the image of the n-th diﬀerential map in a free resolution of M .
(2) For an integer n ≥ −1 we say that R satisfies Serre’s condition (Rn ) if the local ringRp is regular for all prime ideals p of R with ht p ≤ n.
(3) The singular locus Sing R of R is by definition the set of prime ideals p of R such thatthe local ring Rp is nonregular.
(4) Let M be an R-module. The nonfree locus NF(M ) (respectively, the infinite projectivedimension locus IPD(M )) of M is defined as the set of prime ideals p of R such that theRp -module Mp is nonfree (respectively, is of infinite projective dimension).
(5) Let V be a closed subset of Spec R. Then we set codim V = d − dim V and call thisthe codimension of V . The codimension codim M of an R-module M is defined as thecodimension of Supp M , whence codim M = d − dim M .
Remark 2.2. (1) If X, Y are n-th syzygies of an R-module M , then X ⊕ F ∼= Y ⊕ G forsome free R-modules F, G.
(2) By definition R always satisfies (R−1 ).

MCM APPROXIMATIONS AND SERRE’S CONDITION

3(3) It is well-known and easy to see that the nonfree locus and the infinite projective
dimension locus of an R-module are always closed subsets of Spec R in the Zariski topology.
(4) If M is a MCM R-module, then NF(M ) is contained in Sing R.
In the following proposition we study how to represent each MCM module as a direct
summand of a syzygy of a certain CM module. This result will become a basis of ourmain results.
Proposition 2.3. Let M be a MCM R-module. Then for each integer 0 ≤ c ≤ codim NF(M )there exists a CM R-module N such that
(1) codim N = c,(2) IPD(N ) = NF(M ) and(3) M is isomorphic to a direct summand of a c-th syzygy of N .
Proof. By virtue of [5, Remark 5.2(1)], there exists an ideal I of R with NF(M ) = V(I)
such that I · ExtiR (M, X) = 0 for all integers i > 0 and all R-modules X. As
dim NF(M ) = dim R/I = d − ht I,

we have ht I = codim NF(M ) ≥ c, and can take an R-sequence x = x1 , . . . , xc in I. Setting
N = M/xM , we see from [7, Proposition 2.2] that M is isomorphic to a direct summandof Ωc N . The condition (3) is thus satisfied, and it is observed that N is a CM R-modulewith codim N = d − dim N = c.
Now it remains to verify that N satisfies the condition (2). Fix a prime ideal p in theunion IPD(N ) ∪ NF(M ). Then it is easily observed that p contains the sequence x. Henceby [3, Exercise 1.3.6] the equalities
pdRp Np = pdRp Mp /xMp = pdRp Mp + chold. This shows that the Rp -module Np has infinite projective dimension if and only if
so does Mp . Since M is a MCM R-module, the Auslander–Buchsbaum formula implies
IPD(M ) = NF(M ). Therefore we obtain IPD(N ) = NF(M ).
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As an immediate consequence of the above proposition, the following holds.
Corollary 2.4. Let M be a MCM R-module whose nonfree locus has dimension n. Then
there exists a CM R-module N of dimension n such that M is isomorphic to a direct
summand of Ωd−n N .
Proof. We have codim NF(M ) = d − n. Apply Proposition 2.3 to c := d − n.

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Applying the above corollary to n = 0, we obtain the following result, which recovers[7, Corollary 2.6].
Corollary 2.5. Let M be a MCM R-module which is locally free on the punctured spectrum of R. Then there exists an R-module N of finite length such that M is isomorphic
to a direct summand of Ωd N .
Next we establish a criterion for R to satisfy Serre’s condition (Rn ) in terms of thecodimensions of the nonfree loci of MCM R-modules.
Proposition 2.6. The following are equivalent for each 0 ≤ c ≤ d.

