arXiv:1711.00652v1 [] 2 Nov 2017

ULRICH MODULES OVER COHEN–MACAULAY LOCAL RINGS
WITH MINIMAL MULTIPLICITY
TOSHINORI KOBAYASHI AND RYO TAKAHASHI
Abstract. Let R be a Cohen–Macaulay local ring. In this paper we study the structure of Ulrich
R-modules mainly in the case where R has minimal multiplicity. We explore generation of Ulrich Rmodules, and clarify when the Ulrich R-modules are precisely the syzygies of maximal Cohen–Macaulay
R-modules. We also investigate the structure of Ulrich R-modules as an exact category.

Introduction
The notion of an Ulrich module, which is also called a maximally generated (maximal) Cohen–Macaulay
module, has first been studied by Ulrich [30], and widely investigated in both commutative algebra and
algebraic geometry; see [2, 4, 5, 9, 10, 12, 20, 23] for example. A well-known conjecture asserts that
Ulrich modules exist over any Cohen–Macaulay local ring R. Even though the majority seem to believe
that this conjecture does not hold true in full generality, a lot of partial (positive) solutions have been
obtained so far. One of them states that the conjecture holds whenever R has minimal multiplicity ([2]).
Thus, in this paper, mainly assuming that R has minimal multiplicity, we are interested in what we can
say about the structure of Ulrich R-modules.
We begin with exploring the number and generation of Ulrich modules. The following theorem is a
special case of our main results in this direction (Ω denotes the first syzygy).
Theorem A. Let (R, m, k) be a d-dimensional complete Cohen–Macaulay local ring.
(1) Assume that R is normal with d = 2 and k = C and has minimal multiplicity. If R does not have a
rational singularity, then there exist infinitely many indecomposable Ulrich R-modules.
(2) Suppose that R has an isolated singularity. Let M, N be maximal Cohen–Macaulay R-modules with
ExtiR (M, N ) = 0 for all 1 6 i 6 d − 1. If either M or N is Ulrich, then so is HomR (M, N ).
(3) Let x = x1 , . . . , xd be a system of parameters of R such that m2 = xm. If M is an Ulrich R-module,
then so is Ω(M/xi M ) for all 1 6 i 6 d. If one chooses M to be indecomposable and not to be a direct
summand of Ωd k, then one finds an indecomposable Ulrich R-module not isomorphic to M among
the direct summands of the modules Ω(M/xi M ).
Next, we relate the Ulrich modules with the syzygies of maximal Cohen–Macaulay modules. To state
our result, we fix some notation. Let R be a Cohen–Macaulay local ring with canonical module ω. We
denote by mod R the category of finitely generated R-modules, and by Ul(R) and ΩCM× (R) the full
subcategories of Ulrich modules and first syzygies of maximal Cohen–Macaulay modules without free
summands, respectively. Denote by (−)† the canonical dual HomR (−, ω). Then Ul(R) is closed under
(−)† , and contains ΩCM× (R) if R has minimal multiplicity. The module Ωd k belongs to ΩCM× (R), and
hence Ωd k, (Ωd k)† belong to Ul(R). Thus it is natural to ask when the conditions in the theorem below
hold, and we actually answer this.
Theorem B. Let R be a d-dimensional singular Cohen–Macaulay local ring with residue field k and
canonical module ω, and assume that R has minimal multiplicity. Consider the following conditions.
(1) The equality Ul(R) = ΩCM× (R) holds.
(2) The category ΩCM× (R) is closed under (−)† .
(3) The module (Ωd k)† belongs to ΩCM× (R).
(4) One has Tor1 (Tr(Ωd k)† , ω) = 0.
2010 Mathematics Subject Classification. 13C14, 13D02, 13H10.
Key words and phrases. Ulrich module, Cohen–Macaulay ring/module, minimal multiplicity, syzygy.
The second author was partly supported by JSPS Grant-in-Aid for Scientific Research 16H03923 and 16K05098.
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TOSHINORI KOBAYASHI AND RYO TAKAHASHI

d †
(5) One has Extd+1
R (Tr(Ω k) , R) = 0 and R is locally Gorenstein on the punctured spectrum.
(6) There is an epimorphism ω ⊕n → Ωd k for some n > 0.
(7) There is an isomorphism Ωd k ∼
= (Ωd k)† .
(8) The local ring R is almost Gorenstein.
Then (1)–(6) are equivalent and (7) implies (1). If d > 0 and k is infinite, then (1) implies (8). If d = 1
and k is infinite, then (1)–(8) are equivalent. If R is complete normal with d = 2 and k = C, then (1)–(7)
are equivalent unless R has a cyclic quotient singularity.

Finally, we study the structure of the category Ul(R) of Ulrich R-modules as an exact category in the
sense of Quillen [25]. We prove that if R has minimal multiplicity, then Ul(R) admits an exact structure
with enough projective/injective objects.
Theorem C. Let R be a d-dimensional Cohen–Macaulay local ring with residue field k and canonical
module, and assume that R has minimal multiplicity. Let S be the class of short exact sequences 0 → L →
M → N → 0 of R-modules with L, M, N Ulrich. Then (Ul(R), S) is an exact category having enough
projective objects and enough injective objects with proj Ul(R) = add Ωd k and inj Ul(R) = add(Ωd k)† .
The organization of this paper is as follows. In Section 1, we deal with a question of Cuong on the
number of indecomposable Ulrich modules. We prove the first assertion of Theorem A to answer this
question in the negative. In Section 2, we consider how to generate Ulrich modules from given ones,
and prove the second and third assertions of Theorem A. In Section 3, we compare Ulrich modules with
syzygies of maximal Cohen–Macaulay modules, and prove Theorem B; in fact, we obtain more equivalent
and related conditions. The final Section 4 is devoted to giving applications of the results obtained in
Section 3. In this section we study the cases of dimension one and two, and exact structures of Ulrich
modules, and prove the rest assertions of Theorem B and Theorem C.
Convention
Throughout, let (R, m, k) be a Cohen–Macaulay local ring of Krull dimension d. We assume that all
modules are finitely generated and all subcategories are full. A maximal Cohen–Macaulay module is
simply called a Cohen–Macaulay module. For an R-module M we denote by ΩM the first syzygy of M ,
that is, the kernel of the first differential map in the minimal free resolution of M . Whenever R admits
a canonical module ω, we denote by (−)† the canonical dual functor HomR (−, ω). For an R-module M
we denote by e(M ) and µ(M ) the multiplicity and the minimal number of generators of M , respectively.
1. A question of Cuong
In this section, we consider a question raised by Cuong [6] on the number of Ulrich modules over
Cohen–Macaulay local rings with minimal multiplicity. First of all, let us recall the definitions of an
Ulrich module and minimal multiplicity.
Definition 1.1. (1) An R-module M is called Ulrich if M is Cohen–Macaulay with e(M ) = µ(M ).
(2) The ring R is said to have minimal multiplicity if e(R) = edim R − dim R + 1.
An Ulrich module is also called a maximally generated (maximal) Cohen–Macaulay module. There is
always an inequality e(R) > edim R − dim R + 1, from which the name of minimal multiplicity comes. If
k is infinite, then R has minimal multiplicity if and only if m2 = Qm for some parameter ideal Q of R.
See [3, Exercise 4.6.14] for details of minimal multiplicity.
The following question has been raised by Cuong [6].
Question 1.2 (Cuong). If R is non-Gorenstein and has minimal multiplicity, then are there only finitely
many indecomposable Ulrich R-modules?
To explore this question, we start by introducing notation, which is used throughout the paper.
Notation 1.3. We denote by mod R the category of finitely generated R-modules. We use the following
subcategories of mod R:
CM(R) = {M ∈ mod R | M is Cohen–Macaulay},
Ul(R) = {M ∈ CM(R) | M is Ulrich},

