R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
D MITRI A LEXEEV
On quasi-projective uniserial modules
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DMITRI
ALEXEEV (*)
ABSTRACT - Let R be a valuation domain with maximal ideal P . We
study quasi- projective
uniserial modules over R .By making
use of the absence of «shrin- kable» uniserial modules over R we prove our main result: a characterization ofquasi-projectivity
of a uniserial module U over R in terms oflifting
of endo-morphisms
of factors of U.Using
this characterization allows us to describequasi-projective
ideals of R in terms ofcompleteness
of certain localizations offactor-rings
of R . Archimedean ideals of R admit the bestpossible description
from this
point
of view. We show that anon-principal
archimedean ideal of R isquasi-projective
if andonly
if R/K iscomplete
in theR/K-topology
for eacharchimedean ideal P .
Finally,
we show thattaking
tensorproducts
witharchimedean ideals preserves
quasi-projectivity.
1. Uniserial modules.
Let us
begin
with necessary definitions.DEFINITIONS. A module over a
ring
is called uniserial if its submo- dules form a chain under inclusion. A commutativeintegral
domain R is a valuation domain if it is a uniserial module over itself.From now on, the letter R will denote a valuation domain with maxi- mal ideal P and
quotient field Q,
unless stated otherwise. We refer the reader to Fuchs and Salce[6]
for a treatment of modules over valuation domains.Submodules and factor modules of uniserial modules are likewise
(*) Indirizzo dell’A.:
Department
of Mathematics, TulaneUniversity,
New Orleans, LA 70118; e-mail: alexeev@math.tulane.eduThis paper is a part of the author’s Ph.D. thesis written under
supervision
ofProfessor Laszlo Fuchs at Tulane
University.
uniserial. The
simplest examples
of uniserial modules are besides the va-luation
rings themselves,
theirrings
ofquotients,
theircyclic modules,
and more
generally,
the R-modules of the formIIJ
where J I are R- submodulesof Q.
Uniserial modules of the latter kind are called stan- dard uniserial. The existence of non-standard uniserial modules has been first establishedby
Shelah in[10].
Standard uniserial modules are
completely
classifiedby
thefollowing proposition.
PROPOSITION 1.1
(Shores
and Lewis[11]).
Two standard uniserials modules IIJ and I ’ /J’ over a valuation domain R areisomorphic if
and
only if
there exists an element 0 ~ q EQ
such that I =qI ’
andJ = qJ’ .
Several
properties
of a uniserial moduledepend
on itstype.
DEFINITIONS. Let R be a valuation domain with the maximal ideal P . We use the
following
notation from[6].
If U is a torsion uniserial and 0 ~we set
The fractional ideal I is called the
height
ideal of u , and we say that U is oftype [I/J] (the isomorphy
classof I/J), t( U) _ [I/J].
Thetype t( U)
doesnot
depend
on the choice of 0 # u E U. For a uniserial module U we defi-ne ideals
It is easy to
verify
that bothU
=I
and U =J
areprime
ideals of Rcontaining
the annihilator Ann U . An ideal I of R is called archimedeanif7"=P.
It is a
good
time to introduce the mainobjects
of ourstudy:
thequasi- projective
modules. In thefollowing definition,
R may be anarbitrary ring.
DEFINITION. An R-module U is called
quasi-projective
if it isprojec-
tive relative to all exact sequences of the form
0 -~ V -~ U ~ U/V -~ 0 ,
where V is a submodule of U and jr is the canonical
projection.
Thatis,
for every
homomorphism f :
U ~ U/V there exists a mapf ’ :
U ~ U suchthat the
following diagram
commutes:For more on
quasi-projective
modules via relativeprojectivity
see
[7].
As the
following proposition states, possibilities
ofendomorphisms
ofquasi-projective
uniserials are limited.PROPOSITION 1.2. A
surjective endomorphism of
aquasi-projective
uniserial moduLe U over any
ring R is
anautomorphism.
PROOF. Uniserial modules are
indecomposable.
