arXiv:1704.00719v1 [] 3 Apr 2017

LOCAL RINGS WITH QUASI-DECOMPOSABLE
MAXIMAL IDEAL
SAEED NASSEH AND RYO TAKAHASHI
Abstract. Let (R, m) be a commutative noetherian local ring. In this paper,
we prove that if m is decomposable, then for any finitely generated R-module
M of infinite projective dimension m is a direct summand of (a direct sum of)
syzygies of M . Applying this result to the case where m is quasi-decomposable,
we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of R.

1. Introduction
Let R be a commutative noetherian local ring with maximal ideal m. First, we
investigate the structure of syzygies of finitely generated R-modules in the case
where m is decomposable as an R-module. Our main result in this direction is
Theorem 3.6, which includes the following remarkable statement.
Theorem A. Let R be a commutative noetherian local ring with decomposable
maximal ideal m. Let M be a finitely generated R-module with infinite projective
dimension. Then m is a direct summand of Ω3 M ⊕ Ω4 M ⊕ Ω5 M .
Here, Ωn (−) stands for the n-th syzygy in the minimal free resolution. This result
(Theorem 3.6 strictly) recovers and refines main theorems of the first author and
Sather-Wagstaff [14] on the vanishing of Tor and Ext modules over fiber products.
Next, we apply the above theorem to classification problems of subcategories over
a Cohen-Macaulay local ring with quasi-decomposable maximal ideal. Here, we say
that an ideal I of R is quasi-decomposable if there exists an R-regular sequence
x = x1 , . . . , xn in I such that I/(x) is decomposable. Our main result in this
direction is Theorem 4.5, which especially yields a complete classification of the
thick subcategories of the singularity category Dsg (R) of such R:
Theorem B. Let R be a Cohen-Macaulay singular local ring whose maximal ideal
is quasi-decomposable. Suppose that on the punctured spectrum R is either locally a
hypersurface or locally has minimal multiplicity. Then taking the singular supports
induces a one-to-one correspondence between the thick subcategories of Dsg (R) and
the specialization-closed subsets of Sing R.
Here, Sing R stands for the singular locus of R. The singularity category Dsg (R),
which is also called the stable derived category of R, is a triangulated category
that has been introduced by Buchweitz [3] and Orlov [16], and studied deeply and
2010 Mathematics Subject Classification. 13C60, 13D02, 13D09, 13H10.
Key words and phrases. Cohen-Macaulay ring, derived category, direct summand, fiber product, finite Cohen-Macaulay representation type, hypersurface, maximal Cohen-Macaulay module,
minimal multiplicity, resolving subcategory, singularity category, syzygy, thick subcategory.
Takahashi was partly supported by JSPS Grants-in-Aid for Scientific Research 16K05098.
1

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SAEED NASSEH AND RYO TAKAHASHI

widely so far; see [10] and references therein. The singular support of an object C
of Dsg (R) is defined as the set of prime ideals p such that Cp is non-zero in Dsg (Rp ).
The inverse map of the bijection in the above theorem can also be given explicitly.
There are indeed a lot of examples satisfying the assumptions of Theorems A
and B (more precisely, Theorems 3.6 and 4.5), and we shall present some of them.
The organization of this paper is as follows. Section 2 is devoted to notation,
definitions and some basic properties which are used in later sections. In Section 3,
we prove a structure result of syzygies of modules over a local ring with decomposable maximal ideal, which gives rise to Theorem A as an immediate corollary. In
Section 4, for a Cohen-Macaulay local ring with quasi-decomposable maximal ideal,
we classify resolving/thick subcategories of module/derived/singularity categories,
including Theorem B. Sections 5 and 6 state several other applications of our main
results, including the main theorems of [14].
2. Background and conventions
This section contains the terminology and some of the definitions and their basic
properties that will tacitly be used in this paper. For more details see [18, 19, 20].
2.1. Throughout this paper, R is a commutative noetherian local ring with maximal
ideal m and residue field k, and all modules are finitely generated.
2.2. Let M be an R-module. All syzygies of M are calculated by using the minimal
free resolution of M , and the i-th syzygy of M is denoted by ΩiR M . The minimal
number of generators for M is denoted by νR (M ). For an integer i ≥ 0 the ith Betti
number of M is denoted by βiR (M ); note by definition that βiR (M ) = νR (ΩiR M ).
2.3. We denote by Sing(R) the singular locus of R, namely, the set of prime ideals
p of R for which the local ring Rp is singular (i.e. non-regular). Also, NonGor(R)
stands for the non-Gorenstein locus of R, that is, the set of prime ideals p of R
such that Rp is not Gorenstein.
2.4. We denote the punctured spectrum Spec R\{m} of R by Spec0 R. Recall that R
is said to have an isolated singularity if Rp is a regular local ring for all p ∈ Spec0 R,
in other words, if Sing R ⊆ {m}.
2.5. A subset S of Sing R is called specialization-closed provided that for prime
ideals p ⊆ q of R if p is in S, then so is q.
2.6. Throughout this paper, all subcategories are full and closed under isomorphism. We denote by mod R the category of (finitely generated) R-modules and
by PD(R) the subcategory of mod R consisting of modules of finite projective dimension. Also CM(R) is the subcategory of mod R consisting of maximal CohenMacaulay R-modules. By add R we denote the subcategory of mod R consisting of
direct summands of finite direct sums of copies of R.
2.7. We denote by Db (R) the bounded derived category of mod R, and by Dsg (R)
the singularity category of R, that is, the Verdier quotient of Db (R) by perfect
complexes. We denote by π : Db (R) → Dsg (R) the canonical functor.
For a subcategory X of Db (R) we denote by πX the subcategory of Dsg (R)
consisting of objects M such that M ∼
= πX for some X ∈ X . For a subcategory
Y of Dsg (R) we denote by π −1 Y the subcategory of Db (R) consisting of objects N
such that πN is in Y.

