Ranks of indecomposable modules over rings of infinite Cohen-Macaulay type

0.24 MB·English

by  A. Crabbe

RANKS OF INDECOMPOSABLE MODULES OVER
RINGS OF INFINITE COHEN-MACAULAY TYPE
ANDREWCRABBEANDSILVIA SACCON
Abstract. Let(R,m,k)beaone-dimensionalanalyticallyunramifiedlocalringwithminimalprimeidealsP1,
2 ..., Ps. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R.
1 Such behavior isencoded bythe monoid C(R) ofisomorphismclasses of maximalCohen-Macaulay R-modules:
0 the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is
2 uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid
giveameasureofhowbadlythispropertyfails. Thekeytounderstanding themonoidC(R) isdeterminingthe
n
a ranksofindecomposable maximalCohen-Macaulaymodules. Ourmaintechnical resultshowsthatifR/P1 has
J infinite Cohen-Macaulay type and the residue field k is infinite, then there exist |k| pairwise non-isomorphic
indecomposablemaximalCohen-MacaulayR-modulesofrank(r1,...,rs)providedr1≥ri foralli∈{1,...,s}.
6 b
ThisresultallowsustodescribethemonoidC(R)whenR/QhasinfiniteCohen-Macaulaytypeforeveryminimal
1 b
primeidealQofthem-adiccompletionR.

]
C
A
1. Introduction
.
h
t AcommutativeringRislocal ifRisaNoetherianringwithexactlyonemaximalideal,andalocalring(R,n)
a
m is analytically unramified if the n-adic completion R of R is reduced. If R is complete, then the Krull-Remak-
Schmidt property holds for the class of finitely generated R-modules; that is, direct-sum decompositions of
[
finitely generatedR-modulesare unique (see [13, Thbeorem5.20]). However,there areexamples ofnon-complete

1 local rings for which direct-sum decomposition is not unique (see for example [9, Section 1] or [16, Sections 1
v
and 2]). Thus it is natural to ask how modules behave over R, especially when R is a non-complete local ring.
1
In this paper, we restrict our attention to one-dimensional analytically unramified local rings (R,m,k), and
4
1 to the class of maximal Cohen-Macaulay R-modules. In this context, a maximal Cohen-Macaulay R-module is
3 exactly a non-zero finitely generated torsion-free R-module. We say R has finite Cohen-Macaulay type if there
. exist only finitely many indecomposable maximal Cohen-Macaulay R-modules, up to isomorphism. Otherwise,
1
0 we say R has infinite Cohen-Macaulay type.
2 One approach to the study of direct-sum behavior of modules over R is to describe the monoid C(R) of
1 isomorphismclassesof maximalCohen-MacaulayR-modules (together with [0]), with operationgivenby [M]+
:
v [N] = [M ⊕N]. This monoid is free if and only if the Krull-Remak-Schmidt property holds for the class of
i maximal Cohen-Macaulay R-modules; moreover, some invariants of this monoid give a measure of how badly
X
the Krull-Remak-Schmidt property can fail.
r
a The notion of rank of a module plays a fundamental role in studying the monoid C(R), and thus in un-
derstanding the direct-sum behavior of modules over R. The rank of an R-module M is the tuple consisting
of the vector-space dimensions of M over R , where P ranges over the minimal prime ideals of R. More
P P
precisely, let P , ..., P denote the minimal prime ideals of R. The rank of M at the minimal prime ideal P
1 s i
is r = dim M , the dimension of the vector space M over the field R . The rank of M is the s-tuple
i RPi Pi Pi Pi
(r ,...,r ). In this paper, we are primarily interested in the following question: What ranks occur for inde-
1 s
composable maximalCohen-MacaulayR-modules? By answeringthis question, we willbe able to givea precise
description of the monoid C(R).
When R has finite Cohen-Macaulay type, N. Baeth and M. Luckas give a complete answer to this question
in[4,MainTheorem]. BaethandLuckasclassifyallpossibleranksofindecomposablemaximalCohen-Macaulay
R-modules depending on the number of minimal prime ideals (over such rings there are at most three minimal
Date:December20,2011.
2000 Mathematics Subject Classification. Primary13C14. Secondary13C05.
Key words and phrases. InfiniteCohen-Macaulaytype, indecomposablemaximalCohen-Macaulaymodule,rankofamodule.
PartsofthisworkappearinSaccon’s Ph.D.dissertationattheUniversityofNebraska–Lincoln, underthesupervisionofRoger
Wiegand.
1
2 ANDREWCRABBEANDSILVIASACCON
prime ideals). Their work builds upon results of Green and Reiner [10, pages 76-77, 81-82], R. Wiegand and
S.Wiegand[18,Theorem3.9],C¸imen[6,Theorem2.2],Baeth[2,Theorem4.2],andArnavut,LuckasandS.Wie-
gand[1,Theorem2.3]. Moreover,in [3, Theorem3.4],BaethandLuckas use the listofranksofindecomposable
maximal Cohen-Macaulay R-modules to give a description of the monoid C(R).
The question is open when R has infinite Cohen-Macaulay type. In [14, Section 2], R. Wiegand constructs
an indecomposable maximal Cohen-Macaulay module of constant rank (r,...,r) for each positive integer r.
In [17, Lemma 2.2], he shows that if R/P satisfies a certain specific condition (corresponding to Case (1)
1
of Remark 3.2), and if r > 0 and r ≥ r for all i ∈ {1,...,s}, then there is an indecomposable maximal
1 1 i
Cohen-Macaulay R-module of rank (r ,...,r ).
1 s
Inthis article,weproveastrongerresult: we generalizeWiegand’stheoremandwe determine the cardinality
of the set of isomorphismclasses of indecomposable maximal Cohen-MacaulayR-modules. Our goal is to prove
the following theorem.