ULRICH MODULES AND MINIMAL MULTIPLICITY

3



M is the kernel of an epimorphism from a
,
ΩCM(R) = M ∈ CM(R)
free module to a Cohen–Macaulay module
ΩCM× (R) = {M ∈ ΩCM(R) | M does not have a (nonzero) free summand}.
Remark 1.4. (1) The subcategories CM(R), Ul(R), ΩCM(R), ΩCM× (R) of mod R are closed under finite direct sums and direct summands.
(2) One has ΩCM(R) ∪ Ul(R) ⊆ CM(R) ⊆ mod R.
Here we make a remark to reduce to the case where the residue field is infinite.
Remark 1.5. Consider the faithfully flat extension S := R[t]mR[t] of R. Then we observe that:
(1) If X is a module in ΩCM× (R), then X ⊗R S is in ΩCM× (S).
(2) A module Y is in Ul(R) if and only if Y ⊗R S is in Ul(S) (see [13, Lemma 6.4.2]).
The converse of (1) also holds true; we prove this in Corollary 3.4.
If R has minimal multiplicity, then all syzygies of Cohen–Macaulay modules are Ulrich:
Proposition 1.6. Suppose that R has minimal multiplicity. Then ΩCM× (R) is contained in Ul(R).
Proof. By Remark 1.5 we may assume that k is infinite. Since R has minimal multiplicity, we have
m2 = Qm for some parameter ideal Q of R. Let M be a Cohen–Macaulay R-module. There is a short
exact sequence 0 → ΩM → R⊕n → M → 0, where n is the minimal number of generators of M . Since M
is Cohen–Macaulay, taking the functor R/Q ⊗R − preserves the exactness; we get a short exact sequence
f

0 → ΩM/QΩM −
→ (R/Q)⊕n → M/QM → 0.
The map f factors through the inclusion map X := m(R/Q)⊕n → (R/Q)⊕n , and hence there is an
injection ΩM/QΩM → X. As X is annihilated by m, so is ΩM/QΩM . Therefore mΩM = QΩM , which
implies that ΩM is Ulrich.

As a direct consequence of [7, Corollary 3.3], we obtain the following proposition.
Proposition 1.7. Let R be a 2-dimensional normal excellent henselian local ring with algebraically closed
residue field of characteristic 0. Then there exist only finitely many indecomposable modules in ΩCM(R)
if and only if R has a rational singularity.
Combining the above propositions yields the following result.
Corollary 1.8. Let R be a 2-dimensional normal excellent henselian local ring with algebraically closed
residue field of characteristic 0. Suppose that R has minimal multiplicity and does not have a rational
singularity. Then there exist infinitely many indecomposable Ulrich R-modules. In particular, Quenstion
1.2 has a negative answer.
Proof. Proposition 1.7 implies that ΩCM(R) contains infinitely many indecomposable modules, and so
does Ul(R) by Proposition 1.6.

Here is an example of a non-Gorenstein ring satisfying the assumption of Corollary 1.8, which concludes
that the question of Cuong is negative.
Example 1.9. Let B = C[x, y, z,
ring with deg x = deg t = 3, deg y = 5 and deg z = 7.
 xt]ybe za polynomial

Consider the 2 × 3-matrix M = y z x3 −t3 over B, and let I be the ideal of B generated by 2 × 2-minors

of M . Set A = B/I. Then A is a nonnegatively graded C-algebra as I is homogeneous. By virtue of
the Hilbert–Burch theorem ([3, Theorem 1.4.17]), A is a 2-dimensional Cohen–Macaulay ring, and x, t
is a homogeneous system of parameters of A. Directly calculating the Jacobian ideal J of A, we can
verify that A/J is Artinian. The Jacobian criterion implies that A is a normal domain. The quotient
ring A/tA is isomorphic to the numerical semigroup ring C[H] with H = h3, 5, 7i. Since this ring is not
Gorenstein (as H is not symmetric), neither is A. Let a(A) and F (H) stand for the a-invariant of A and
the Frobenius number of H, respectively. Then
a(A) + 3 = a(A) + deg(t) = a(A/tA) = F (H) = 4,

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TOSHINORI KOBAYASHI AND RYO TAKAHASHI

where the third equality follows from [26, Theorem 3.1]. Therefore we get a(A) = 1 6< 0, and A does not
have a rational singularity by the Flenner–Watanabe criterion (see [21, Page 98]).
Let A′ be the localization of A at A+ , and let R be the completion of the local ring A′ . Then R
is a 2-dimensional complete (hence excellent and henselian) normal non-Gorenstein local domain with
residue field C. The maximal ideal m of R satisfies m2 = (x, t)m, and thus R has minimal multiplicity.
Having a rational singularity is preserved by localization since A has an isolated singularity, while it is
also preserved by completion. Therefore R does not have a rational singularity.
We have seen that Question 1.2 is not true in general. However, in view of Corollary 1.8, we wonder
if having a rational singularity is essential. Thus, we pose a modified question.
Question 1.10. Let R be a 2-dimensional normal local ring with a rational singularity. Then does R
have only finitely many indecomposable Ulrich modules?
Proposition 1.7 leads us to an even stronger question:
Question 1.11. If ΩCM(R) contains only finitely many indecomposable modules, then does Ul(R) so?
2. Generating Ulrich modules
In this section, we study how to generate Ulrich modules from given ones. First of all, we consider
using the Hom functor to do it.
Proposition 2.1. Let M, N be Cohen–Macaulay R-modules. Suppose that on the punctured spectrum of
R either M is locally of finite projective dimension or N is locally of finite injective dimension.
(1) ExtiR (M, N ) = 0 for all 1 6 i 6 d − 2 if and only if HomR (M, N ) is Cohen–Macaulay.
(2) Assume ExtiR (M, N ) = 0 for all 1 6 i 6 d − 1. If either M or N is Ulrich, then so is HomR (M, N ).
Proof. (1) This follows from the proof of [14, Proposition 2.5.1]; in it the isolated singularity assumption
is used only to have that the Ext modules have finite length.
(2) By (1), the module HomR (M, N ) is Cohen–Macaulay. We may assume that k is infinite by Remark
1.5(2), so that we can find a reduction Q of m which is a parameter ideal of R.
First, let us consider the case where N is Ulrich. Take a minimal free resolution F = (· · · → F1 →
F0 → 0) of M . Since ExtiR (M, N ) = 0 for all 1 6 i 6 d − 1, the induced sequence
f