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

3

2.8. For an object M ∈ Dsg (R), the singular support ssupp M of M is defined as
the set of prime ideals p of R such that Mp ∼
6 0 in Dsg (Rp ).
=
For a subcategory
X
of
D
(R),
the
singular
support ssupp X of X is defined by
sg
S
ssupp X := X∈X ssupp X.
For a subset S of Spec R, we denote by ssupp−1 S the subcategory of Dsg (R)
consisting of objects whose singular supports are contained in S.
2.9. For an R-module M , the non-free locus NF(M ) (resp. infinite projective dimension locus IPD(M )) of M is defined as the set of prime ideals p of R such that
Mp is non-free (resp. of infinite projective dimension) over Rp .
For a subcategory X of mod R, the non-free locus NF(X ) S
(resp. infinite projective dimension
locus
IPD(X
))
of
X
is
defined
by
NF(X
)
:=
X∈X NF(X) (resp.
S
IPD(X ) := X∈X IPD(X)).
−1
(S)) the subcateFor a subset S of Spec R, we denote by NF−1
CM (S) (resp. IPD
gory of CM(R) (resp. mod R) consisting of modules whose non-free (resp. infinite
projective dimension) loci are contained in S.
2.10. There are three types of restriction maps rest that we will consider in this
paper. We clarify each one for the convenience of the reader.
• For a subcategory X of mod R, we set restCM(R) (X ) := X ∩ CM(R).
• For a subcategory X of Db (R), we set restmod R (X ) := X ∩ mod R.
• For a subcategory X of Dsg (R), we set restCM(R) (X ) := π −1 X ∩ CM(R).
2.11. Let A be an abelian category with enough projective objects. We say that
a subcategory X of A is resolving if it contains the projective objects and is closed
under direct summands, extensions, and kernels of epimorphisms. Note that a
resolving subcategory of mod R is nothing but a subcategory containing R and
closed under direct summands, extensions and syzygies.
For an object M ∈ A, we denote by res M the smallest resolving subcategory of
A that contains M . This subcategory is called the resolving closure of M .
2.12. Let A (resp. T ) be an abelian category (resp. a triangulated category). A
subcategory X of A (resp. T ) is called thick provided that X is closed under direct
summands, and for any exact sequence 0 → L → M → N → 0 in A (resp. exact
triangle L → M → N
in T ), if two of L, M, N are in X , then so is the third.
The thick closure of a subcategory X of A (resp. T ), denoted by thickA X (resp.
thickT X ), is by definition the smallest thick subcategory of A (resp. T ) containing
X . Note that thickmod R R = PD(R), and that thickDb (R) R consists of the perfect
complexes, that is, bounded complexes of free R-modules (of finite rank). For a
subcategory X of Db (R), we simply write thickDsg (R) X for thickDsg (R) (πX ).
b is isomorphic
2.13. We say that R is a hypersurface if its m-adic completion R
to the quotient of a regular local ring by an element. (Hence, by definition, any
regular local ring is a hypersurface.)
2.14. Let R be a Cohen-Macaulay local ring. Then R satisfies the inequality
e(R) ≥ edim R − dim R + 1,

(2.14.1)

where e(R) and edim R stand for the multiplicity of R and the embedding dimension
of R, respectively. We say that R has minimal multiplicity (or maximal embedding
dimension) if the equality of (2.14.1) holds. If the residue field k of R is infinite,

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SAEED NASSEH AND RYO TAKAHASHI

then this is equivalent to the existence of a system of parameters x = x1 , . . . , xd of
R such that m2 = xm; see [2, Exercise 4.6.14].
2.15. Let R be a Cohen-Macaulay local ring. We say that Spec0 R is a hypersurface
(resp. has minimal multiplicity, has finite Cohen-Macaulay representation type) if
the Cohen-Macaulay local ring Rp is a hypersurface (resp. has minimal multiplicity,
has finite Cohen-Macaulay representation type) for every p ∈ Spec0 R. A CohenMacaulay local ring with an isolated singularity is a typical example satisfying all
of these three conditions, so they are mild conditions.
3. Local rings with decomposable maximal ideal
This section is devoted to the proof of Theorem A. This theorem is a consequence
of Theorem 3.6, which is the main result of this section.
The following fact was established by Ogoma [15, Lemma 3.1] and gives a characterization of local rings with decomposable maximal ideal in terms of fiber products.
Fact 3.1. The maximal ideal m has a direct sum decomposition m = I ⊕ J in
which I, J are non-zero ideals of R if and only if R is isomorphic to the non-trivial
fiber product (R/I) ×k (R/J), that is, R is isomorphic to the pull-back of the
πJ
πI
R/J. This isomorphism is naturally defined by
k ←−−
natural surjections R/I −→
r 7→ (r + I, r + J) for r ∈ R. In this case, by [4, Remark 3.1] we have the equality
depth R = min{depth R/I, depth R/J, 1}.

(3.1.1)

In particular, if m is decomposable and M is an R-module, then it follows from the
Auslander-Buchsbaum formula that pdR M ≥ 2 if and only if pdR M = ∞.
Lemma 3.2. Suppose that m has a direct sum decomposition m = I ⊕ J. Let N
be an R/I-module, and let n := νR/I (N ) = νR (N ). Then there is an isomorphism
ΩR N ∼
= I ⊕n ⊕ ΩR/I N of R-modules.
Proof. There is a commutative diagram

0

0

0


I ⊕n


I ⊕n


/ ΩR N

λ


/ R⊕n

0

/ ΩR/I N

0

/N

/0

/N

/0

ε

ζ



ρ

θ


/ (R/I)⊕n

π


0

with exact rows and columns. Now we have
ΩR/I N = Ker π ⊆ m(R/I)⊕n = (m/I)⊕n ∼
= J ⊕n ⊆ R⊕n .
Call this composite injection η : ΩR/I N → R⊕n . Then we have θ = εη, and ρη =
πεη = πθ = 0. Hence, there is a homomorphism ξ : ΩR/I N → ΩR N such that

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

5

η = λξ. Since θζξ = ελξ = εη = θ and θ is injective, we get ζξ = 1. Therefore, the
left column in the above diagram splits and we get ΩR N ∼

= I ⊕n ⊕ ΩR/I N .
Lemma 3.3. Suppose that m has a direct sum decomposition m = I ⊕ J such that
J 6= 0. Then the following hold.
(1) If R/I is a discrete valuation ring, then there is an isomorphism R/I ∼
= J of
R-modules.
(2) Conversely, if J is a free R/I-module, then R/I is a discrete valuation ring.
Proof. Note that m/I ∼
= J as R-modules.
(1) By assumption we have m/I ∼
= R/I. Hence, the assertion follows.
(2) By assumption, m/I is a free R/I-module. Hence, the global dimension of
R/I is at most one. If it is zero, then R/I is a field and we have I = m, which
contradicts J being non-zero. Thus, R/I is a discrete valuation ring.

The following lemma is shown by a similar argument as in [14, Lemma 2.5].
Lemma 3.4. Let I be a proper ideal of R. Let M be an R-module with pdR M ≥ 2.
If Ω2R M is R/I-free, then R/I has positive depth.
Proof. By assumption there are injections R/I → Ω2R M → mR⊕e = m⊕e for some
e > 0, and call this composition φ : R/I → m⊕e . Set ξ = φ(1) ∈ m⊕e . If R/I has
depth zero, then it contains a non-zero socle s, and we have φ(s) = sξ = 0, which
contradicts the injectivity of the map φ. Hence, R/I has positive depth.