Main Theorem. Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime
ideals P , ..., P . Assume R/P has infinite Cohen-Macaulay type. Let (r ,...,r ) be a non-trivial s-tuple of
1 s 1 1 s
non-negative integers with r ≥r for all i∈{1,...,s}.
1 i
(1) There exists an indecomposable maximal Cohen-Macaulay R-module of rank (r ,...,r ).
1 s
(2) If the residue field k is infinite, then the set of isomorphism classes of indecomposable maximal Cohen-
Macaulay R-modules of rank (r ,...,r ) has cardinality |k|.
1 s
In the proof of this result, we put a specific construction due to R. Wiegand in a general context. This
introduces various technical difficulties, particularly in the cases when the multiplicity of the ring is three, or
when the residue field is not of characteristic zero. There are two key ingredients in the proof, namely:
(a) Proposition 3.1 that allows us to focus our attention to finite dimensional k-algebras of dimension at
least four, and
(b) the construction of a family of modules of a certain rank—we show that these modules are indecompo-
sable in all the cases that can arise for a finite-dimensional k-algebra of dimension at least four. This
new construction (depending on a parameter t∈k) allows us to determine the cardinality of the set of
isomorphism classes of indecomposable maximal Cohen-Macaulay modules.
In Section 2, we provide some backgroundand some knownresults. In Section 3, we provethe main theorem
and use this result to describe the monoid C(R) when R/Q has infinite Cohen-Macaulay type for all minimal
prime ideals Q of R. In Section 4, we study a particular ring R with exactly two minimal prime ideals P and
1
P such that R/P has infinite Cohen-Macaulay type anbd R/P has finite Cohen-Macaulay type; we show that
2 1 2
certain ranks cannobt occur for this example.
2. Background
One-dimensional reduced local rings (R,m) of finite Cohen-Macaulay type are completely characterized as
those rings satisfying the following two conditions of Drozd and Ro˘ıter (see [8]):
(DR1) theintegralclosureRofRinthetotalquotientringQ(R)ofRisgeneratedbyatmostthreeelements
as an R-module;
(DR2) (mR+R)/R is a cyclic R-module.
These two conditions were first introduced in [8] by Drozd and Ro˘ıter, who proved the theorem in the special
case of a ring which is a localization of a module-finite Z-algebra. R. Wiegand proved necessity of the two
conditions in [14] and sufficiency, except in the case when k is imperfect of characteristic 2, in [14] and [15].
C¸imen proved the remaining case of the theorem in his Ph.D. dissertation [6].
Themaintoolforconstructingindecomposable(maximalCohen-Macaulay)modulesisthe Artinianpair(see
for example [14, Section 1]).
Definition 2.1. An Artinian pair A֒→B is a module-finite extension of commutative Artinian rings.
Definition 2.2. A module over the Artinian pair A֒→B is a pair V ֒→W such that
(1) W is a finitely generated projective B-module,
(2) V is an A-submodule of W and
(3) BV =W.
RANKS OF INDECOMPOSABLE MODULES 3
Definition 2.3. A homomorphism from an (A ֒→B)-module V ֒→W to an (A֒→ B)-module V ֒→W is a
1 1 2 2
B-module homomorphism ϕ: W →W such that ϕ(V )⊆V .
1 2 1 2
Thustherearenotionsofisomorphismof(A֒→B)-modules,endomorphismringsEnd (V,W)ofan(A֒→B)-
B
module V ֒→W and direct-sum decompositions of V ֒→W.
Definition 2.4. We say that an Artinian pair A֒→B has finite representation type if there exist only finitely
many indecomposable (A֒→B)-modules, up to isomorphism.
Given a one-dimensional analytically unramified local ring (R,m,k) with integral closure R 6= R, we can
associate to R an Artinian pair R in the following way. Let c={r∈R|rR⊆R} denote the conductor of R,
art
i.e., the largestidealof R containedin R. As c contains a non-zero-divisorof R, both R/cand R/c are Artinian
rings, and R/c ֒→ R/c is an Artinian pair, denoted R . We have a pullback diagram, called the conductor
art
square of R:
(cid:31)(cid:127) //
R R
(cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15)
R(cid:31)(cid:127) // R
.
c c
Given a maximal Cohen-Macaulay R-module M, we can associate to M an R -module M = (M/cM ֒→
art art
RM/cM), where RM denotes the R-submodule of Q(R)⊗ M generated by the image of M.
R
For a one-dimensional analytically unramified local ring (R,m,k), the Drozd and Ro˘ıter conditions can be
interpretedinterms ofthe ArtinianpairR =(A֒→B), where(A,n,k) isa localring. Inthis case,the Drozd
art
and Ro˘ıter conditions (DR1) and (DR2) are equivalent to:
B
(dr1) dim ≤3,
k
nB
nB+A
(dr2) dim ≤1.
k n2B+A
(cid:18) (cid:19)
That is, R satisfies (DR1) and (DR2) if and only if R satisfies (dr1) and (dr2).
art
In order to study maximal Cohen-Macaulay R-modules, it is convenient to work in the category of R -
art
modules. The following theorems summarize some facts about the relationships between R and R , and the
art
modules M and M .
art
Theorem 2.5 ([14],Propositions1.5-1.9). Let (R,m,k) be a one-dimensional analytically unramified local ring,
and let R denote the corresponding Artinian pair. Let R denote the integral closure of R in its total quotient
art
ring, andassumeR6=R. LetM,N bemaximalCohen-MacaulayR-modules,andletV ֒→W beanR -module.
art
(1) (V ֒→ W) ∼= X for some maximal Cohen-Macaulay R-module X if and only if W ∼= F/cF for some
art
finitely generated projective R-module F.