0 → HomR (M, N ) → HomR (F0 , N ) −
→ · · · → HomR (Fd−1 , N ) → HomR (Ωd M, N ) → ExtdR (M, N ) → 0
is exact. Note that ExtdR (M, N ) has finite length. By the depth lemma, the image L of the map f is
Cohen–Macaulay. An exact sequence 0 → HomR (M, N ) → HomR (F0 , N ) → L → 0 is induced, and the
application of the functor − ⊗R R/Q to this gives rise to an injection
HomR (M, N ) ⊗R R/Q ֒→ HomR (F0 , N ) ⊗R R/Q.
Since N is Ulrich, the module HomR (F0 , N )⊗R R/Q is annihilated by m, and so is HomR (M, N )⊗R R/Q.
Therefore HomR (M, N ) is Ulrich.
Next, we consider the case where M is Ulrich. As x is an M -sequence, there is a spectral sequence
E2pq = ExtpR (R/Q, ExtqR (M, N )) =⇒ H p+q = Extp+q
R (M/QM, N ).
The fact that x is an R-sequence implies E2pq = 0 for p > d. By assumption, E2pq = 0 for 1 6 q 6 d − 1.
Hence an exact sequence 0 → E2d0 → H d → E20d → 0 is induced. Since M/QM is annihilated by m, so is
H d = ExtdR (M/QM, N ), and so is E2d0 . Note that
∼ Hd (x, HomR (M, N )) ∼
E d0 = Extd (R/Q, HomR (M, N )) =
= HomR (M, N ) ⊗R R/Q,
2

R

where H∗ (x, −) stands for the Koszul cohomology. If follows that m kills HomR (M, N ) ⊗R R/Q, which
implies that HomR (M, N ) is Ulrich.

As an immediate consequence of Proposition 2.1(2), we obtain the following corollary, which is a special
case of [9, Theorem 5.1].
Corollary 2.2. Suppose that R admits a canonical module. If M ∈ Ul(R), then M † ∈ Ul(R).
Next, we consider taking extensions of given Ulrich modules to obtain a new one.

ULRICH MODULES AND MINIMAL MULTIPLICITY

5

Proposition 2.3. Let Q be a parameter ideal of R which is a reduction of m. Let M, N be Ulrich Rmodules, and take any element a ∈ Q. Let σ : 0 → M → E → N → 0 be an exact sequence, and consider
the multiplication aσ : 0 → M → X → N → 0 as an element of the R-module Ext1R (N, M ). Then X is
an Ulrich R-module.
Proof. It follows from [27, Theorem 1.1] that the exact sequence
aσ ⊗R R/aR : 0 → M/aM → X/aX → N/aN → 0
∼ M/aM ⊕ N/aN . Applying the functor − ⊗R/aR R/Q, we get
splits; we have an isomorphism X/aX =
∼ M/QM ⊕ N/QN . Since M, N are Ulrich, the modules M/QM, N/QN are
an isomorphism X/QX =
k-vector spaces, and so is X/QX. Hence X is also Ulrich.

As an application of the above proposition, we give a way to make an Ulrich module over a Cohen–
Macaulay local ring with minimal multiplicity.
Corollary 2.4. Let Q be a parameter ideal of R such that m2 = Qm. Let M be an Ulrich R-module.
Then for each R-regular element a ∈ Q, the syzygy Ω(M/aM ) is also an Ulrich R-module.
Proof. There is an exact sequence σ : 0 → ΩM → R⊕n → M → 0, where n is a minimal number of
generators of M . We have a commutative diagram

σ:

0

aσ :

0

/ ΩM
/ ΩM

0O

0O

M/aM
O

M/aM
O

/ R⊕n
O

/M
O

/0

/X
O

/M
O

/0

0

0

a

with exact rows and columns. Since the minimal number of generators of M/aM is equal to n, the middle
column shows X ∼

= Ω(M/aM ). Propositions 1.2 and 2.3 show that X is Ulrich, and we are done.
Remark 2.5. In Corollary 2.4, if the parameter ideal Q annihilates the R-module Ext1R (M, ΩM ), then
we have aσ = 0, and Ω(M/aM ) ∼
= M ⊕ ΩM . Hence, in this case, the operation M 7→ Ω(M/aM ) does
not produce an essentially new Ulrich module.
Next, we investigate the annihilators of Tor and Ext modules.
Proposition 2.6. For an R-module M one has
annR Ext1R (M, ΩM ) =
= annR TorR
1 (M, TrM ) =
Proof. It is clear that
I :=
J :=

T

i>0, N ∈mod R

T

i>0, N ∈mod R

T

T

i>0, N ∈mod R

annR ExtiR (M, N )

i>0, N ∈mod R

annR TorR
i (M, N ).

annR ExtiR (M, N ) ⊆ annR Ext1R (M, ΩM )
R
annR TorR
i (M, N ) ⊆ annR Tor1 (M, TrM ).

It is enough to show that ann Ext1 (M, ΩM ) ∪ ann Tor1 (M, TrM ) is contained in I ∩ J.
(1) Take any element a ∈ annR Ext1R (M, ΩM ). The proof of [16, Lemma 2.14] shows that the multia

f

a

π

plication map (M −
→ M ) factors through a free module, that is, (M −
→ M ) = (M −
→F −
→ M ) with F
free. Hence, for all i > 0 and N ∈ mod R we have commutative diagrams:
a

/ Tori (M, N )
Tori (M, N )
◆◆◆
7
◆◆◆
♣♣♣
Tori (f,N )
'
♣♣♣Tori (π,N )
Tori (F, N )

Exti (M, N )
◆◆◆
◆◆&
i
Ext (π,N )

As Tori (F, N ) = Exti (F, N ) = 0, the element a is in I ∩ J.

a

/ Exti (M, N )
♣♣8
♣♣♣ i

Exti (F, N )

Ext (f,N )

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TOSHINORI KOBAYASHI AND RYO TAKAHASHI

(2) Let b ∈ annR TorR
1 (M, TrM ). By [32, Lemma (3.9)], the element b annihilates HomR (M, M ).
b
Hence the map b · idM , which is nothing but the multiplication map (M −
→ M ), factors through a free
R-module. Similarly to (1), we get b is in I ∩ J.

Definition 2.7. We denote by annh M the ideal in the above proposition.
Note that annh M = R if and only if M is a free R-module.
For an R-module M we denote by add M the subcategory of mod R consisting of direct summands of
finite direct sums of copies of M .
With the notation of Remark 2.5, we are interested in when the operation M 7→ Ω(M/aM ) actually
gives rise to an essentially new Ulrich module. The following result presents a possible way: if we choose
an indecomposable Ulrich module M that is not a direct summand of Ωd k, then we find an indecomposable
Ulrich module not isomorphic to M among the direct summands of the modules Ω(M/xi M ).
Proposition 2.8. Suppose that R is henselian. Let Q = (x1 , . . . , xd ) be a parameter ideal of R which is a
reduction of m. Let M be an indecomposable Ulrich R-module. If M is a direct summand of Ω(M/xi M )
for all 1 6 i 6 d, then M is a direct summand of Ωd k.
Proof. For all integer 1 6 i 6 d the module Ext1R (M, ΩM ) is a direct summand of Ext1R (Ω(M/xi M ), ΩM ).
The latter module is annihilated by xi since it is isomorphic to Ext2R (M/xi M, ΩM ). Hence Q is contained
in annR Ext1R (M, ΩM ) = annh M , and therefore Q Ext>0
R (M, N ) = 0 for all N ∈ mod R. It follows from
[29, Corollary 3.2(1)] that M is a direct summand of Ωd (M/QM ). As M is Ulrich, the module M/QM
is a k-vector space, and Ωd (M/QM ) belongs to add(Ωd k), whence so does M . Since R is henselian and
M is indecomposable, the Krull–Schmidt theorem implies that M is a direct summand of Ωd k.