Proposition 3.5. Suppose that m has a direct sum decomposition m = I ⊕ J in
which I, J are non-zero ideals of R. Let M be an R-module with pdR M ≥ 2. If
either R/I or R/J has depth zero, then Ω2R M is neither R/I-free nor R/J-free.
Proof. Assume that depth R/I = 0. (The case where depth R/J = 0 is handled
similarly.) It follows from Lemma 3.4 that Ω2R M is not R/I-free. Suppose that
Ω2R M is R/J-free. Then again by Lemma 3.4 the local ring R/J has positive
(cij )

depth. There is an exact sequence 0 → (R/J)⊕a → R⊕b −−−→ R⊕d → M → 0
where each cij is an element of m and a, b, d are positive integers. Applying the
functor HomR (k, −) to this, we have an exact sequence
0

0 → HomR (k, (R/J)⊕a ) → HomR (k, R⊕b ) −
→ HomR (k, R⊕d ).
As R/J has positive depth, HomR (k, (R/J)⊕a ) = 0. We see from this exact sequence that HomR (k, R⊕b ) = 0, and hence depth R > 0. This is a contradiction
because by (3.1.1) we must have depth R = 0. Thus, Ω2R M is not R/J-free.

Theorem A is a consequence of the following result, which is the main result of
this section.
Theorem 3.6. Let (R, m) be a local ring, and suppose that there is a non-trivial
direct sum decomposition m = I ⊕ J. Let M be an R-module with pdR M ≥ 2.
(1) One of the following holds true.
(i) m is a direct summand of Ω3R M or Ω4R M .
(ii) m is a direct summand of Ω5R M , and Ω2R M is free over R/I or R/J.
(iii) m is a direct summand of Ω3R M ⊕ Ω4R M , and either R/I is a discrete
valuation ring and Ω2R M is R/J-free, or R/J is a discrete valuation ring
and Ω2R M is R/I-free.

6

SAEED NASSEH AND RYO TAKAHASHI

(2) If either R/I or R/J has depth 0, then neither (ii) nor (iii) occurs.
Proof. (1) It follows from [6] (see also [13, Proposition 4.2]) that
Ω4R M ∼
Ω3R M ∼
Ω2R M ∼
=X ⊕Y
= V ⊕ W,
= A ⊕ B,
for some R/I-modules A, V, X and R/J-modules B, W, Y . By Lemma 3.2, we have1
ΩR V
ΩR A ∼
= I ⊕ν(A) ⊕ ΩR/I A,
⊕ν(B)
∼
⊕ ΩR/J B, ΩR W
ΩR B = J

∼
ΩR X
= I ⊕ν(V ) ⊕ ΩR/I V,
⊕ν(W )
∼
⊕ ΩR/J W, ΩR Y
=J

∼
= I ⊕ν(X) ⊕ ΩR/I X,
∼
= J ⊕ν(Y ) ⊕ ΩR/J Y.

We consider the following five cases. Note that since pdR M ≥ 2, the second
Betti number β := β2R (M ) is non-zero.
Case 1: A 6= 0 6= B. In this case, I and J are direct summands of ΩR A and
ΩR B, respectively. Hence, m = I ⊕ J is a direct summand of ΩR A ⊕ ΩR B = Ω3R M .
Case 2: A = 0 = V . In this case, Ω2R M = B and Ω3R M = W are both R/Jmodules, and annihilated by the ideal J of R. Hence, by the short exact sequence
0 → Ω3R M → R⊕β → Ω2R M → 0 we observe that J 2 annihilates R⊕β . Since β 6= 0
we get J 2 = 0. Therefore, mJ = (I + J)J = 0. This implies that J is a k-vector
space. Since B = Ω2R M 6= 0, the ideal J is a direct summand of ΩR B = Ω3R M .
Thus k is a direct summand of Ω3R M , and m is a direct summand of Ω4R M .
Case 3: B = 0 = W . This case is treated similar to Case 2, and we see that m
is a direct summand of Ω4R M .
Case 4: A = 0 6= V . In this case, we must have B 6= 0, and hence, n := ν(B) > 0.
If W 6= 0, then V 6= 0 6= W , and Case 1 shows that m is a direct summand of Ω4R M .
Thus, we may assume that W = 0. Then we have Ω2R M ∼
= B and
⊕n
3
V ∼
= J ⊕ ΩR/J B.
= ΩR B ∼
=Ω M ∼
R

Since V is annihilated by I, so is the direct summand ΩR/J B. As ΩR/J B is also
annihilated by J, and by m = I ⊕ J, we see that ΩR/J B is a k-vector space. If
ΩR/J B is non-zero, then k is a direct summand of ΩR/J B, and hence, of Ω3R M . In
this case m is a direct summand of Ω4R M , and we are done. Thus, we may assume
that ΩR/J B = 0. In this case, Ω2R M ∼
=B∼
= (R/J)⊕n and we have
V ∼
= Ω3R M ∼
= ΩR B ∼
= J ⊕n .

(3.6.1)

If Y = 0, then W = 0 = Y , and Case 3 shows that m is a direct summand of Ω5R M .
Hence, we may assume Y 6= 0. If X 6= 0 6= Y , then Case 1 shows that m is a direct
summand of Ω5R M . Therefore, we may assume X = 0. Now we have
∼ Ω4 M ∼
Y =
= ΩR V ∼
= I ⊕m ⊕ ΩR/I V
R

where m := ν(V ). We make a similar argument as above. Since Y is annihilated by
J, so is ΩR/I V , which is also annihilated by I. Hence, ΩR/I V is a k-vector space. If
ΩR/I V is non-zero, then k is a direct summand of ΩR/I V , and thus, of Ω4R M . In this
case m is a direct summand of Ω5R M . So, we may assume ΩR/I V = 0. Under this
assumption, V is a free R/I-module and we get V ∼
= (R/I)⊕m . Therefore, by (3.6.1)
⊕m
⊕n ∼
we have J
= (R/I) . Since n > 0 we see that J is a free R/I-module. Now
Lemma 3.3(2) implies that R/I is a discrete valuation ring. Then J is isomorphic
to R/I by Lemma 3.3(1). We obtain isomorphisms Ω3R M ∼
= (R/I)⊕n and
= J ⊕n ∼
4
⊕n
3
∼
ΩR M = I . Therefore, m = I ⊕ J is a direct summand of ΩR M ⊕ Ω4R M .
1We use ν instead of ν to keep the notation simpler.
R

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

7

Case 5: B = 0 6= W . This case is treated similar to Case 4, and we see that
Ω2R M is R/I-free and that either m is a direct summand of Ω5R M , or R/J is a
discrete valuation ring and m is a direct summand of Ω3R M ⊕ Ω4R M .
(2) This part follows from Proposition 3.5.