(2) (M ⊕N) ∼=M ⊕N .
art art art
(3) M ∼=N if and only if M ∼=N.
art art
(4) The Krull-Remak-Schmidt theorem holds for direct-sum decompositions of modules over R .
art
(5) R has finite Cohen-Macaulay type if and only if R has finite representation type.
art
Theorem 2.6 ([18], Proposition 2.2). With the notation as in Theorem 2.5, assume R is a direct product of
local rings.
(1) ThefunctorM 7→M determinesabijection betweenthesetofisomorphism classes ofmaximalCohen-
art
Macaulay R-modules and the set of isomorphism classes of R -modules.
art
(2) M is indecomposable if and only if M is indecomposable.
art
Proposition2.7. Let(R,m,k)beaone-dimensionalanalyticallyunramifiedlocalringwithminimalprimeideals
P , ..., P . Let R denote the integral closure of R in its total quotient ring, and assume R6=R. Set R =R/P
1 s i i
and B =R /cR for i∈{1,...,s}. If V ֒→W is an indecomposable R -module with W =B(r1)×···×B(rs),
i i i art 1 s
then (V ֒→W)∼=Xart for an indecomposable maximal Cohen-Macaulay R-module X of rank (r1,...,rs).
Proof. Let F =R(r1)×···×R(rs) be a projective R-module. Since W ∼=F/cF, Theorem 2.5(2) guarantees the
1 s
existence of a maximal Cohen-Macaulay R-module X such that X ∼=(V ֒→W).
art
4 ANDREWCRABBEANDSILVIASACCON
Suppose X = X1⊕X2 for R-modules X1 and X2. Then Xart ∼= X1,art⊕X2,art, and since Xart is indecom-
posable, X = 0 or X = 0. Without loss of generality, assume X = (X /cX ֒→ RX /cX ) = 0; in
1,art 2,art 1,art 1 1 1 1
particular, the first component X /cX of X is zero. Since c ⊆m, by Nakayama’s lemma we conclude that
1 1 1,art
X1 =0. Thus X is indecomposable. Since RX ∼=F, we see that rank(X)=(r1,...,rs). (cid:3)
It is often useful to study the Artinian pair k ֒→ D obtained from the Artinian pair A ֒→ B by reducing
modulo the maximal ideal of A. We use the following results in [14].
Lemma 2.8 ([14],Lemma2.4). Let A֒→B bean Artinian pair, and let I bea nilpotent ideal of B. The functor
from the category of (A֒→B)-modules to the category of (A/(I ∩A)֒→B/I)-modules defined by
V +IW W
(V ֒→W)7→ ֒→
IW IW
(cid:18) (cid:19)
is surjective on isomorphism classes and reflects indecomposable objects.
Lemma 2.9 ([14], Lemma 2.4). Let A ֒→ B be an Artinian pair, and let C be a ring between A and B. The
functor from the category of (A֒→C)-modules to the category of (A֒→B)-modules defined by
(V ֒→W)7→(V ֒→B⊗ W)
C
is full and faithful, and preserves and reflects indecomposable objects.
3. Indecomposable maximal Cohen-Macaulay modules of a given rank
ItisusefultoreducetheconstructionofindecomposablemodulesoverRtotheconstructionofindecomposable
modules over the Artinian pair R . The following proposition allows one to replace the Artinian pair R =
art art
(A ֒→ B) with the Artinian pair k ֒→ D, where k is a field and D is a finite-dimensional k-algebra satisfying
certain conditions.
Proposition 3.1. Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime
ideals P , ..., P . Assume R/P has infinite Cohen-Macaulay type. Let R = (A ֒→B) be the corresponding
1 s 1 art
Artinian pair. Then there is a ring C between A and B such that, reducing modulo the maximal ideal of A, we
have
(cid:31)(cid:127) //
A C
(cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15)
(cid:31)(cid:127) //
k D,
where D =D ×···×D is a product of finite-dimensional k-algebras and either
1 s
dimkD1 ≥4 or D1 ∼=k[X,Y]/(X2,XY,Y2).
Moreover, if there exists an indecomposable (k ֒→ D)-module V ֒→ W, where W = D(r1) ×···×D(rs), then
1 s
there exists an indecomposable maximal Cohen-Macaulay R-module of rank (r ,...,r ).
1 s
Proof. Letc be the conductorof R, andlet R denote the integraldomainR/P for i∈{1,...,s}. Considerthe
i i
conductor square of R:
(cid:31)(cid:127) // (cid:31)(cid:127) //
R R ×···×R R ×···×R =R
1 s 1 s
(cid:15)(cid:15) (cid:15)(cid:15)
R(cid:31)(cid:127) //// R1 Rs (cid:31)(cid:127) // R
A= ×···× B ×···×B = =B,
1 s
c cR cR c
1 s
RANKS OF INDECOMPOSABLE MODULES 5
where B = R /cR , i ∈ {1,...,s}. Now consider the conductor square for R = R/P , and let c denote the
i i i 1 1 1
conductor of R (note that c ⊇cR =cR ):
1 1 1 1
(cid:31)(cid:127) //
R R
1 1
(cid:15)(cid:15) (cid:15)(cid:15)
R1 (cid:31)(cid:127) //// R1
A = =B
1 1
cR cR
1 1
(cid:15)(cid:15) (cid:15)(cid:15)
R1(cid:31)(cid:127) //// R1
A = =B .