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HIROKI MATSUI AND RYO TAKAHASHI(1) The ring R satisfies (Rc−1 ).
(2) One has codim Sing R ≥ c.
(3) One has codim NF(M ) ≥ c for all MCM R-modules M .
Proof. (1) ⇒ (2): Let p be a prime ideal in Sing R. As R satisfies (Rc−1 ), the height of pis at least c, whence dim R/p ≤ d − c. Therefore dim Sing R ≤ d − c, which means thatSing R has codimension at least c.
(2) ⇒ (3): Since NF(M ) is contained in Sing R, we have dim NF(M ) ≤ dim Sing R.
Hence the (in)equalities
codim NF(M ) = d − dim NF(M ) ≥ d − dim Sing R = codim Sing R ≥ cfollow.
(3) ⇒ (1): Let p be a prime ideal of R with ht p ≤ c−1. Let M be a d-th syzygy of the Rmodule R/p. Then M is a MCM R-module, and by assumption we have codim NF(M ) ≥ c,or equivalently,
dim NF(M ) ≤ d − c.
Suppose that Rp is not regular. Then the Rp -module Mp is not free, for it is a d-th syzygyof the Rp -module κ(p). Hence p belongs to NF(M ), and there are inequalities
dim NF(M ) ≥ dim R/p ≥ d − c + 1.
This contradiction shows that Rp is regular.

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Let us now state and prove the main result of this section, which characterizes CM
local rings satisfying Serre’s (Rn )-condition.
Theorem 2.7. For every integer 0 ≤ c ≤ d the following are equivalent.

(1) The ring R satisfies (Rc−1 ).
(2) Every MCM R-module is isomorphic to a direct summand of a c-th syzygy of a CM
R-module of codimension c.
(3) Every MCM R-module is isomorphic to a direct summand of some syzygy of an Rmodule of codimension at least c.
Proof. Propositions 2.3 and 2.6 show that (1) implies (2), and it is obvious that (2) implies(3). Assume that (3) holds, and take any MCM R-module M . By assumption, there arean R-module N with codim N ≥ c and an integer b ≥ 0 such that M is isomorphic toa direct summand of Ωb N . Then we have inclusions NF(M ) ⊆ NF(Ωb N ) ⊆ Supp N ofclosed subsets of Spec R, which implies
dim NF(M ) ≤ dim NF(Ωb N ) ≤ dim Supp N = dim N ≤ d − c.
Hence NF(M ) has codimension at least c, and it is deduced from Proposition 2.6 that R
satisfies (Rc−1 ).
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3. MCM modules that are direct summands of MCM approximations
Throughout this section, our ring R is further assumed to be Gorenstein. The following
is a celebrated result due to Auslander and Buchweitz [2, Theorem 1.8].

MCM APPROXIMATIONS AND SERRE’S CONDITION

5Theorem 3.1 (Auslander–Buchweitz). For each R-module M there exists an exact sequence(3.1.1)0→Y →X→M →0of R-modules such that X is MCM and Y has finite projective dimension.
Definition 3.2. A MCM R-module X admitting an exact sequence of the form (3.1.1)
is called a MCM approximation of M .
For an R-module M we denote by TrM the (Auslander) transpose of M , that is, thecokernel of the R-dual of the first diﬀerential map in a free resolution of M . We denote byMCM(R) the stable category of MCM R-modules. This is defined as follows: the objectsof MCM(R) are precisely the MCM R-modules, and the hom-set HomMCM(R) (M, N ) ofobjects M, N in MCM(R) is the quotient of HomR (M, N ) by the R-submodule consisting ofhomomorphisms factoring through free R-modules. Since R is assumed to be Gorenstein,MCM(R) is a triangulated category, and taking a syzygy and a transpose defines anautoequivalence and a duality of MCM(R), respectively.
∼=Ω : MCM(R) −→ MCM(R),∼=Tr : MCM(R) −→ MCM(R)op .
For details, we refer the reader to [1] and [4].
One can describe a MCM approximation by using syzygies and transposes:
Lemma 3.3. For any R-module M , the R-moduleTrΩn TrΩn M
is a MCM approximation of M for all n ≥ d − depth M .
Proof. Note that Ωn M is a MCM R-module. Since both Ω and Tr preserve the MCM
property, the R-module X = TrΩn Tr(Ωn M ) is also a MCM module. It follows from [1,Proposition (2.21) and Corollary (4.22)] that there exists an exact sequence(3.3.1)