In the remainder of this section, we present several examples showing that each
of the cases (i)-(iii) in Theorem 3.6 occurs and these conditions are independent.
Convention 3.7. In the following examples, k will be a field as before, and X, Y, Z
will be indeterminates over k. We will use the lower-case letters x, y, z to represent
the residues of the variables X, Y, Z in the ring R.
We start by providing an example that satisfies (i), but not (ii) or (iii).
Example 3.8. Consider the fiber product
k[[X, Y ]]
k[[X]]
×k k[[Y ]] ∼
R=
=
(X 2 )
(X 2 , XY )
and set I = (x) and J = (y). Then we have m = I ⊕ J. Since R/J has depth zero,
it follows from Theorem 3.6 that neither (ii) nor (iii) holds. Let M = R/J. The
minimal free resolution of M is


x y 0
0 0 x



(x,y)

x

y

→R−
→ R → M → 0.
· · · → R −−−−−→ R2 −−−→ R −
3
∼
It follows from this that ΩR M = m = k ⊕ (R/I), and we observe that m is a direct
summand of ΩiR M for all i ≥ 3. In particular, statement (i) holds true.
3

The following is an example satisfying (iii), but not (i) or (ii).
Example 3.9. Consider the fiber product
k[[X, Y ]]
R = k[[X]] ×k k[[Y ]] ∼
=
(XY )
and set I = (x) and J = (y). We then have m = I ⊕ J. We also have R/J ∼
= k[[x]]
and R/I ∼
= k[[y]], which are discrete valuation rings. Take M = R/I. Then
(
R/J if i ≥ 0 is odd,
ΩiR M ∼
=
R/I if i ≥ 0 is even.
In particular, Ω2R M is R/I-free, and Ω3R M ⊕Ω4R M ∼
= m. Thus, (iii) holds. Moreover,
m is not a direct summand of ΩiR M for all i ≥ 0. Hence neither (i) nor (ii) holds.
The following is also an example that satisfies (iii), but not (i) or (ii). Comparing
with Example 3.9, this example has the advantage that R/J is not a discrete
valuation ring.
Example 3.10. Consider the fiber product
k[[X, Y, Z]]
.
R = k[[X]] ×k k[[Y, Z]] ∼
=
(XY, XZ)
Setting I = (y, z) and J = (x), we have m = I ⊕ J. The ring R/I = k[[x]] is a
discrete valuation ring, while R/J = k[[y, z]] is not. One can easily check (by hand)
that the minimal free resolution of the R-module R/(y) has the form:
y z 0 0 0
0 0y z 0
0 0 0 0x

!



x 0 z
0 x −y



(y,z)

x

y

→R−
→ R → R/(y) → 0
· · · → R −−−−−−−−→ R −−−−−−−→ R2 −−−→ R −
5

3

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SAEED NASSEH AND RYO TAKAHASHI

Note that it contains a minimal free presentation of I and ΩR I ∼
= (R/I)⊕2 ⊕ R/(x).
Let M be the (Auslander) transpose of I, that is, the R-module M defined by the
following exact sequence


x 0
0 x
z −y



0 → I ∗ → R2 −−−−−−→ R3 → M → 0
where I ∗ := HomR (I, R). Then Ω2R M = I ∗ . From the natural short exact sequence
0 → I → R → R/I → 0, we have an exact sequence
f

0 → (R/I)∗ −
→ R∗ → I ∗ → Ext1R (R/I, R) → 0.
Note that the map f can be described by the inclusion map J = (0 : I) → R, and
hence, there is an exact sequence
0 → R/J → I ∗ → Ext1R (R/I, R) → 0.

(3.10.1)

We claim that Ext1R (R/I, R) = 0. Indeed, from the natural short exact sequence
0 → R → (R/I) ⊕ (R/J) → k → 0 we get an exact sequence
g

(R/I)∗ ⊕ (R/J)∗ −
→ R∗ → Ext1R (k, R) → Ext1R ((R/I) ⊕ (R/J) , R) → 0.
Since the map g can be described by the inclusion map
m = J ⊕ I = (0 : I) ⊕ (0 : J) → R
its cokernel is isomorphic to k. The element x − y ∈ R is R-regular and annihilates
k. Using [2, Lemma 3.1.16] and the isomorphism R/(x − y) ∼
= k[[y, z]]/(y 2, yz),
1
∼
∼
we easily see that ExtR (k, R) = HomR/(x−y) (k, R/(x − y)) = k. Thus, we obtain
Ext1R ((R/I) ⊕ (R/J) , R) = 0, which implies that Ext1R (R/I, R) = 0, as we claimed.
This claim and (3.10.1) imply that Ω2R M = I ∗ ∼
= R/J. Combining this with the
isomorphism Ω3R M ⊕ Ω4R M ∼
= m shows that (iii) holds.
On the other hand, none of the syzygies Ω3R M = J, Ω4R M = I, and Ω5R M =
ΩR I = (R/I)2 ⊕(R/J) contain m as a direct summand. In fact, taking into account
the minimal numbers of generators, we see that m is not a direct summand of J or
I. Also, m ∼
= (R/I)⊕2 ⊕ (R/J) if we assume that m is a direct summand of ΩR I.
⊕2
Then, however, localization at the prime ideal I yields RI = mRI ∼
= (RI /IRI )
which contradicts the fact that the local ring RI is indecomposable as a module
over itself. Consequently, neither (i) nor (ii) holds.
The following is an example that satisfies (ii), but not (i) or (iii).
Example 3.11. Let R, I, J be as in Example 3.10, and let M = R/(y). Then
Ω3R M = I,

Ω4R M = (R/I)⊕2 ⊕ (R/J),

Ω5R M = I ⊕2 ⊕ J.

By the argument in Example 3.10, m is not a direct summand of Ω3R M or Ω4R M ,
while it is a direct summand of Ω5R M . Therefore, (i) does not hold, but (ii) holds.
We also have Ω2R M = R/I, which is not R/J-free, and R/J = k[[y, z]] is not a
discrete valuation ring. Thus, (iii) does not hold.
The next example satisfies half of (i), but does not satisfy the other half. Moreover, it does not satisfy (ii) or (iii), either.
Example 3.12. Let R, I, J be as in Example 3.10, and let M = (y) ∼
= R/(x). Then
Example 3.10 shows that m is not a direct summand of Ω3R M = (R/I)⊕2 ⊕ (R/J),
but a direct summand of Ω4R M = I ⊕2 ⊕ J. Hence, half of (i) holds, while the

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

9

other half does not. As I 2 = (y 2 , yz, z 2) is non-zero, Ω2R M = I is not R/I-free.
Since R/J ∼
= k[[y, z]] is not a discrete valuation ring, Ω2R M = I is not R/J-free by
Lemma 3.3(2). Thus, neither (ii) nor (iii) holds.
4. Local rings with quasi-decomposable maximal ideal
This section contains the proof of Theorem B. Recall that the definitions and
terminology have been introduced in Section 2.
Definition 4.1. If there exists an R-sequence x of length n ≥ 0 such that m/(x)
is decomposable, then we say that m is quasi-decomposable (with x).
If m is decomposable, then of course m is quasi-decomposable. However, the
converse is far from being true. We shall give several examples of a local ring with
indecomposable quasi-decomposable maximal ideal at the end of this section.
We prove the following lemmas for the future use.
Lemma 4.2. Let M be an R-module, and let x be an R-sequence of length n ≥ 0
annihilating M . Then for all t ≥ 0 and u ≥ n there exists v ≥ 0 such that
∼ Ωu+t M ⊕ R⊕v .
M=
Ωu Ωt
R

R/(x)

R

Proof. There is an exact sequence
0 → ΩtR/(x) M → Ft−1 → Ft−2 → · · · → F1 → F0 → M → 0
of R/(x)-modules with each Fi free over R/(x). Taking the u-th syzygies over R,
we get an exact sequence
0 → ΩuR ΩtR/(x) M → ΩuR Ft−1 ⊕ R⊕at−1 → · · ·
→ ΩuR F1 ⊕ R⊕a1 → ΩuR F0 ⊕ R⊕a0 → ΩuR M → 0
for some ai ≥ 0. As u ≥ n, each ΩuR Fi is R-free. This exact sequence then says
that ΩuR ΩtR/(x) M is isomorphic to the direct sum of Ωt ΩuR M = Ωu+t
R M and some
free R-module, as desired.