1 1
c c
1 1
Since R is semilocal and integrally closede, R is a principal ideal rineg. Hence R is a principal ideal domain,
1
and so B is a principal ideal ring. Also, note that A is local with maximal ideal n. Since R has infinite
1 1 1
Cohen-Macaulay type, the Drozd-Ro˘ıter conditions fail for the corresponding Artinian pair A ֒→ B . By
1 1
Propositeion 2.3 in [14], there is a ring C between A aend B such that, setting D =C /nC , we have either
1 1 1 1 1 1
e e
dimkD1e≥4 or eD1 ∼=ke[X,Y]/(X2,XY,Y2).e e e (3.1)
(If (DR1) fails, then dim (B /nB ) ≥ 4 and we take C = B ; otherwise (DR2) fails and we take C =
k 1 1e e 1 1 1
A +nB .)
1 1
Now let f = c /cR denoteetheekernel of the lower-righet verteical map in the diagram above, and let Ce be
1 1 1
tehe ringebetween A and B such that C /f= C . Finally, set D = C /nC , and observe that D and D are
1 1 1 1 1 1 1 1 1
naturally isomorphic.
Then C = C1×B2×···×Bs is a ring betweeen A and B. Reducing modulo the maximal ideal n ofeA, we
have
(cid:31)(cid:127) //
A C
(cid:15)(cid:15)(cid:15)(cid:15) (cid:15)(cid:15)(cid:15)(cid:15)
(cid:31)(cid:127) //
k D,
where D = D ×D ×···×D , D = C /nC and D = B /nB for i ∈ {2,...,s}. In particular, by (3.1), we
1 2 s 1 1 1 i i i
have either dimkD1 ≥4 or D1 ∼=k[X,Y]/(X2,XY,Y2).
By Lemma 2.8, if there exists an indecomposable (k ֒→D)-module V ֒→W with W =D(r1)×···×D(rs),
1 1 1 1 s
then there exists an indecomposable (A֒→C)-module V ֒→W with W =C(r1)×B(r2)×···×B(rs). Hence,
2 2 2 1 2 s
by Lemma 2.9, there exists an indecomposable (A ֒→ B)-module V ֒→ W with W = B(r1) ×···×B(rs). By
1 s
Proposition 2.7, there exists an indecomposable maximal Cohen-Macaulay R-module of rank (r ,...,r ). (cid:3)
1 s
The following remark will be useful in the proof of the main theorem.
Remark 3.2. Let k be a field, and let D be a finite-dimensional k-algebrawith dim D ≥4. The following are
k
all the cases that can arise (see the proof of [14, Proposition 2.6] and [11, pages 5-7]).
(1) There exists an element a∈D such that {1,a,a2} is linearly independent over k.
(2) For all a∈D, {1,a,a2} is linearly dependent over k. In this case, one of the following holds.
(a) Thereexistelementsa,b∈Dsuchthat{1,a,b}islinearlyindependentoverkanda2 =ab=b2 =0.
(b) There existelements a, b∈D suchthat {1,a,ab,b}is linearly independent overk and a2 =b2 =0.
(c) The characteristic of k is 2, (D,m ,K ) is a local ring, K is a purely inseparable extension of k
D D D
of degree at least four and a2 ∈k for all a∈K .
D
(d) The field k has cardinality 2 and D =D ×···×D , where l ≥4 and each (D ,m ) is an Artinian
1 l i i
local ring such that D /m =k.
i i
Observe that in case (1), there are elements a, b ∈ D such that {1,a,a2,b} is linearly independent over k. In
case (2c), there are elements a, b ∈ K such that {1,a,ab,b} is linearly independent over k and a2, b2 ∈ k
D
(indeed, choose a, b such that [k(a,b):k]=4).
6 ANDREWCRABBEANDSILVIASACCON
3.1. Building modules over Artinian pairs. Let k be a field, and let D = D × ···× D , where each
1 s
D is a finite-dimensional k-algebra. Assume dim D ≥ 3. Let (r ,...,r ) be a non-trivial s-tuple of non-
i k 1 1 s
negative integers with r ≥ r for all i ∈ {1,...,s}. We shall build explicit (k ֒→ D)-modules V ֒→ W, where
1 i
W =D(r1)×···×D(rs). Thefirstconstructionisageneralizationofconstructionsfoundin[14,Construction2.5]
1 s
and [17, Lemma 2.2]. The second construction is a generalization of Dade’s construction (see [7] and [14,
Theorem 2.9]).
Construction 3.3. Let W =D1(r1)×···×Ds(rs), and let k(r1) be the k-vector space of r1×1 column vectors.
Let ∂: k(r1) →W be the “truncated diagonal map”, given by:
c c c c
1 1 1 1
. . . .
. 7→ . , . ,..., . .
 .   .   .   . 
c c c c
 r1  r1  r2  rs
       
Choose elements a , b ∈ D such that {1,a ,b } is linearly independent over k. Let a = (a ,0,...,0) and
1 1 1 1 1 1
b = (b ,0,...,0) be elements in D. For i ∈ {1,...,s}, let H be the r ×r nilpotent matrix with 1 on the
1 i i i
superdiagonal and 0 elsewhere.
For t∈k, let V be the k-subspace of W consisting of elements of the form
t
∂(x)+(a+tb)∂(y)+b∂(H y),
1
where x and y range over k(r1). Let erj,i ∈ Dj(rj) denote the standard unit vector in Dj(rj) with 1 in the
i-th position. We see that every element in W is a D-linear combination of elements of the form e =
W
(0,...,erj,i,0,...,0), where i ∈ {1,...,rj} and j ∈ {1,...,s}. Note that we can consider erj,i ∈ k(rj) as
an element of k(r1) (by appending zeros if r1 > ri). Let εj ∈ D be an idempotent with support {j}. Then
e =ε ∂(e )∈DV . Hence V ֒→W is a (k ֒→D)-module with W =D(r1)×···×D(rs).
W j rj,i t t 1 s
Construction 3.4. Assume D = D ×D ×···×D , where l ≥ 4 and each (D ,m ) is an Artinian
1 1,1 1,2 1,l 1,i 1,i
local ring such that D /m =k.