Fuchs andRanga-
swamy show in
[5]
that if a factor U/V of aquasi-projective
module U isisomorphic
to a summand ofU,
then V is alsoisomorphic
to a summandof U.
Thus, if cp :
U- U is asurjective endomorphism
with non-trivial kernelV,
then U/V = U and V must beisomorphic
to U. Since V is a pro- per submodule ofU,
it has to be standard uniserial.Thus,
U is standard uniserial either.Suppose
that U = IIJ for some J Then V =h /J
for some
h
such that JI,
I.By Proposition 1,
there are non-zero ele-ments p,
qin Q
such thatThus,
we have = J =pql1,
whichimplies
I =I1.
This renders= U
impossible,
a contradiction.The
following corollary
is an immediate consequence ofProposi-
tion 1.2.
COROLLARY 1.3. Let R be a valuation
ring. If U is a quasi-projecti-
ve uniserials module over
R,
thenUn. Moreover, if
U is non-stan-dard,
thenU
=~7~.
PROOF.
Suppose
that U is aquasi-projective
uniserial R-module. Ifr E R then
multiplication by r
is asurjective endomorphism
of U if andonly
if r f1.U
a and aninjective endomorphism
if andonly
if Thefirst statement follows now from
Proposition
1.2.If U is non-standard
quasi-projective,
then every monicendomorphi-
sm of U has to be an
isomorphism
because all proper submodules of Uare standard uniserials. This
implies U p ;
U aand, consequently, UP
=- Up.
The
following
Lemma uses a resultby
Facchini and Salce[3]. They
call a uniserial module U over an
arbitrary ring
shrinkable if U =V/W
for some proper submodules 0 W V U. It isproved
in[3]
that the-re are no shrinkable uniserial modules over commutative or Noetherian
rings.
We use the result to prove a moregeneral
statement.LEMMA 1.4. Let U and V be two Uniserial moduLes over a valua- tion domain R.
If
there exist both amonomorphism f
and anepimor- phism
gfrom
U to V then at least oneof f
and g is anisomorphisrn.
Mo-reover,
if
bothU,
V are standard and(a)
Vthen g is
anisomorphism;
(b) V
thenf
is anisomorphism.
PROOF. There are four
possibilities depending
on whether U and Vare standard or not. If both U and V are
non-standard,
then every mono-morphism f :
U- V has to be onto. If U is standard and V isnot,
then noepimorphism
g : U ~ V exists.Similarly,
if U is non-standard and V is standard then nomonomorphism f
exists. It remains to consider the caseof standard U and V.
There is
nothing
to proveif f or g
is anisomorphism. Suppose that f
maps
monomorphically
onto a proper submodule of Vand 9
has a non-tri-vial kernel. We claim that this is
impossible.
To prove theclaim,
we writeU = I/J and V =
I’IJ’,
with submodulesJ I , J’ I ’ of Q .
Then Imf *
==Ií/J’
and forappropriate J1
andI1
tin Q.
We haveisomorphisms
implying
that V isshrinkable,
which isimpossible by [3].
This proves thefirst statement.
We conclude that whenever there is a
pair f, g satisfying
thehypo- theses,
then eitherJ1
= J’ or I ’ =Ii .
Thesepossibilities
are definedby
the inclusions of
the «sharps»
of V andimply
thateither g
is monic orf is
epic, accordingly.
We are
going
to show that thequasi-projective property
of uniserial modules isclosely
related to theproperty
oflifting endomorphisms.
Thefollowing
definition does notrequire R
to be a valuation domain.DEFINITION. Let M be an R-module and N its submodule. We say that an
endomorphism h
of M/N can besifted
orlifts
toM,
if there existsan
endomorphism
h ’ of Mmaking
thefollowing diagram
commutative:Here Jr denotes the canonical
projection.
If all theendomorphisms
of ea-ch factor of a module M
lift,
then M will be calledweakly quasi-projecti-
ve. This term was introduced
by Rangaswamy
andVanaja
in[9].