Lemma 4.3. Let M be an R-module, and let x be an M -sequence of length n ≥ 0.
Then ΩiR (M/xM ) belongs to the resolving closure res M of M for all i ≥ n.
Proof. Since x is an M -sequence, the Koszul complex K(x, M ) is acyclic with
H0 (K(x, M )) = M/xM . Each K(x, M )i is a direct sum of copies of M , which
belongs to res M . Hence, ΩnR (M/xM ) is in res M by [18, Lemma 4.3]. Therefore,
ΩiR (M/xM ) is in res M for all i ≥ n.

Lemma 4.4. Let R be a d-dimensional Cohen-Macaulay local ring with quasidecomposable maximal ideal. Let X be a resolving subcategory of mod R contained
in CM(R). If X contains a non-free R-module, then X also contains ΩdR k.
Proof. Let x be an R-sequence of length n ≥ 0 such that m/(x) is decomposable,
and let X ∈ X be a non-free R-module. Then X is maximal Cohen-Macaulay, and
x is an X-sequence. Hence, pdR/(x) X/xX = pdR X = ∞ by [2, Lemma 1.3.5].
Theorem A implies that ΩR/(x) k = m/(x) is a direct summand of
Ω3R/(x) (X/xX) ⊕ Ω4R/(x) (X/xX) ⊕ Ω5R/(x) (X/xX) = Y /xY
where Y := Ω3R X ⊕ Ω4R X ⊕ Ω5R X ∈ X .

10

SAEED NASSEH AND RYO TAKAHASHI

∼ d
If n ≤ d − 1, then by Lemma 4.2 we have Ωd−1
R ΩR/(x) k = ΩR k ⊕ F for some free
d−1
R-module F . The module ΩR ΩR/(x) k is a direct summand of Ωd−1
R (Y /xY ). As
Y is maximal Cohen-Macaulay, x is a Y -sequence. It follows from Lemma 4.3 that
d−1
d
Ωd−1
R (Y /xY ) is in X . Hence, so is ΩR ΩR/(x) k, and therefore, ΩR k is in X .
Now let us consider the case n = d. Similarly as above, we see that ΩdR ΩR/(x) k ∼
=
d+1
ΩR k ⊕ F ′ for some free R-module F ′ , that ΩdR ΩR/(x) k is a direct summand of
ΩdR (Y /xY ), and that ΩdR (Y /xY ) is in X . We also see that Ωd+1
R k belongs to
d+1
X . Since depthR ΩR k = d, it follows from [5, Proposition 4.2] that ΩdR k is in
d+1
d+1
d
res(Ωd+1

R k ⊕ ΩR k) = res ΩR k. Hence, ΩR k is in X , as desired.
Now we can state and prove the main result of this section.
Theorem 4.5. Let (R, m) be a d-dimensional Cohen-Macaulay singular local ring
and assume that m is quasi-decomposable.
(1) Suppose that Spec0 R is a hypersurface or has minimal multiplicity. Then one
has the following commutative diagram of mutually inverse bijections:



Resolving subcategories of
o
mod R contained in CM(R)

NF

/

NF−1
CM



Specialization-closed
subsets of Sing R
O
IPD

IPD−1






Thick subcategories of thickmod R / Thick subcategories of
o
CM(R) containing R
mod R containing R
restCM(R)
O
O
restCM(R)



thickDsg (R)



Thick subcategories of
o
Dsg (R)

restmod R
π −1
π

thickDb (R)




/ Thick subcategories of
Db (R) containing R

(2) Assume that R is excellent and has a canonical module ω. Suppose that Spec0 R
has finite Cohen-Macaulay representation type. Then one has the following
commutative diagram of mutually inverse bijections:


 Resolving subcategories 
o
of mod R contained in


CM(R) and containing ω

NF
NF−1
CM



 Specialization-closed 
/
subsets of Sing R


containing NonGor R
O
IPD

IPD−1






Thick subcategories of thickmod R Thick subcategories of
/
o
CM(R) containing
mod R containing

 restCM(R) 

R and ω
R and ω
O
O
restCM(R)

thickDsg (R)




Thick subcategories of
o
Dsg (R) containing ω

restmod R

π −1
π

thickDb (R)




Thick subcategories of
/
Db (R) containing


R and ω

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

11

Proof. Note that NonGor(R) = NF(ω) = IPD(ω) for part (2). Hence, by [19, Lemma
2.6] we observe that all the twelve maps in Theorem 4.5 are well-defined, and that
NF(NF−1
CM (W )) = W,

IPD(IPD−1 (W )) = W

(4.5.1)

for each specialization-closed subset W of Spec R contained in Sing R.
We first show that
X = thickmod R (restCM(R) X ) = thickmod R (X ∩ CM(R))

(4.5.2)

for each thick subcategory X of mod R containing R. Since X is thick and contains
X ∩ CM(R), we see that X contains thickmod R (X ∩ CM(R)). Take any module
X ∈ X . Then there is an exact sequence 0 → Ωd X → Fd−1 → · · · → F0 → X → 0
of R-modules with each Fi free. Note that Ωd X and Fi for 0 ≤ i ≤ d − 1 are in
X ∩ CM(R). Thus, these modules belong to thickmod R (X ∩ CM(R)). Therefore, X
is also in thickmod R (X ∩ CM(R)), and we have the equalities in (4.5.2).
Next we show that
IPD(thickmod R X ) = NF(X )
(4.5.3)
for any subcategory X of CM(R). It is straightforward to check that
IPD(thickmod R X ) = IPD(X ) ⊆ NF(X ).
Let p be a prime ideal such that Xp is a non-free Rp -module for some X ∈ X . Since
the Rp -module Xp is maximal Cohen-Macaulay, it has infinite projective dimension.
Hence, NF(X ) is contained in IPD(X ). Therefore, we have the equality in (4.5.3).
In view of (4.5.1), (4.5.2) and (4.5.3), we show the following three statements.
(a) NF−1
CM · NF = 1.
(b) IPD−1 · IPD = 1.
(c) The equalities in the diagrams hold.
For (a) fix a resolving subcategory X of mod R contained in CM(R). If X =
add R, then NF(X ) is empty, and we have NF−1
CM (NF(X )) = add R = X . Thus,
we may assume that X 6= add R. Then X contains a non-free R-module, and it
follows from Lemma 4.4 that X contains Ωd k. If Spec0 R is a hypersurface (resp.
has minimal multiplicity), then by virtue of [18, Theorem 5.13] (resp. [19, Theorem
0
5.6]) we obtain the equality NF−1
CM (NF(X )) = X . If R is excellent, Spec R has finite
Cohen-Macaulay representation type, and X contains a canonical module, then the
equality NF−1
CM (NF(X )) = X follows from [19, Theorem 6.11 and Corollary 6.12].
For (b) let X be a thick subcategory of mod R containing R. Then X is contained
in IPD−1 (IPD(X )). Now for an R-module M in IPD−1 (IPD(X )) we claim that
NF(ΩdR M ) ⊆ NF(X ∩ CM(R)).
Indeed, let p be a prime ideal such that (ΩdR M )p is non-free. Then the Rp -module
ΩdRp (Mp ) is non-free, and Mp has infinite projective dimension. Hence, p is in
IPD(M ), which is contained in IPD(X ). This implies that Xp has infinite projective
dimension for some X in X . Therefore, (ΩdR X)p is non-free, which says that p
belongs to NF(ΩdR X). Since ΩdR X is in X ∩ CM(R), the claim follows.
This claim implies that Ωd M belongs to NF−1
CM (NF(X ∩ CM(R))). Note that X ∩
CM(R) is a resolving subcategory contained in CM(R). In each of the cases where
Spec0 R is a hypersurface, where Spec0 R has minimal multiplicity, and where R is
excellent, Spec0 R has finite Cohen-Macaulay representation type and X contains