3. Comparison of Ul(R) with ΩCM× (R)
In this section, we study the relationship of the Ulrich R-modules with the syzygies of Cohen–Macaulay
R-modules. We begin with giving equivalent conditions for a given Cohen–Macaulay module to be a
syzygy of a Cohen–Macaulay module, after stating an elementary lemma.
Lemma 3.1. Let M, N be R-modules. The evaluation map ev : M ⊗R HomR (M, N ) → N is surjective
if and only if there exists an epimorphism (f1 , . . . , fn ) : M ⊕n → N .
Proof. The “only if” part follows by taking an epimorphism R⊕n → HomR (M, N ) and tensoring M .
To show the “if” part, pick any element
y ∈ N . Then we have y = f1 (x1 ) + · · · + fn (xn ) for some
Pn
x1 , . . . , xn ∈ M . Therefore y = ev( i=1 xi ⊗ fi )), and we are done.

Proposition 3.2. Let R be a Cohen–Macaulay local ring with canonical module ω. Then the following
are equivalent for a Cohen–Macaulay R-module M .

(1)
(2)
(3)
(4)
(5)
(6)
(7)

M ∈ ΩCM(R).
HomR (M, ω) = 0.
There exists a surjective homomorphism ω ⊕n → HomR (M, ω).
The natural homomorphism Φ : ω ⊗R HomR (ω, HomR (M, ω)) → HomR (M, ω) is surjective.
M is torsionless and TrΩTrM is Cohen–Macaulay.
Ext1R (TrM, R) = Ext1R (TrΩTrM, ω) = 0.
TorR
1 (TrM, ω) = 0.

Proof. (1) ⇒ (2): By the assumption, there is an exact sequence 0 → M → F → N → 0 such that N is
Cohen–Macaulay and F is free. Take f ∈ HomR (M, ω). There is a commutative diagram
0

/M

0

/ ω

f

/F

/W

/N

/0

/N

/0

with exact rows. Since N is Cohen–Macaulay, we have Ext1R (N, ω) = 0. Hence the second row splits,
and f factors through F . This shows HomR (M, ω) = 0.

ULRICH MODULES AND MINIMAL MULTIPLICITY

7

f

(2) ⇒ (1): There is an exact sequence 0 → M −
→ ω ⊕m → N → 0 such that N is Cohen–Macaulay.
⊕m
⊕m
Since HomR (M, ω ) = HomR (M, ω)
= 0, there are a free R-module F , homomorphisms g : M → F
and h : F → ω ⊕m such that f = hg. We get a commutative diagram
0

/M

0

/M

g

f

/F
h
/ ω ⊕m

/L

/0


/N

/0

with exact rows. The secound square is a pullback-pushout diagram, which gives an exact sequence
0 → F → L ⊕ ω ⊕m → N → 0. This shows that L is Cohen–Macaulay, and hence M ∈ ΩCM(R).
(2) ⇔ (7): This equivalence follows from [32, Lemma (3.9)].
(1) ⇒ (3): Let 0 → M → R⊕n → N → 0 be an exact sequence with F free. Applying (−)† , we have
an exact sequence 0 → N † → ω ⊕n → M † → 0.
(3) ⇒ (1): There is an exact sequence 0 → K → ω ⊕n → M † → 0. It is seen that K is Cohen–Macaulay.
Taking (−)† gives an exact sequence 0 → M → R⊕n → K † → 0, which shows M ∈ ΩCM(R).
(3) ⇔ (4): This follows from Lemma 3.1.
(5) ⇔ (6): The module TrΩTrM is Cohen–Macaulay if and only if ExtiR (TrΩTrM, ω) = 0 for all i > 0.
One has Ext1R (TrM, R) = 0 if and only if M is torsionless, if and only if M ∼
= ΩTrΩTrM up to free
summands; see [1, Theorem (2.17)]. Hence ExtiR (TrΩTrM, ω) = Exti−1
R (M, ω) = 0 for all i > 1.
(1) ⇔ (5): This equivalence follows from [18, Lemma 2.5] and its proof.

Remark 3.3. The equivalence (1) ⇔ (5) in Proposition 3.2 holds without the assumption that R admits
a canonical module. Indeed, its proof does not use the existence of a canonical module.
The property of being a syzygy of a Cohen–Macaulay module (without free summand) is preserved
under faithfully flat extension.
Corollary 3.4. Let R → S be a faithfully flat homomorphism of Cohen–Macaulay local rings. Let M be
a Cohen–Macaulay R-module. Then M ∈ ΩCM× (R) if and only if M ⊗R S ∈ ΩCM× (S).
Proof. Using Remark 3.3, we see that M ∈ ΩCM(R) if and only if Ext1R (TrR M, R) = 0 and TrR ΩR TrR M
is Cohen–Macaulay. Also, M has a nonzero R-free summand if and only if the evaluation map M ⊗R
HomR (M, R) → R is surjective by Lemma 3.1. Since the latter conditions are both preserved under
faithfully flat extension, they are equivalent to saying that M ⊗R S ∈ ΩCM(S) and that M ⊗R S has a
nonzero S-free summand, respectively. Now the assertion follows.

Next we state and prove a couple of lemmas. The first one concerns Ulrich modules and syzygies of
Cohen–Macaulay modules with respect to short exact sequences.
Lemma 3.5. Let 0 → L → M → N → 0 be an exact sequence of R-modules.
(1) If L, M, N are in Ul(R), then the equality µ(M ) = µ(L) + µ(N ) holds.
(2) Suppose that L, M, N are in CM(R). Then:
(a) If M is in Ul(R), then so are L and N .
(b) If M is in ΩCM× (R), then so is L.
Proof. (1) We have µ(M ) = e(M ) = e(L) + e(N ) = µ(L) + µ(N ).
(2) Assertion (a) follows by [2, Proposition (1.4)]. Let us show (b). As M is in ΩCM× (R), there is an
β

γ

exact sequence 0 → M −
→ R⊕a −
→ C → 0 with C Cohen–Macaulay. As M has no free summand, γ is a
minimal homomorphism. In particular, µ(C) = a. The pushout of β and γ gives a commutative diagram

0

/L

0

/L

0

/M
β
/ R⊕a
γ

δ

0

/N

/0


/D

/0


C


C


0


0

8

TOSHINORI KOBAYASHI AND RYO TAKAHASHI

with exact rows and columns. We see that a = µ(C) 6 µ(D) 6 a, which implies that δ is a minimal
homomorphism. Hence L = ΩD ∈ ΩCM× (R).

The following lemma is used to reduce to the case of a lower dimensional ring.
Lemma 3.6. Let Q = (x1 , . . . , xd ) be a parameter ideal of R that is a reduction of m. Let M be a Cohen–
Macaulay R-module. Then M is an Ulrich R-module if and only if M/xi M is an Ulrich R/xi R-module.
Proof. Note that Q/xi R is a reduction of m/xi R. We see that (m/xi R)(M/xi M ) = (Q/xi R)(M/xi M )
if and only if mM = QM . Thus the assertion holds.