0→K→X→M →0of R-modules such that K has projective dimension at most n − 1. Consequently, X is a
MCM approximation of M .
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A MCM approximation version of Corollary 2.4 also holds true:
Proposition 3.4. (1) Let M be a MCM R-module with n-dimensional nonfree locus.
Then there exists an n-dimensional CM R-module N such that M is isomorphic to a
direct summand of a MCM approximation of N .
(2) Let M be a MCM R-module that is locally free on the punctured spectrum of R. Then
there exists an R-module N of finite length such that M is isomorphic to a direct
summand of a MCM approximation of N .
Proof. (1) It is easy to see that NF(Ωd−n M ) coincides with NF(M ). Applying Corollary
2.4 to the MCM module Ωd−n M , we find a CM module N of dimension n such that

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HIROKI MATSUI AND RYO TAKAHASHIΩd−n M is isomorphic to a direct summand of Ωd−n N . Taking TrΩd−n Tr yields that M isisomorphic to a direct summand ofX := TrΩd−n TrΩd−n N ⊕ F
for some free R-module F . Using Lemma 3.3, we easily see that X is a MCM approximation of N .
(2) The assertion follows from applying (1) to n = 0.
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The main result of this section is the following characterization of Gorenstein local rings
satisfying Serre’s condition (Rn ). This result can be viewed as a MCM approximation
version of Theorem 2.7.
Theorem 3.5. The following are equivalent for each 0 ≤ c ≤ d.

(1) R satisfies (Rc−1 ).
(2) Every MCM R-module is isomorphic to a direct summand of a MCM approximationof a CM R-module of codimension c.
(3) Every MCM R-module is isomorphic to a direct summand of a MCM approximationof an R-module of codimension at least c.
Proof. (1) ⇒ (2): Let M be a MCM R-module. Using Theorem 2.7 for the MCM Rmodule Ωc M , we get a CM R-module N of codimension c such that Ωc M is isomorphic toa direct summand of Ωc N . Then applying TrΩc Tr to this relation shows that TrΩc TrΩc Mis isomorphic to a direct summand of X := TrΩc TrΩc N up to free summands. By Lemma
3.3 the module X is a MCM approximation of N . Since we have a duality∼=TrΩc : MCM(R) −→ MCM(R),the R-module TrΩc TrΩc M is isomorphic to M up to free summands. Therefore M isisomorphic to a direct summand of X ⊕ F for some free R-module F . It is easy to seethat X ⊕ F is also a MCM approximation of N .
(2) ⇒ (3): This implication is obvious.
(3) ⇒ (1): Let M be a MCM R-module. Then N := TrΩd TrM is also a MCM Rmodule. Applying the condition (3) to N , we observe that there exists an R-module Lof codimension at least c such that N is isomorphic to a direct summand of a MCM
approximation X of L. It follows from [2, Theorem B] and Lemma 3.3 that the R-module
X is isomorphic to TrΩd TrΩd L up to free summands. The functor∼=TrΩd Tr : MCM(R) −→ MCM(R)is an equivalence, so we see that M is isomorphic to a direct summand of Ωd L up to freesummands. Thus Theorem 2.7 implies that R satisfies Serre’s condition (Rc−1 ).
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Proof of Theorem 1.2. The assertion follows by combining Theorems 2.7 and 3.5.

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Acknowlegments. The authors are grateful to Olgur Celikbas for his helpful comments.
The authors also thank the referee for his/her careful reading.

MCM APPROXIMATIONS AND SERRE’S CONDITION

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No. 38 (1989), 5–37.
[3] W. Bruns; J. Herzog, Cohen–Macaulay rings, revised edition, Cambridge Studies in Advanced
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[4] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings,
Preprint (1986), http://hdl.handle.net/1807/16682.
[5] H. Dao; R. Takahashi, The dimension of a subcategory of modules, Preprint (2012),
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Adv. Math. 225 (2010), no. 4, 2076–2116.
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
Email address: m14037f@math.nagoya-u.ac.jp
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
Email address: takahashi@math.nagoya-u.ac.jp
URL: http://www.math.nagoya-u.ac.jp/~takahashi/