12

SAEED NASSEH AND RYO TAKAHASHI

a canonical module, it follows from (a) that NF−1
CM (NF(X ∩ CM(R))) = X ∩ CM(R).
Hence, Ωd M is in X , and so is M as X is thick and contains R.
Finally, for (c) let X be a resolving subcategory of mod R contained in CM(R).
−1
By (a) we have X = NF−1
CM (NF(X )); setting W = NF(X ), we have X = NFCM (W ).
Let 0 → A → B → C → 0 be an exact sequence of maximal Cohen-Macaulay
R-modules. If A and B are in X , then NF(A) and NF(B) are contained in W , and
we see that NF(C) is also contained in W by the Auslander-Buchsbaum formula.
Hence C is in X , which shows that X is a thick subcategory of CM(R). The proof
of (c) is now complete.
The one-to-one correspondence (restmod R , thickDb (R) ) in (1) and (2) follows from
[9, Theorem 1], while the one-to-one correspondence (π −1 , π) is shown by noting
that for two objects M, N ∈ Db (R) with πM ∼
= πN one has M ∈ thickDb (R) {R, N }.
It is straightforward that restCM(R) X = restCM(R) (restmod R (π −1 X )) for a (thick)
subcategory X of Dsg (R). Let Y be a thick subcategory of CM(R) containing
R. Then π(thickDb (R) (thickmod R Y)) = π(thickDb (R) Y) is a thick subcategory of
Dsg (R) containing πY. If Z is a thick subcategory of Dsg (R) containing πY, then
Y is contained in π −1 Z, and so is thickDb (R) Y. It follows that π(thickDb (R) Y) =
thickDsg (R) Y. Thus, the lower squares in the diagrams in (1) and (2) are commutative. This completes the proof of Theorem 4.5.

Our Theorem B can now be deduced from Theorem 4.5.
4.6 (Proof of Theorem B). Let X be a thick subcategory of Dsg (R), and let W be
a specialization-closed subset of Sing R. According to Theorem 4.5, it suffices to
show the equalities:
(a) IPD(restmod R (π −1 X )) = ssupp X ; and
(b) π(thickDb (R) (IPD−1 W )) = ssupp−1 W .
For (a), pick a prime ideal p ∈ IPD(restmod R (π −1 X )) = IPD(π −1 X ∩ mod R).
Then there is an R-module M such that X := πM ∈ X and pdRp Mp = ∞.
Hence, Xp ∼
6 0 in Dsg (Rp ), which implies that p is in ssupp X . Conversely, pick
=
a prime ideal p ∈ ssupp X . Then there is an object X ∈ X such that Xp ∼
6 0
=
in Dsg (Rp ). Choose an object Y ∈ Db (R) with πY = X. There exists an exact
triangle P → Y → M [n]
in Db (R) such that P ∈ thickDb (R) R, M ∈ mod R, and
n ∈ Z. Hence X = πY = (πM )[n], which implies that πM is in X , and hence, M
is in π −1 X ∩ mod R = restmod R (π −1 X ). Since p belongs to IPD(M ), it belongs to
IPD(restmod R (π −1 X )).
For (b), let A be an object in π(thickDb (R) (IPD−1 W )). Then A ∼
= πB for
some B ∈ thickDb (R) (IPD−1 W ). Hence, ssupp A = ssupp(πB) = IPD(B). If p is
a prime ideal outside W , then (IPD−1 (W ))p is contained in PD(Rp ), and so is
(thickDb (R) (IPD−1 (W )))p , which contains Bp . Hence, Bp has finite projective dimension as an Rp -module, which shows that IPD(B) is contained in W . Therefore,
A belongs to ssupp−1 W . Conversely, let A be an object in ssupp−1 W , and write
A = πB with B ∈ Db (R). Then it is easy to see that IPD(B) is contained in W ,
and B belongs to thickDb (R) (IPD−1 W ). Hence, A is in π(thickDb (R) (IPD−1 W )). 
We close this section by presenting several examples of a Cohen-Macaulay nonGorenstein local ring with quasi-decomposable maximal ideal; actually, it turns out
that there are plenty of such examples.

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

13

Example 4.7. Let R be a d-dimensional Cohen-Macaulay local ring which is not
a hypersurface. Suppose that k is infinite and that R has minimal multiplicity.
Then m is quasi-decomposable. In fact, we find a system of parameters x of R with
m2 = xm. Then we have m(m/(x)) = 0, which means that m/(x) is a k-vector
space. Since R is not a hypersurface, the dimension of m/(x) as a k-vector space
is at least two. Hence m/(x) is decomposable.
Example 4.8. Let H = hpq + p + 1, 2q + 1, p + 2i be the numerical semigroup
with p, q > 0 and gcd(p + 2, 2q + 1) = 1, and let R = k[[H]] be (the completion
of) the numerical semigroup ring of H over a field k. Then R is a non-Gorenstein
almost-Gorenstein (see [8]) Cohen-Macaulay local domain of dimension 1, and m is
quasi-decomposable. In fact, by virtue of [12, Proposition 2(1)], we have
R∼
=

k[[x, y, z]]
x y z
I2 y p z q x

(4.8.1)