Now we explore syzygies of the residue field of a Cohen–Macaulay local ring with minimal multiplicity.
Lemma 3.7. Assume that R is singular and has minimal multiplicity.
(1) One has ΩdR k ∈ ΩCM× (R). In particular, ΩdR k is an Ulrich R-module.
∼ d ⊕n for some n > 0.
(2) There is an isomorphism Ωd+1
R k = (ΩR k)
(3) Let Q = (x1 , . . . , xd ) be a parameter ideal of R with m2 = Qm, and suppose that d > 1. Then
d−1
1
∼ d
Ω1R (ΩiR/(x1 ) k) ∼
= Ωi+1
R k for all i > 0. In particular, ΩR (ΩR/(x1 ) k) = ΩR k.
(4) For each M ∈ Ul(R) there exists a surjective homomorphism (ΩdR k)⊕n → M for some n > 0.
Proof. (1)(2) We may assume that k is infinite; see Remark 1.5. So we find a parameter ideal Q =
(x1 , . . . , xd ) of R with m2 = Qm. The module m/Q is a k-vector space, and there is an exact sequence
0 → k ⊕n → R/Q → k → 0. Taking the dth syzygies gives an exact sequence
0 → (Ωd k)⊕n → R⊕t → Ωd k → 0.
Since Ωd k has no free summand by [28, Theorem 1.1], we obtain Ωd k ∈ ΩCM× (R) and (Ωd k)⊕n ∼
= Ωd+1 k.
The last assertion of (1) follows from this and Proposition 1.6.
(3) Set x = x1 . We show that Ω(ΩiR/xR k) ∼
= Ωi+1 k for all i > 0. We may assume i > 1; note then that
i
x is Ω k-regular. By [28, Corollary 5.3] we have an isomorphism Ωi k/xΩi k ∼
= ΩiR/xR k ⊕ Ωi−1
R/xR k. Hence
(3.7.1)

Ωi k ⊕ Ωi+1 k ∼
= Ω(Ωi k/xΩi k) ∼
= Ω(ΩiR/xR k) ⊕ Ω(Ωi−1
R/xR k),

where the first isomorphism follows from the proof of Corollary 2.4. There is an exact sequence 0 →
ΩiR/xR k → (R/xR)⊕ai−1 → · · · → (R/xR)⊕a0 → k → 0 of R/xR-modules, which gives an exact sequence
0 → Ω(ΩiR/xR k) → R⊕bi−1 → · · · → R⊕b0 → Ωk → 0
of R-modules. This shows Ω(ΩiR/xR k) ∼
= Ωi+1 k ⊕ R⊕u for some u > 0, and similarly we have an
⊕v
∼ i
isomorphism Ω(Ωi−1
for some v > 0. Substituting these in (3.7.1), we see u = v = 0
R/xR k) = Ω k ⊕ R
i
and obtain an isomorphism Ω(ΩR/xR k) ∼
= Ωi+1 k.
(4) According to Lemma 3.1 and Remark 1.5, we may assume that k is infinite. Take a parameter
ideal Q = (x1 , . . . , xd ) of R with m2 = Qm. We prove this by induction on d. If d = 0, then M is a
k-vector space, and there is nothing to show. Assume d > 1 and set x = x1 . Clearly, R/xR has minimal
multiplicity. By Lemma 3.6, M/xM is an Ulrich R/xR-module. The induction hypothesis gives an exact
⊕n
sequence 0 → L → (Ωd−1
→ M/xM → 0 of R/xR-modules. Lemma 3.5(2) shows that L is also
R/xR k)
an Ulrich R/xR-module, while Lemma 3.5(1) implies
⊕n
µR/xR (L) + µR/xR (M/xM ) = µR/xR ((Ωd−1
).
R/xR k)

Note that µR (X) = µR/xR (X) for an R/xR-module X. Thus, taking the first syzygies over R, we get an
exact sequence of R-modules:
⊕n
→ Ω(M/xM ) → 0.
0 → ΩL → Ω(Ωd−1
R/xR k)

From the proof of Corollary 2.4 we see that there is an exact sequence 0 → ΩM → Ω(M/xM ) → M → 0,
d
d ⊕n
while Ω(Ωd−1
→ M. 
R/xR k) is isomorphic to Ω k by (3). Consequently, we obtain a surjection (Ω k)
We have reached the stage to state and prove the main result of this section.
Theorem 3.8. Let R be a d-dimensional Cohen–Macaulay local ring with residue field k and canonical
module ω. Suppose that R has minimal multiplicity. Then the following are equivalent.

ULRICH MODULES AND MINIMAL MULTIPLICITY

9

(1) The equality ΩCM× (R) = Ul(R) holds.
(2) For an exact sequence M → N → 0 in CM(R), if M ∈ ΩCM× (R), then N ∈ ΩCM× (R).
(3) The category ΩCM× (R) is closed under (−)† .
(4) The module (Ωd k)† belongs to ΩCM× (R). (4’) The module (Ωd k)† belongs to ΩCM(R).
(5) One has HomR ((Ωd k)† , ω) = 0.
d †
(6) One has TorR
1 (Tr((Ω k) ), ω) = 0.
d+1
(7) One has ExtR (Tr((Ωd k)† ), R) = 0 and R is locally Gorenstein on the punctured spectrum.
(8) The natural homomorphism ω ⊗R HomR (ω, Ωd k) → Ωd k is surjective.
(9) There exists a surjective homomorphism ω ⊕n → Ωd k.
If d is positive, k is infinite and one of the above nine conditions holds, then R is almost Gorenstein.
Proof. (1) ⇒ (2): This follows from Lemma 3.5(2).
(2) ⇒ (3): Let M be an R-module in ΩCM× (R). Then M ∈ Ul(R) by Proposition 1.6, and hence
†
M ∈ Ul(R) by Corollary 2.2. It follows from Lemma 3.7(4) that there is a surjection (Ωd k)⊕n → M † .
Since (Ωd k)⊕n is in ΩCM× (R) by Lemma 3.7(1), the module M † is also in ΩCM× (R).
(3) ⇒ (4): Lemma 3.7(1) says that Ωd k is in ΩCM× (R), and so is (Ωd k)† by assumption.
(4) ⇒ (1): The inclusion ΩCM× (R) ⊆ Ul(R) follows from Proposition 1.6. Take any module M in
Ul(R). Then M † is also in Ul(R) by Corollary 2.2. Using Lemma 3.7(4), we get an exact sequence
0 → X → (Ωd k)⊕n → M † → 0 of Cohen–Macaulay modules, which induces an exact sequence 0 → M →
(Ωd k)†⊕n → X † → 0. The assumption and Lemma 3.5(2) imply that M is in ΩCM× (R).
(4) ⇔ (4’): As R is singular, by [28, Corollary 4.4] the module (Ωd k)† does not have a free summand.
(4’) ⇔ (5) ⇔ (6) ⇔ (8) ⇔ (9): These equivalences follow from Proposition 3.2.
(4’) ⇔ (7): We claim that, under the assumption that R is locally Gorenstein on the punctured
d †
d †
spectrum, (Ωd k)† ∈ ΩCM(R) if and only if Extd+1
R (Tr((Ω k) ), R) = 0. In fact, since (Ω k) is Cohen–
i
Macaulay, it satisfies Serre’s condition (Sd ). Therefore it is d-torsionfree, that is, ExtR (Tr((Ωd k)† ), R) = 0
d †
d †
for all 1 6 i 6 d; see [22, Theorem 2.3]. Hence, Extd+1
R (Tr((Ω k) ), R) = 0 if and only if (Ω k) is (d + 1)torsionfree, if and only if it belongs to ΩCM(R) by [22, Theorem 2.3] again. Thus the claim follows.
According to this claim, it suffices to prove that if (4’) holds, then R is locally Gorenstein on the
punctured spectrum. For this, pick any nonmaximal prime ideal p of R. There are exact sequences
⊕a