1,i 1,i
Let I =(m ×···×m )×(0)×···×(0), and let E =D/I =(k×···×k)×D ×···×D . We construct
1,1 1,l 2 s
a (k ֒→E)-module as follows. For i∈{2,...,s}, let ∂i: k(r1) →Di(ri) denote the map
c c
1 1
. .
. 7→ . .
 .   . 
c c
 r1  ri
   
Let W =(k(r1)×···×k(r1))×D2(r2)×···×Ds(rs). Let V be the k-subspace of W consisting of elements of the
form
(x,y,x+y,x+H y,x,...,x,∂ (x),...,∂ (x)),
1 2 s
where x and y range over k(r1). Note that V contains (x,0,x,x,x,...,x,∂2(x),...,∂s(x)) for all x∈k(r1), and
(0,y,y,H1y,0,...,0) for all y ∈ k(r1). Using idempotents of E, we see that EV = W. Hence V ֒→ W is a
(k ֒→E)-module with W =(k(r1)×···×k(r1))×D2(r2)×···×Ds(rs).
3.2. Main Result. Using the constructions described above, we now state and prove the main theorem.
Main Theorem. Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime
ideals P , ..., P . Assume R/P has infinite Cohen-Macaulay type. Let (r ,...,r ) be a non-trivial s-tuple of
1 s 1 1 s
non-negative integers with r ≥r for all i∈{1,...,s}.
1 i
(1) There exists an indecomposable maximal Cohen-Macaulay R-module of rank (r ,...,r ).
1 s
(2) If the residue field k is infinite, then the set of isomorphism classes of indecomposable maximal Cohen-
Macaulay R-modules of rank (r ,...,r ) has cardinality |k|.
1 s
Proof. With the notation as in Proposition3.1, it is enough to construct the desired modules over the Artinian
pair k ֒→D, where D =D1×···×Ds and either dimkD1 ≥4 or D1 ∼=k[X,Y]/(X2,XY,Y2).
Let W =D(r1)×···×D(rs). Choose a , b ∈D such that {1,a ,b } is linearly independent over k, and let
1 s 1 1 1 1 1
V ֒→W be the (k ֒→D)-module given in Construction 3.3.
t
RANKS OF INDECOMPOSABLE MODULES 7
Observethat,whenk isinfinite,theset{1,a ,b ,(a +tb )2}islinearlyindependentoverk forallbutfinitely
1 1 1 1
many t∈k; moreover,for a fixed t, the set {1,a ,b ,(a +tb )(a +ub )} is linearly independent over k for all
1 1 1 1 1 1
but finitely many u∈k. We show the following:
(i) there is t ∈ k such that V ֒→ W is indecomposable; moreover, if k is infinite, then V ֒→ W is
t t
indecomposable for all but finitely many t;
(ii) if u 6= t and both {1,a ,b ,(a +tb )2} and {1,a ,b ,(a +tb )(a +ut )} are linearly independent
1 1 1 1 1 1 1 1 1 1
over k, then V ֒→ W is not isomorphic to V ֒→ W. By the previous observation, when k is infinite,
t u
there are |k| pairwise non-isomorphic indecomposable (k ֒→D)-modules.
Consider a morphism ϕ: (Vt ֒→ W) →(Vu ֒→W). Since ϕ(Vt) ⊆ Vu and ∂(x) ∈Vt for all x ∈ k(r1), we can
write
′ ′′ ′′
ϕ(∂(x))=∂(x)+(a+ub)∂(x )+b∂(H x )
1
for elements x′, x′′ ∈ k(r1). By linear independence of {1,a1,b1} over k, the elements x′ and x′′ are uniquely
and linearly determined by x. Thus there exist r ×r matrices σ and τ with entries in k such that
1 1
ϕ(∂(x))=∂(σx)+(a+ub)∂(τx)+b∂(H τx) (3.2)
1
for all x∈k(r1). Similarly, there are r ×r matrices µ and ρ with entries in k such that
1 1
ϕ((a+tb)∂(y)+b∂(H y))=∂(µy)+(a+ub)∂(ρy)+b∂(H ρy) (3.3)
1 1
for all y ∈k(r1). Since ϕ is D-linear, we can rewrite (3.3) as:
−∂(µy)+a∂((σ−ρ)y)+b∂((tσ−uρ+σH −H ρ)y)+(a+tb)(a+ub)∂(τy)
1 1
+ab∂((H τ +τH )y)+b2∂((tH τ +uτH +H τH )y)=0 (3.4)
1 1 1 1 1 1
for all y ∈k(r1). Moreover,if u=t and ϕ is an idempotent, then the equation ϕ2(∂(x))=ϕ(∂(x)) gives:
∂(σ2x)+a∂((τσ+στ)x)+b∂((t(τσ+στ)+H τσ+σH τ)x)+(a+tb)b∂((H τ2+τH τ)x)
1 1 1 1
+(a+tb)2∂(τ2x)+b2∂(H τH τx)=∂(σx)+a∂(τx)+b∂((tI+H )τx) (3.5)
1 1 1
for all x∈k(r1).
First suppose {1,a ,b } is linearly independent over k and a2 = a b = b2 = 0. (Note that if D ∼=
1 1 1 1 1 1 1
k[X,Y]/(X2,XY,Y2), then this hypothesis is clearly satisfied.) Equation (3.4) simplifies to
−∂(µy)+a∂((σ−ρ)y)+b∂((tσ−uρ+σH −H ρ)y)=0. (3.6)
1 1
Using linear independence of {1,a ,b } over k, we obtain the equations
1 1
µ=0, σ =ρ, tσ−uρ+σH −H ρ=0,
1 1
and hence
σ[(t−u)I+H ]=H σ. (3.7)
1 1
Ifϕisanisomorphism,thenfromequation(3.2)weseethatσ isinvertible(usethefactsthata ,b arenilpotent
1 1
elements and that if α=β+γ for a unit α and a nilpotent element γ, then β is a unit). This is a contradiction
to equation (3.7) if u6=t. Hence, if u6=t, then (Vt ֒→W)6∼=(Vu ֒→W).