where I2 (−) stands for the ideal generated by 2 × 2-minors. This implies R/(z) ∼
=
k[[x, y]]/(x2 , xy, y p+1 ), and hence we obtain a non-trivial direct sum decomposition
m/(z) ∼
= x(R/(z)) ⊕ y(R/(z)).
Example 4.9. Let R = k[[H]] be a non-Gorenstein almost-Gorenstein numerical semigroup ring with embedding dimension 3 and multiplicity ≤ 6. Then m is
quasi-decomposable. Indeed, by [12, Proposition 2(2)] we again have the isomorphism (4.8.1) with p, q > 0, and a similar argument as in Example 4.8 applies.
Example 4.10. For each of the Cohen-Macaulay local rings R given in [19, Examples 7.1, 7.2, 7.4 and 7.5] the maximal ideal is quasi-decomposable, since the
quotient of a suitable system of parameters is isomorphic to the ring
k[s, t]/(s2 , st, t2 ) = k[s]/(s2 ) ×k k[t]/(t2 ),
where s, t are indeterminates over the field k. We should notice that these four
examples also explain the independence of the assumptions of Theorem 4.5 on the
punctured spectrum. (None of them has an isolated singularity.)
5. More on thick subcategories of mod R
As another application of Theorem 3.6, in this section we study the thick subcategories of mod R that contain a module of infinite projective dimension where m
is quasi-decomposable. We begin with making a lemma.
Lemma 5.1. Let M be an R-module and x = x1 , . . . , xn be an R-sequence. Then
x is an ΩnR M -sequence.
Proof. We use induction on n. The assertion is trivial when n = 0. Let n > 0.
The induction hypothesis implies that the sequence x′ := x1 , . . . , xn−1 is regular
on both Ωn−1 M and Ωn M = Ωn−1 (ΩM ). The exact sequence
0 → Ωn M → F → Ωn−1 M → 0
in which F is free induces an exact sequence
0 → Ωn M/x′ Ωn M → F/x′ F → Ωn−1 M/x′ Ωn−1 M → 0.
This shows that the multiplication map of Ωn M/x′ Ωn M by xn is injective. Thus,
x = x1 , . . . , xn is regular on Ωn M .


14

SAEED NASSEH AND RYO TAKAHASHI

The next proposition plays a central role in this section.
Proposition 5.2. Suppose that m is quasi-decomposable with an R-sequence x
of length n. If M is an R-module with pdR M ≥ n + 2, then k is in X :=
thickmod R {R, M }.
Proof. We consider the following cases.
Case 1: n = 0. In this case m is decomposable, and we can apply Theorem 3.6.
Since X contains all the syzygies of M and their direct summands, we are done.
Case 2: n > 0. It follows from Lemma 5.1 that x is an ΩnR M -sequence. Using
[2, Lemma 1.3.5], we have
pdR/(x) ΩnR M/xΩnR M = pdR ΩnR M ≥ (n + 2) − n = 2.
By Case 1 we have k is in thickmod R/(x) {R/(x), ΩnR M/xΩnR M }. Therefore, k is
in thickmod R {R/(x), ΩnR M/xΩnR M }. The modules R/(x) and ΩnR M/xΩnR M are in
thickmod R {R, ΩnR M } = X . Therefore, k belongs to X , as desired.

Corollary 5.3. Suppose that R has an isolated singularity, and that m is quasidecomposable with an R-sequence x of length n.
(1) If X is a thick subcategory of mod R containing R and at least one module of
projective dimension ≥ n + 2, then X = mod R.
(2) There exists no thick subcategory X with PD(R) ( X ( mod R.
Proof. (1) Since we assume that R has an isolated singularity, by [17, Theorem
VI.8] (see also [9, Proposition 9]) we have thickmod R {R, k} = mod R. The assertion
now follows from Proposition 5.2.
(2) If X is a thick subcategory strictly containing PD(R), then X contains a
module of infinite projective dimension, and X = mod R by (1).

Remark 5.4. Corollary 5.3(2) holds whenever R is a hypersurface with an isolated
singularity. In fact, since R is a hypersurface, by [20, Theorem 5.1(1)] there is a oneto-one correspondence between the thick subcategories of mod R containing R and
the specialization-closed subsets of Sing R. As R has an isolated singularity, Sing R
is trivial, and thus there is no thick subcategory X with PD(R) ( X ( mod R.
The following is also a corollary of Proposition 5.2 (whence one of Theorem 3.6),
which is a variant of [19, Proposition 5.2].
Corollary 5.5. Let R be a Cohen-Macaulay non-Gorenstein local ring with minimal multiplicity and infinite residue field. Let M be an R-module with infinite
projective dimension. Then k belongs to thickmod R {R, M }.
Proof. In view of Example 4.7, the maximal ideal of R is quasi-decomposable.
Hence the assertion follows from Proposition 5.2.

6. Vanishing of Ext and Tor
In this section we give some Ext and Tor vanishing results over local rings with
quasi-decomposable maximal ideal as applications of Theorem 3.6. Some of these
results refine the existing results of [14] as we explain below. We start by a lemma.
Lemma 6.1. Let M and N be R-modules. Let t ≥ 0 be an integer.
(1) Suppose that m is a direct summand of Ωt M .
(a) If TorR
ℓ (M, N ) = 0 for some ℓ ≥ t + 1, then pdR N < ∞.

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

15

(b) If ExtℓR (M, N ) = 0 for some ℓ ≥ t + max{1, depthR N }, then idR N < ∞.
(2) Suppose that m is a direct summand of Ωt M ⊕ Ωt+1 M .
R
(a) If TorR
ℓ (M, N ) = Torℓ+1 (M, N ) = 0 for some ℓ ≥ t + 1, then pdR N < ∞.
ℓ
(b) If ExtR (M, N ) = Extℓ+1
R (M, N ) = 0 for some ℓ ≥ t + max{1, depthR N },
then idR N < ∞.
R
t
∼
Proof. (1a) Since ℓ−t > 0, we have TorR
ℓ (M, N ) = Torℓ−t (Ω M, N ). By assumption
R
R
we get Torℓ−t+1 (k, N ) ∼
= Torℓ−t (m, N ) = 0. Hence, pdR N < ∞.
t
(1b) As ℓ − t > 0, we have ExtℓR (M, N ) ∼
= Extℓ−t
R (Ω M, N ). The assumption
ℓ−t+1
ℓ−t
∼
implies ExtR
(k, N ) = ExtR (m, N ) = 0. Since ℓ − t + 1 > depth N , by [7,
Theorem (1.1)] we conclude that idR N < ∞.
(2a) Since ℓ − t > 0, we have
0 = TorR (M, N ) ⊕ TorR (M, N ) ∼
= TorR (Ωt M ⊕ Ωt+1 M, N ),
ℓ

TorR
ℓ−t (m, N )

ℓ+1

ℓ−t

∼
= TorR
ℓ−t+1 (k, N ) as a direct summand. Hence N has

which contains
finite projective dimension.
(2b) This follows from a similar argument to the proofs of (2a) and (1b).