0 → (Ωd k)p → Rp d−1 → · · · → Rp⊕a0 → 0.
P
i
d
We observe that (Ωd k)p is a free Rp -module with rankRp ((Ωd k)p ) = d−1
i=0 (−1) ad−1−i = rankR (Ω k).
d
d
The module Ω k has positive rank as it is torsionfree, and we see that (Ω k)p is a nonzero free Rp -module.
Since we have already shown that (4’) implies (9), there is a surjection ω ⊕n → Ωd k. Localizing this at
p, we see that ωp⊕n has an Rp -free summand, which implies that the Rp -module Rp has finite injective
dimension. Thus Rp is Gorenstein.
So far we have proved the equivalence of the conditions (1)–(9). It remains to prove that R is almost
Gorenstein under the assumption that d is positive, k is infinite and (1)–(9) all hold. We use induction
on d.
Let d = 1. Let Q be the total quotient ring of R, and set E = EndR (m). Let K be an R-module with
K∼
= ω and R ⊆ K ⊆ R in Q, where R is the integral closure of R. Using [24, Proposition 2.5], we have:
0 → Ωd k → R⊕ad−1 → · · · → R⊕a0 → k → 0,

(3.8.1)

m∼
= HomR (m, R) = E

and m† ∼
= HomR (m, K) ∼
= (K :Q m).

By (4) the module m† belongs to ΩCM× (R). It follows from [19, Theorem 2.14] that R is almost
Gorenstein; note that the completion of R also has Gorenstein punctured spectrum by (4’).
Let d > 1. Since (Ωd k)† ∈ ΩCM(R), there is an exact sequence 0 → (Ωd k)† → R⊕m → N → 0 for
some m > 0 and N ∈ CM(R). Choose a parameter ideal Q = (x1 , . . . , xd ) of R satisfying the equality
m2 = Qm, and set (−) = (−) ⊗R R/(x1 ). An exact sequence
0 → (Ωd k)† → R

⊕m

→N →0
x

is induced, which shows that (Ωd k)† is in ΩCM(R). Applying (−)† to the exact sequence 0 → Ωd k −
→
Ωd k → Ωd k → 0 and using [3, Lemma 3.1.16], we obtain isomorphisms
(Ωd k)† ∼
= Ext1R (Ωd k, ω) ∼
= HomR (Ωd k, ω).

10

TOSHINORI KOBAYASHI AND RYO TAKAHASHI

d−1
k is a direct summand of Ωd k by [28, Corollary 5.3], and hence HomR (Ωd−1
k, ω) is
The module ΩR
R

a direct summand of HomR (Ωd k, ω). Summarizing these, we observe that HomR (Ωd−1
k, ω) belongs to
R
ΩCM(R). Since R has minimal multiplicity, we can apply the induction hypothesis to R to conclude that
R is almost Gorenstein, and so is R by [11, Theorem 3.7].

Remark 3.9. When d > 2, it holds that
d−1
d †
d
∼
Extd+1
R (Tr((Ω k) ), R) = ExtR (HomR (ω, Ω k), R).
d
Thus Theorem 3.8(7) can be replaced with the condition that Extd−1
R (HomR (ω, Ω k), R) = 0.
Indeed, using the Hom-⊗ adjointness twice, we get isomorphisms
∼ HomR (ω, (Ωd k)†† ) ∼
HomR (ω, Ωd k) =
= HomR ((Ωd k)† ⊗R ω, ω) ∼
= HomR ((Ωd k)† , ω † ) ∼
= (Ωd k)†∗ ,

and (Ωd k)†∗ is isomorphic to Ω2 Tr((Ωd k)† ) up to free summand.
We have several more conditions related to the equality ΩCM× (R) = Ul(R).
Corollary 3.10. Let R be as in Theorem 3.8. Consider the following conditions:
(1) (Ωd k)† ∼
= Ωd k, (2) (Ωd k)† ∈ add(Ωd k), (3) annh (Ωd k)† = m, (4) ΩCM× (R) = Ul(R).
It then holds that (1) =⇒ (2) ⇐⇒ (3) =⇒ (4).
Proof. The implications (1) ⇒ (2) ⇒ (3) are obvious. The proof of Proposition 2.8 shows that if an
Ulrich R-module M satisfies annh M = m, then M is in add(Ωd k). This shows (3) ⇒ (2). Proposition
3.7(1) says that Ωd k is in ΩCM× (R), and so is (Ωd k)† by assumption. Theorem 3.8 shows (2) ⇒ (4). 
We close this section by constructing an example by applying the above corollary.
Example 3.11. Let S = C[[x, y, z]] be a formal power series ring. Let G be the cyclic group 12 (1, 1, 1),
and let R = S G be the invariant (i.e. the second Veronese) subring of S. Then ΩCM× (R) = Ul(R). In
fact, by [32, Proposition (16.10)], the modules R, ω, Ωω are the nonisomorphic indecomposable Cohen–
Macaulay R-modules and (Ωω)† ∼
= Ωω. By [28, Theorem 4.3] the module Ω2 C does not have a nonzero
free or canonical summand. Hence Ω2 C is a direct sum of copies of Ωω, and thus (Ω2 C)† ∼
= Ω2 C. The
×
equality ΩCM (R) = Ul(R) follows from Corollary 3.10.
4. Applications
This section is devoted to stating applications of our main theorems obtained in the previous section.
4.1. The case of dimension one. We begin with studying the case where R has dimension 1.
Theorem 4.1. Let (R, m, k) be a 1-dimensional Cohen–Macaulay local ring with k infinite and canonical
module ω. Suppose that R has minimal multiplicity, and set (−)† = HomR (−, ω). Then
∼ m ⇐⇒ R is almost Gorenstein.
ΩCM× (R) = Ul(R) ⇐⇒ m† ∈ ΩCM× (R) ⇐⇒ m† =
Proof. Call the four conditions (i)–(iv) from left to right. The implications (i) ⇔ (ii) ⇒ (iv) are shown by
Theorem 3.8, while (iii) ⇔ (iv) by [19, Theorem 2.14] and (3.8.1). Lemma 3.7(1) shows (iii) ⇒ (ii). 
Now we pose a question related to Question 1.2.
Question 4.2. Can we classify 1-dimensional Cohen–Macaulay local rings R with minimal multiplicity
(and infinite residue field) satisfying the condition # ind Ul(R) < ∞?
According to Proposition 1.6, over such a ring R we have the property that # ind ΩCM(R) < ∞, which
is studied in [18]. If R has finite Cohen–Macaulay representation type (that is, if # ind CM(R) < ∞),
then of course this question is affirmative. However, we do not have any partial answer other than this.
The reader may wonder if the condition # ind Ul(R) < ∞ implies the equality ΩCM× (R) = Ul(R). Using
the above theorem, we observe that this does not necessarily hold:
Example 4.3. Let R = k[[t3 , t7 , t8 ]] be (the completion of) a numerical semigroup ring, where k is an
algebraically closed field of characteristic zero. Then R is a Cohen–Macaulay local ring of dimension 1
with minimal multiplicity. It follows from [12, Theorem A.3] that # ind Ul(R) < ∞. On the other hand,
R is not almost-Gorenstein by [8, Example 4.3], so ΩCM× (R) 6= Ul(R) by Theorem 4.1.