ToshowthatV ֒→W isindecomposable,takeu=tandassumeϕisanidempotent. Equation(3.5)simplifies
t
to
∂(σ2x)+a∂((τσ+στ)x)+b∂((t(τσ+στ)+H τσ+σH τ)x)=∂(σx)+a∂(τx)+b∂((tI+H )τx).
1 1 1
Comparing the coefficients of 1 and a, we have σ2 = σ and τσ +στ = τ. Since u = t, equation (3.7) yields
σH = H σ. Thus σ is either zero or the identity, and, in either cases, τ = 0. Equation (3.2) shows that ϕ is
1 1
either zeroortheidentity; i.e., ϕisa trivialidempotent. HenceV ֒→W isanindecomposable(k ֒→D)-module
t
with W =D(r1)×···×D(rs).
1 s
For the rest of the proof, assume dim D ≥4. By Remark 3.2, we need to consider the following four cases.
k 1
Case (1): {1,a ,a2,b } is linearly independent over k.
1 1 1
In equation(3.4), we use descending induction to show that HjτHi =0 for all i, j ≥0 (cf. the proof of
1 1
Theorem1.4in[11]ortheproofofLemma2.2in[17]). NotethatHjτHi =0fori≥r andj ≥r . Thus
1 1 1 1
we may assume inductively that Hj+1τHi = 0 and HjτHi+1 = 0. Set H =(H ,...,H )∈ End (W),
1 1 1 1 1 s D
8 ANDREWCRABBEANDSILVIASACCON
and observe that H∂(y)= ∂(H y) for all y ∈ k(r1). Replace y by Hiy in (3.4), apply Hj to both sides
1 1
of the resulting equation and use the inductive hypothesis to obtain
−∂(HjµHiy)+a∂(Hj(σ−ρ)Hiy)+b∂(Hj(tσ−uρ+σH −H ρ)Hiy)
1 1 1 1 1 1 1 1
+(a+tb)(a+ub)∂(HjτHiy)=0.
1 1
If {1,a ,b ,(a +tb )(a +ub )} is linearly independent over k, then we conclude that HjτHi = 0 for
1 1 1 1 1 1 1 1
all i, j ≥0. Setting i=j =0, we obtain τ =0.
Thus equation (3.4) simplifies to equation (3.6), and by linear independence of {1,a ,b } over k,
1 1
we obtain equation (3.7). As τ = 0, from equation (3.2) we have ϕ(∂(x)) = ∂(σx); thus, if ϕ is an
isomorphism, then σ is invertible, which contradicts equation (3.7) if u 6= t. Hence, if u 6= t and
{1,a1,b1,(a1+tb1)(a1+ub1)} is linearly independent over k, then (Vt ֒→W)6∼=(Vu ֒→W).
ToshowthatV ֒→W isindecomposable,takeu=tandassumeϕisanidempotent. Inequation(3.4),
t
use descending induction and the fact that {1,a ,b ,(a +tb )2} is linearly independent over k to show
1 1 1 1
thatτ =0. Weobtainσ2 =σfromequation(3.5),andσH =H σfromequation(3.7). Thusσiseither
1 1
zero or the identity, and we conclude that ϕ is a trivial idempotent. Hence, if {1,a ,b ,(a +tb )2} is
1 1 1 1
linearly independent over k, then V ֒→ W is an indecomposable (k ֒→ D)-module with W = D(r1) ×
t 1
···×D(rs).
s
Case (2b): {1,a ,a b ,b } is linearly independent over k and a2 =b2 =0.
1 1 1 1 1 1
Equation (3.4) simplifies to
−∂(µy)+a∂((σ−ρ)y)+b∂((tσ−uρ+σH −H ρ)y)+ab∂(((u+t)τ +H τ +τH )y)=0. (3.8)
1 1 1 1
Assuming u 6= −t, use descending induction in the equation above and the fact that {1,a ,a b ,b } is
1 1 1 1
linearlyindependentoverktoshowthatτ =0(inparticular,observethatτ =0ifu=tandchark 6=2).
Proceed as in the previous case to conclude that if u6=±t, then (Vt ֒→W)6∼=(Vu ֒→W).
To show that V ֒→ W is indecomposable, take u = t and assume ϕ is an idempotent. If chark 6= 2,
t
then τ = 0 as previously observed. If chark = 2, then from equation (3.8) we see that σH = H σ
1 1
and τH = H τ. Thus σ, τ ∈ k[H ], and in particular στ = τσ. Comparing the coefficients of a in
1 1 1
equation (3.5), we conclude that τ = στ +τσ = 0. In either case, τ = 0, and from equations (3.5)
and (3.7), we conclude that ϕ is a trivial idempotent. Hence V ֒→ W is an indecomposable (k ֒→ D)-
t
module with W =D(r1)×···×D(rs).
1 s
Case (2c): The characteristic of k is 2, (D ,m ,K ) is a local ring, K is a purely inseparable extension
1 1 1 1
of degree at least four and a2 ∈k for all a∈K .
1
LetI =m ×(0)×···×(0),andpasstotheArtinianpairk֒→D/I,whereD/I =K ×D ×···×D . By
1 1 2 s
Lemma 2.8, it is enough to construct |k| pairwise non-isomorphic indecomposable (k ֒→ D/I)-modules
V ֒→W, where W =K(r1)×D(r2)×···×D(rs).