Our Theorem 3.6 recovers [14, Theorems 1.1] as we state next.
Corollary 6.2 (Nasseh and Sather-Wagstaff). Assume that R = S ×k T is a fiber
product, where S and T are local rings with common residue field k and S 6= k 6= T .
Let M and N be R-modules.
(1) If S or T has depth 0 and TorR
ℓ (M, N ) = 0 for some ℓ ≥ 5, then M or N is
R-free.
R
(2) If TorR
ℓ (M, N ) = Torℓ+1 (M, N ) = 0 for some ℓ ≥ 5, then pdR M ≤ 1 or
pdR N ≤ 1.
Proof. (1) By Fact 3.1 the ring R has depth 0. If M is not R-free, then pdR M = ∞.
Theorem 3.6 implies that m is a direct summand of Ω3R M or Ω4R M . Lemma 6.1(1)
now implies that pdR N < ∞, and N is R-free.
(2) By Fact 3.1 the ring R has depth ≤ 1. Theorem 3.6 implies that m is a direct
summand of Ω3R M ⊕ Ω4R M or Ω5R M . Assume that pdR M ≥ 2. Note that ℓ ≥ 3 + 1
and ℓ + 1 ≥ 5 + 1. It is observed from (1a) and (2a) in Lemma 6.1 that pdR N < ∞.
Hence, pdR N ≤ 1.

An analogous argument to the proof of Corollary 6.2 yields its Ext version. Note
that this highly refines the Ext vanishing results in [14].
Corollary 6.3. Assume that R = S ×k T is a fiber product, where S and T are local
rings with common residue field k and S 6= k 6= T . Let M and N be R-modules.
(1) If S or T has depth 0 and ExtℓR (M, N ) = 0 for some ℓ ≥ 4+max{1, depthR N },
then M is R-free or N is R-injective.
(2) If ExtℓR (M, N ) = Extℓ+1
R (M, N ) = 0 for some ℓ ≥ 4 + max{1, depthR N }, then
pdR M ≤ 1 or idR N ≤ 1.
To get our next applications, we prepare a lemma.
Lemma 6.4. Let x = x1 , . . . , xn be an R-sequence such that m/(x) is decomposable.
Then n is equal to either depth R − 1 or depth R.
Proof. By Fact 3.1 we have depth R/(x) ≤ 1. This shows the assertion.



16

SAEED NASSEH AND RYO TAKAHASHI

The following two results are regarded as generalizations of Corollaries 6.2 and
6.3 to a Cohen-Macaulay local ring with quasi-decomposable maximal ideal.
Corollary 6.5. Let R be a d-dimensional Cohen-Macaulay local ring such that
m/(x) is decomposable for some R-sequence x = x1 , . . . , xn . Let M and N be Rmodules, and set s := 2d−depthR M −depthR N (≥ 0). Suppose that there exists an
integer t ≥ max{5, n+1} such that TorR
i (M, N ) = 0 for all t+s−n ≤ i ≤ t+s−n+d.
Then pdR M < ∞ or pdR N < ∞.
Proof. Let a := d − depthR M and b := d − depthR N and note that t − n > 0.
We see that X := Ωa M and Y := Ωb N are maximal Cohen-Macaulay and that
TorR
i (X, Y ) = 0 for all t − n ≤ i ≤ t − n + d. Using the long exact sequences arising
from the short exact sequences
xj

0 → Y /(x1 , . . . , xj−1 )Y −→ Y /(x1 , . . . , xj−1 )Y → Y /(x1 , . . . , xj )Y → 0
for 1 ≤ j ≤ n, we observe that TorR
i (X, Y /xY ) = 0 for all t ≤ i ≤ t − n + d.
R/(x)
(X/xX, Y /xY ) = 0 for all
Applying [11, §18, Lemma 2(3)], we obtain Tori
t ≤ i ≤ t − n + d.
According to Lemma 6.4, the integer n is either d or d − 1. If n = d, then
R/(x)
Tort
(X/xX, Y /xY ) = 0 and R/(x) has depth zero. Fact 3.1 and Corollary
6.2(1) imply that either X/xX or Y /xY is R/(x)-free. If n = d − 1, then we have
R/(x)
R/(x)
Tort
(X/xX, Y /xY ) = Tort+1 (X/xX, Y /xY ) = 0. Fact 3.1 and Corollary
6.2(2) imply that pdR/(x) (X/xX) ≤ 1 or pdR/(x) (Y /xY ) ≤ 1. Hence, pdR X < ∞
or pdR Y < ∞ by [2, Lemma 1.3.5]. Therefore, pdR M < ∞ or pdR N < ∞.

Corollary 6.6. Let R be a d-dimensional Cohen-Macaulay local ring such that
m/(x) is decomposable for some R-sequence x = x1 , . . . , xn . Let M and N be Rmodules, and set s := d − depth M (≥ 0). Suppose that there exists an integer t ≥ 5
such that ExtiR (M, N ) = 0 for all t + s ≤ i ≤ t + s + d. Then pdR M < ∞ or
idR N < ∞.
Proof. Without loss of generality we may assume that R is complete, so that R admits a canonical module. Let L := ΩsR M and note then that L is maximal CohenMacaulay. We now have ExtiR (L, N ) = 0 for all t ≤ i ≤ t + d. By virtue of [1,
Theorem 1.1], we get an exact sequence 0 → Y → X → N → 0 in which X is maximal Cohen-Macaulay and Y has finite injective dimension. Since Extm
R (L, Y ) = 0
for all m > 0, we get ExtiR (L, X) = 0 for all t ≤ i ≤ t + d. Using the long exact
sequences arising from the short exact sequences
xj

0 → L/(x1 , . . . , xj−1 )L −→ L/(x1 , . . . , xj−1 )L → L/(x1 , . . . , xj )L → 0
for 1 ≤ j ≤ n, we see that ExtiR (L/xL, X) = 0 for all t + n ≤ i ≤ t + d. It follows
from [11, §18, Lemma 2(i)] that ExtiR/(x) (L/xL, X/xX) = 0 for all t ≤ i ≤ t+d−n.
Lemma 6.4 says that n = d or n = d − 1, and note that
5 = 4 + 1 = 4 + max{1, depthR/(x) X/xX}.
When n = d, the ring R/(x) has depth zero and we have ExttR/(x) (L/xL, X/xX) =
0. Combining Fact 3.1 and Corollary 6.3(1), we conclude that L/xL is R/(x)-free
or X/xX is R/(x)-injective. When n = d − 1, we have ExttR/(x) (L/xL, X/xX) =
Extt+1
R/(x) (L/xL, X/xX) = 0. In this case, Fact 3.1 and Corollary 6.3(2) imply that

LOCAL RINGS WITH QUASI-DECOMPOSABLE MAXIMAL IDEAL

17

pdR/(x) (L/xL) ≤ 1 or idR/(x) (X/xX) ≤ 1. Therefore, pdR L < ∞ or idR X < ∞;
see [2, Lemma 1.3.5 and Corollary 3.1.15]. Thus, pdR M < ∞ or idR N < ∞. 
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Department of Mathematical Sciences, Georgia Southern University, Statesboro,
Georgia 30460, U.S.A.
E-mail address: snasseh@georgiasouthern.edu
URL: https://cosm.georgiasouthern.edu/math/saeed.nasseh
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
E-mail address: takahashi@math.nagoya-u.ac.jp
URL: http://www.math.nagoya-u.ac.jp/~takahashi/