ULRICH MODULES AND MINIMAL MULTIPLICITY

11

4.2. The case of dimension two. From now on, we consider the case where R has dimension 2. We
recall the definition of a Cohen–Macaulay approximation. Let R be a Cohen–Macaulay local ring with
canonical module. A homomorphism f : X → M of R-modules is called a Cohen–Macaulay approximation
(of M ) if X is Cohen–Macaulay and any homomorphism f ′ : X ′ → M with X ′ being Cohen–Macaulay
factors through f . It is known that f is a (resp. minimal) Cohen–Macaulay approximation if and only if
there exists an exact sequence
g

f

0→Y −
→X −
→M →0
of R-modules such that X is Cohen–Macaulay and Y has finite injective dimension (resp. and that X, Y
have no common direct summand along g). For details of Cohen–Macaulay approximations, we refer the
reader to [21, Chapter 11].
The module E appearing in the following remark is called the fundamental module of R.
Remark 4.4. Let (R, m, k) be a 2-dimensional Cohen–Macaulay local ring with canonical module ω.
(1) There exists a nonsplit exact sequence
(4.4.1)

0→ω→E→m→0

which is unique up to isomorphism. This is because Ext1R (m, ω) ∼
= k.
= Ext2R (k, ω) ∼
(2) The module E is Cohen–Macaulay and uniquely determined up to isomorphism.
(3) The sequence (4.4.1) gives a minimal Cohen–Macaulay approximation of m.
(4) There is an isomorphism E ∼
= E † . In fact, applying (−)† to (4.4.1) induces an exact sequence
0 → m† → E † → R → Ext1R (m, ω) → Ext1R (E, ω) = 0.
Applying (−)† to the natural exact sequence 0 → m → R → k → 0 yields m† ∼
= ω, while Ext1R (m, ω) ∼
=
†
k. We get an exact sequence 0 → ω → E → m → 0, and the uniqueness of (4.4.1) shows E † ∼
= E.
To prove the main result of this section, we prepare two lemmas. The first one relates the fundamental
module of a 2-dimensional Cohen–Macaulay local ring R with Ul(R) and ΩCM× (R).
Lemma 4.5. Let (R, m, k) be a 2-dimensional Cohen–Macaulay local ring with canonical module ω and
fundamental module E.
(1) Assume that R has minimal multiplicity. Then E is an Ulrich R-module.
(2) For each module M ∈ ΩCM× (R) there exists an exact sequence 0 → M → E ⊕n → N → 0 of
R-modules such that N is Cohen–Macaulay.
Proof. (1) We may assume that k is infinite by Remark 1.5(2). Let Q = (x, y) be a parameter ideal of R
with m2 = Qm. We have m/xm ∼
= m/(x) ⊕ k; see [28, Corollary 5.3]. Note that (m/(x))2 = y(m/(x)). By
[31, Corollary 2.5] the minimal Cohen–Macaulay approximation of m/xm as an R/(x)-module is E/xE.
In view of the proof of [21, Proposition 11.15], the minimal Cohen–Macaulay approximations of m/(x) and
k as R/(x)-modules are m/(x) and HomR/(x) (m/(x), ω/xω), respectively. Thus we get an isomorphism
E/xE ∼
= m/(x) ⊕ HomR/(x) (m/(x), ω/xω).
In particular, E/xE is an Ulrich R/(x)-module by Lemma 3.7(1) and Corollary 2.2. It follows from
Lemma 3.6 that E is an Ulrich R-module.
f
e
(2) Take an exact sequence 0 → M −
→ R⊕n −
→ L → 0 such that L is Cohen–Macaulay. As M has no
free summand, the homomorphism e is minimal. This means that f factors through the natural inclusion
i : m⊕n → R⊕n , that is, f = ig for some g ∈ HomR (M, m⊕n ). The direct sum p : E ⊕n → m⊕n of
copies of the surjection E → m (given by (4.4.1)) is a Cohen–Macaulay approximation. Hence there is a
homomorphism h : M → E ⊕n such that g = ph. We get a commutative diagram
0

/M

0

/M

f

/ R⊕n
O

/L
O

/0

/ E ⊕n

/N

/0

ip

h

with exact rows. This induces an exact sequence 0 → E ⊕n → R⊕n ⊕ N → L → 0, and therefore N is a
Cohen–Macaulay R-module.

A short exact sequence of Ulrich modules is preserved by certain functors:

12

TOSHINORI KOBAYASHI AND RYO TAKAHASHI

Lemma 4.6. Let 0 → X → Y → Z → 0 be an exact sequence of modules in Ul(R). Then it induces
exact sequences of R-modules
(a) 0 → X ⊗R k → Y ⊗R k → Z ⊗R k → 0,
(b) 0 → HomR (Z, k) → HomR (Y, k) → HomR (X, k) → 0, and
(c) 0 → HomR (Z, (Ωd k)† ) → HomR (Y, (Ωd k)† ) → HomR (X, (Ωd k)† ) → 0.
Proof. The sequence X ⊗R k → Y ⊗R k → Z ⊗R k → 0 is exact and the first map is injective by Lemma
3.5(1). Hence (a) is exact, and so is (b) by a dual argument. In what follows, we show that (c) is
exact. We first note that (Ωd k)† is a minimal Cohen–Macaulay approximation of k; see the proof of [21,
Proposition 11.15]. Thus there is an exact sequence 0 → I → (Ωd k)† → k → 0 such that I has finite
injective dimension. As Ul(R) ⊆ CM(R), we have Ext1R (M, I) = 0 for all M ∈ {X, Y, Z}. We obtain a
commutative diagram
0

/ HomR (Y, I)

/ HomR (Y, (Ωd k)† )

/ HomR (Y, k)

0

 α
/ HomR (X, I)

β
/ HomR (X, (Ωd k)† )


/ HomR (X, k)

/0

γ

/0

with exact rows, where α is surjective. The exactness of (b) implies that γ is surjective. By the snake
lemma β is also surjective, and therefore (c) is exact.