1 2 s
Choose a , b ∈ K such that {1,a ,a b ,b } is linearly independent over k, and construct a (k ֒→
1 1 1 1 1 1 1
D/I)-module V ֒→ W, where W = K(r1) ×D(r2) ×···×D(rs), as in Construction 3.3. Using linear
t 1 2 s
independence of {1,a ,a b ,b } over k and the fact that a2, b2 ∈k, compare the coefficients of a, b and
1 1 1 1 1 1
ab in equation (3.4) to obtain:
σ =ρ, σ[(t−u)I+H ]=H σ, (t+u)τ +H τ +τH =0. (3.9)
1 1 1 1
If u 6= t, then u+t 6= 0 (chark = 2), and, by descending induction, from the third equation in (3.9),
it follows that τ = 0. From equation (3.2), we have ϕ(∂(x)) = ∂(σx); thus, if ϕ is an isomorphism,
then σ is invertible, which contradicts the second equation in (3.9) if u 6= t. Hence, if u 6= t, then
(Vt ֒→W)6∼=(Vu ֒→W).
ToshowV ֒→W isindecomposable,takeu=tandassumeϕisanidempotent. Fromthesecondand
t
third equations in (3.9), it follows that σ, τ ∈k[H ]; in particular, στ =τσ. Comparing the coefficients
1
of 1 and a in equation (3.5), we see that σ2 =σ and τ =τσ+στ. It follows that τ =0. As σ ∈k[H ],
1
we conclude that σ is either zero or the identity, and so is ϕ. Thus ϕ is a trivial idempotent. Hence
V ֒→W is an indecomposable (k ֒→D/I)-module with W =K(r1)×D(r2)×···×D(rs).
t 1 2 s
Case (2d): The field k has cardinality 2 and D =D ×···×D , where l ≥4 and each (D ,m ) is
1 1,1 1,l 1,i 1,i
an Artinian local ring such that D /m =k.
1,i 1,i
RANKS OF INDECOMPOSABLE MODULES 9
Let W =(k(r1)×···×k(r1))×D2(r2)×···×Ds(rs), and let V ֒→W be the (k ֒→E)-module given in
Construction 3.4. By Lemma 2.8, it is enough to show that V ֒→W is indecomposable.
Let ϕ∈ End (V,W) be an idempotent endomorphism. Write ϕ= (ϕ ,...,ϕ ,ϕ ,...,ϕ ), where
E 1,1 1,l 2 s
each ϕ , j ∈{1,...,l}, is an r ×r matrix with entries in k and each ϕ , i∈{2,...,s}, is an r ×r
1,j 1 1 i i i
matrix with entries in D . Since ϕ(V)⊆V, there exist r ×r matrices α, β, γ and δ with entries in k
i 1 1
such that
ϕ(x,0,x,...,x,∂ (x),...,∂ (x))
2 s
=(αx,βx,(α+β)x,(α+H β)x,αx,...,αx,∂ (αx),...,∂ (αx)) (3.10)
1 2 s
and
ϕ(0,y,y,H y,0,...,0)=(γy,δy,(γ+δ)y,(γ+H δ)y,γy,...,γy,∂ (γy),...,∂ (γy))
1 1 2 s
for all x, y ∈k(r1). Comparing components, we obtain equations which lead to the conditions:
(i) ϕ =α for all j ∈{1,...,l}, and
1,j
(ii) αH =H α.
1 1
Since ϕ is an idempotent, so is α. As αH = H α, we conclude that α is either zero or the identity.
1 1
From equation (3.10), we see that if α is zero (respectively, the identity), then ϕ is zero (respectively,
i
the identity) for all i∈{2,...,s}. Thus ϕ is a trivial idempotent. Hence V ֒→W is an indecomposable
(k ֒→E)-module with W =(k(r1)×···×k(r1))×D2(r2)×···×Ds(rs). (cid:3)
From the main theorem, we can deduce the structure of the monoid C(R) of isomorphismclasses of maximal
Cohen-Macaulaymodules(togetherwith[0])whenR/QhasinfiniteCohen-Macaulaytypeforallminimalprime
ideals Q of R. The following corollary is a special case of Theorem 3.15 in [5]. Recall that the splitting number
q of R is defined by q = |MinSpec(R)|−|MinSpecb(R)|, where MinSpec(R) and MinSpec(R) denote the set of
minimal primbe ideals of R and R respectively.
b b
Corollary 3.5. Let (R,m,k) be a one-dimensional analytically unramified local ring with splitting number q.
b
Let R denote the m-adic completion of R, and let Λ denote the set of isomorphism classes of indecomposable
maximal Cohen-Macaulay R-modules. Assume R/Q has infinite Cohen-Macaulay type for all minimal prime
idealsbQ of R.
(1) If q =0, then C(R)b∼=C(R)∼=N(Λ). b
(2) If q ≥b 1, then C(R)∼=Ker(A(R))∩N(Λ), where A(R) is a q×|Λ| integer matrix such that every element
of Z(q) occurs |k|·|N| timbes as a column of A(R).
Proof. If q =0, then the natural embedding C(R)֒→C(R) is surjective, and hence an isomorphism.
Assume q ≥1. Let P , ..., P be the minimal prime ideals of R, andlet Q ,..., Q be the minimal prime
1 s i,1 i,ti
ideals of R lying over the minimal prime ideal P of R.bLet l ∈ {0,...,s−1} denote the number of minimal
i
prime ideals of R with t =1. After renumbering (if necessary), assume that P , ..., P are the minimal prime
i 1 l
ideals of Rbwith ti =1, and Pl+1, ..., Ps are the minimal prime ideals of R with ti ≥2.