Now we can state and show our main result in this section.
Theorem 4.7. Let R be a 2-dimensional complete singular normal local ring with residue field C and
having minimal multiplicity. Suppose that R does not have a cyclic quotient singularity. Then:
(Ωd k)† ∼
= Ωd k ⇐⇒ (Ωd k)† ∈ add(Ωd k) ⇐⇒ annh (Ωd k)† = m ⇐⇒ ΩCM× (R) = Ul(R).
Proof. In view of Corollary 3.10, it suffices to show that if R does not have a cyclic quotient singularity,
then the fourth condition implies the first one. By virtue of [32, Theorem 11.12] the fundamental module
α
E is indecomposable. Applying Lemma 4.5(2) to (Ωd k)† , we have an exact sequence 0 → (Ωd k)† −
→
E ⊕n → N → 0 such that N is Cohen–Macaulay. Since E is Ulrich by Lemma 4.5(1), so are all the three
modules in this sequence by Lemma 3.5(2). Thus we can apply Lemma 4.6 to see that the induced map
HomR (α, (Ωd k)† ) : HomR (E ⊕n , (Ωd k)† ) → HomR ((Ωd k)† , (Ωd k)† )
is surjective. This implies that α is a split monomorphism, and (Ωd k)† is isomorphic to a direct summand
of E ⊕n . Since E is indecomposable, it folllows that (Ωd k)† is isomorphic to E ⊕m for some m. We obtain
(Ωd k)† ∼
= Ωd k,
= (Ωd k)†† ∼
= (E † )⊕m ∼
= E ⊕m ∼
where the second isomorphism follows by Remark 4.4(4).



Remark 4.8. Let R be a cyclic quotient surface singularity over C. Nakajima and Yoshida [23, Theorem
4.5] give a necessary and sufficient condition for R to be such that the number of nonisomorphic indecomposable Ulrich R-modules is equal to the number of nonisomorphic nonfree indecomposable special
Cohen–Macaulay R-modules. By [15, Corollary 2.9], the latter is equal to the number of isomorphism
classes of indecomposable modules in ΩCM× (R). Therefore, they actually gives a necessary and sufficient
condition for R to satisfy ΩCM× (R) = Ul(R).
Using our Theorem 4.7, we give some examples of a quotient surface singularity over C to consider
Ulrich modules over them.
Example 4.9. (1) Let S = C[[x, y]] be a formal power series ring. Let G be the cyclic group 13 (1, 1),
and let R = S G be the invariant (i.e. the third Veronese) subring of S. Then ΩCM× (R) = Ul(R). This
follows from [23, Theorem 4.5] and Remark 4.8, but we can also show it by direct caluculation: we have
Cl(R) = {[R], [ω], [p]} ∼
= Z/3Z,
where ω = (x3 , x2 y)R is a canonical ideal of R, and p = (x3 , x2 y, xy 2 )R is a prime ideal of height 1 with
[ω] = 2[p]. Since the second Betti number of C over R is 9, we see Ω2 C ∼
= p⊕3 . As [p† ] = [ω] − [p] = [p],
×
2
† ∼ 2
† ∼
we have p = p and (Ω C) = Ω C. Theorem 4.7 shows ΩCM (R) = Ul(R).

ULRICH MODULES AND MINIMAL MULTIPLICITY

13

(2) Let S = C[[x, y]] be a formal power series ring. Let G be the cyclic group 81 (1, 5), and let R = S G
be the invariant subring of S. With the notation of [23], the Hirzebruch-Jung continued fraction of this
group is [2, 3, 2]. It follows from [23, Theorem 4.5] and Remark 4.8 that ΩCM× (R) 6= Ul(R).
4.3. An exact structure of the category of Ulrich modules. Finally, we consider realization of the
additive category Ul(R) as an exact category in the sense of Quillen [25]. We begin with recalling the
definition of an exact category given in [17, Appendix A].
Definition 4.10. Let A be an additive category. A pair (i, d) of composable morphisms
i

d

X−
→Y −
→Z
is exact if i is the kernel of d and d is the cokernel of i. Let E be a class of exact pairs closed under
isomorphism. The pair (A, E) is called an exact category if the following axioms hold. Here, for each
(i, d) ∈ E the morphisms i and d are called an inflation and a deflation, respectively.
(Ex0) 1 : 0 → 0 is a deflation.
(Ex1) The composition of deflations is a deflation.
(Ex2) For each morphism f : Z ′ → Z and each deflation d : Y → Z, there is a pullback diagram as in
the left below, where d′ is a deflation.
op
(Ex2 ) For each morphism f : X → X ′ and each inflation i : X → Y , there is a pushout diagram as in
the right below, where i′ is an inflation.
Y′

Y

d′

/ Z′
f

d


/Z

X
f 
X′

i

i′

/Y

/ Y′

We can equip a structure of an exact category with our Ul(R) as follows.
Theorem 4.11. Let R be a d-dimensional Cohen–Macaulay local ring with residue field k and canonical
module, and assume that R has minimal multiplicity. Let S be the class of exact sequences 0 → L → M →
N → 0 of R-modules with L, M, N Ulrich. Then Ul(R) = (Ul(R), S) is an exact category having enough
projective objects and enough injective objects with proj Ul(R) = add(Ωd k) and inj Ul(R) = add((Ωd k)† ).
Proof. We verify the axioms in Definition 4.10.
(Ex0): This is clear.
(Ex1): Let d : Y → Z and d′ : Z → W be deflations. Then there is an exact sequence 0 → X →
d′ d

Y −−→ W → 0 of R-modules. Since Y is in Ul(R) and X, W ∈ CM(R), it follows from that X ∈ Ul(R).
Thus this sequence belongs to S, and d′ d is a deflation.
(Ex2): Let f : Z ′ → Z be a homomorphism in Ul(R) and d : Y → Z a deflation in S. Then we get
(d,f )

an exact sequence 0 → Y ′ → Y ⊕ Z ′ −−−→ Z → 0. Since Y ⊕ Z ′ ∈ Ul(R) and Y ′ , Z ∈ CM(R), Lemma
d′

3.5(2) implies Y ′ ∈ Ul(R). Make an exact sequence 0 → X ′ → Y ′ −→ Z ′ → 0. As Y ′ ∈ Ul(R) and
X ′ , Z ′ ∈ CM(R), the module Z ′ is in Ul(R) by Lemma 3.5(2) again. Thus d′ is a deflation.
(Ex2op ): We can check this axiom by the opposite argument to (Ex2).
Now we conclude that (Ul(R), S) is an exact category. Let us prove the remaining assertions. Lemma
4.6(c) yields the injectivity of (Ωd k)† . Since (−)† gives an exact duality of (Ul(R), S), the module
Ωd k is a projective object. We also observe from Lemma 3.7 and Corollary 2.2 that (Ul(R), S) has
enough projective objects with proj Ul(R) = add(Ωd k), and has enough injective objects with inj Ul(R) =
add((Ωd k)† ) by the duality (−)† .

Remark 4.12. Let (R, m) be 1-dimensional Cohen–Macaulay
  local ring with infinite residue field. Let
by [12, Proposition A.1]. This equality
(t) be a minimal reduction of m. Then Ul(R) = CM R mt
m
])
of
categories,
since Hom-sets do not change; see [21,
acturely gives an equivalence Ul(R) ∼
CM(R[
=
t
Proposition 4.14]. Thus the usual exact structure on CM(R[ mt ]) coincides with the exact structure on
Ul(R) given above via this equivalence.
Acknowledgments
The authors are grateful to Doan Trung Cuong for valuable discussion on Ulrich modules. In particular,
his Question 1.2 has given a strong motivation for them to have this paper.

14

TOSHINORI KOBAYASHI AND RYO TAKAHASHI

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Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-8602, Japan
E-mail address: m16021z@math.nagoya-u.ac.jp
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/