ByacorollarytoatheoremduetoLevy-Odenthal[12,Theorem6.2],thereexistsaq×|Λ|integermatrixA(R)
such that C(R)∼= Ker(A(R))∩N(Λ). The columns of the matrix A(R) are indexed by the isomorphism classes
of indecomposable maximal Cohen-Macaulay R-modules; if M is an indecomposable maximal Cohen-Macaulay
R-module of rank (r ,...,r ,...,r ,...,r ), then the column of A(R) indexed by [M] is
1,1 1,t1 s,1 s,ts
b T
r −r ··· r −r ··· r −r ··· r −r .
b l+1,1 l+1,2 l+1,1 l+1,tl+1 s,1 s,2 s,1 s,ts
Let a = a (cid:2) ··· a ··· a ··· a T ∈ Z(q). For each i ∈ {l +(cid:3)1,...,s}, choose an
l+1,2 l+1,tl+1 s,2 s,ts
integer b such that b >max{|a |:2≤j ≤t }. Set
i i i,j i
(cid:2) (cid:3)
1 if i∈{1,...,l},
r =
i,1
(bi if i∈{l+1,...,s},
and for i ∈ {l+1,...,s} and j ∈ {2,...,t }, set r = r −a . Since R/Q has infinite Cohen-Macaulay
i i,j i,1 i,j
type for all minimal prime ideals Q of R, the main theorem guarantees the existence of an indecomposable
maximal Cohen-Macaulay R-module Ma of rank r = (r1,1,...,r1,t1,...,rs,1b,...,rs,ts). By construction, the
column indexed by [Ma] is a. b
b
10 ANDREWCRABBEANDSILVIASACCON
ObservethatthemaintheoremalsoguaranteestheexistenceofanindecomposablemaximalCohen-Macaulay
R-module Ma,n of rank r+(n,...,n), n ∈ N. By construction, the column indexed by [Ma,n] is a, and we
conclude that a appears |N| times as a column of A(R). If k is infinite, then the main theorem guarantees the
ebxistence of|k| pairwisenon-isomorphicindecomposablemaximalCohen-MacaulayR-modulesofthe same rank
r+(n,...,n);notethatthecolumnsindexedbytheseindecomposablemodulesareequaltoa. Henceaappears
|k|·|N| times as a column of A(R). b (cid:3)
4. Ranks of indecomposable maximal Cohen-Macaulay modules over R=k[[x,y]]/((x3−y7)x)
The main theorem describes which tuples can be realized as ranks of indecomposable maximal Cohen-
Macaulay R-modules when R/P has infinite Cohen-Macaulay type. In this case, for every non-trivial s-tuple
1
(r ,...,r )ofnon-negativeintegers,thereexistsanindecomposablemaximalCohen-MacaulayR-moduleofrank
1 s
(r ,...,r ) provided r ≥r for all i∈{1,...,s}. What happens if r <r for some i?
1 s 1 i 1 i
We give an example of a complete local ring R of infinite Cohen-Macaulay type with exactly two minimal
prime idealsP andP suchthatR/P hasinfinite Cohen-Macaulaytype (andR/P hasfinite Cohen-Macaulay
1 2 1 2
type), for which there are no indecomposable maximal Cohen-Macaulay R-modules of rank (1,n) for n≥4.
Consider the hypersurface
R=k[[x,y]]/((x3−y7)x)
with minimal prime ideals P = (x3 −y7)/((x3 −y7)x) and P = (x)/((x3 −y7)x). Observe that R/P ∼=
1 2 1
k[[x,y]]/(x3 −y7) ∼= k[[t7,t3]] has infinite Cohen-Macaulay type, and R/P ∼= k[[y]] is a discrete valuation ring,
2
hence of finite Cohen-Macaulay type.
The integral closure R of R is R=k[[t]]×k[[y]], and the map ϕ: R֒→R is given by:
x7→(t7,0), y 7→(t3,y). (4.1)
Note that R is generated as an R-module by (0,1) and (ti,0) for i∈{0,1,2}. The first Drozd-Ro˘ıter condition
fails, and so R has infinite Cohen-Macaulay type.
The conductor of R is c=(x3,xy4,y7)/((x3−y7)x), and the conductor square is:
k[[x,y]] //
R= k[[t]]×k[[y]]=R
((x3−y7)x)
(cid:15)(cid:15) (cid:15)(cid:15)
R k[[x,y]] // k[[t]] k[[y]] R
A= = × = =B,
c (x3,xy4,y7) (t19) (y7) c
where R =(A֒→B). Write B =B ×B , whereB =k[[t]]/(t19) andB =k[[y]]/(y7). Reducing the Artinian
art 1 2 1 2
pair A ֒→ B modulo the maximal ideal n of A, we obtain the Artinian pair k ֒→ D, where D = D ×D =
1 2
k[[t]]/(t3)×k.
4.1. Matrixrepresentation. Givenan(A֒→B)-moduleV ֒→W,whereW =B(r1)×B(r2),onecanrepresent
1 2
V ֒→W by matrices in the following way. An element w of W can be written as a column vector
w
w= 1 ,
w
2
(cid:20) (cid:21)
where each w is a r ×1 column vector with entries in B . Let {v ,...,v } be a minimal set of generators of
i i i 1 m
V as an A-module. Since V is an A-submodule of W, we can write each generator v as
j
v
v = 1,j .
j v
2,j
(cid:20) (cid:21)
Thus V is the A-column span of the matrix:
v v ··· v
Q= 1,1 1,2 1,m .
v v ··· v
2,1 2,2 2,m
(cid:20) (cid:21)
Note that each row is a block matrix of size r ×m as each v is an r ×1 column vector.
i i,j i
As done in [10], [18] and [6], we can perform row and column operations without changing the module. In
particular, observe the following.