Trivially,
aquasi-projective
module isweakly quasi-projective.
Theconverse is not true even for uniserial modules: the abelian group
~(~ °° )
is an
example
of aweakly quasi-projective
but notquasi-projective
unise-rial module over Z.
Interestingly,
in case of valuationdomains,
thenecessary condition of weak
quasi-projectivity together
with isalso sufficient for U to become
quasi-projective.
THEOREM 1.5. Let R be a valuation domain and U a uniserials R- module. The
following
conditions areequivalent:
(a)
U isweakly quasi-projective
andUO.
(b)
U isquasi-projective.
PROOF OF
(a)
=>(b). Suppose
thatevery h E EndR U/V,
for each VU,
can be lifted to an element ofE ndR
U and U # 5U a.
Let Vbe a submo- dule of Uand f :
U -~ U/V ahomomorphism. Thus,
we aregiven
the solidpart
of thefollowing diagram,
where ,~ 1 is the canonicalprojection:
If we denote the kernel
of f by W,
thenf
factorsthrough
the canonicalprojection
.7r 2: U ~ U/W and an inclusion i : U/V . Since U is uni-serial,
either V =W,
VW,
or W V.If V =
W,
then i eEndR
U/V. It lifts to Uby assumption.
If V
W,
then there is canonicalprojection
U/V-U/W .
Embed-ding
it in theprevious diagram
results in thefollowing:
The
composition z ojrg
is anendomorphism
of U/V and can be lifted to amap
f ’ : U ~ U by
theassumption.
If W
V,
then we have the canonicalepimorphism U/Tl,
and so
by
Lemma1, i
is anisomorphism. Fixing
submodulesI,
Jof Q
such that
t( U) _ [7/J],
we can writeI1 /J
andW= I2 /J,
where J12 h
I and the latterisomorphism
is the restriction of the former.Since U/V =
U/W,
we must haveI/I1 = I/I2. By Proposition 1,
there existsa q E R such that
qI
= I andqI1= I2 .
This means q ~10
=U p ;
There-fore, multiplication q by
q is anautomorphism
of U such thatq V =
W. Itnaturally
induces anisomorphism q :
U/V- U/W . In this case we have thefollowing diagram:
Here, g
is alifting of 4 -
i. The mapf ’ =
is a desiredlifting of f.
(b)
~(a).
Thisimplication
is trivial.In
particular,
for ideals ofR ,
weakquasi-projectivity always implies
quasi-projectivity.
2.
Endomorphisms
of uniserial modules.Since
quasi-projectivity depends
on thelifting property
of endomor-phisms, knowing
the structure ofendomorphism rings
of uniserial mo-dules will enhance our
understanding
of theirquasi-projectivity.
If U is a uniserial module and Ann U is its annihilator
ideal,
then we usethe
following
notation from Fuchs and Salce[6]:
we write Ann U = I if there is a u E U such that Ann u = AnnU,
and we write Ann U = I + otherwise. The module U is calledfinitely
annihilated ornon-finitely annihilated, respectively.
It is known that a uniserial U oftype [7/J]
isnon-finitely
annihilatedexactly
ifJ p I = I .
SeeBazzoni,
Fuchs and Sal-ce
[2]
for details.Thus,
there are twotypes
of uniserial modulesdistinguished by
thestructure of their annihilator ideals. Each
type
has aspecific
kind of theendomorphism ring,
describedby
thefollowing proposition.
THEOREM 2.1
(Shores
and Lewis[11]).
Let R be a vaLuationring
and U a uniserials
(a) If Ann
U =I,
thenUP
a and U carries the natural structu-re
of
an,S-module,
where S is thering
RII localized atU 0 11,
andEndRU=S.
(b)
If Ann U = I+ ,
then U and U carries the natural struc- ture of aT-module,
where T is thering
RII localized at andE ndR
Uis
isomorphic
to thecompletion
T of T in theT-topology.
The
endomorphisms
of U can be viewed asmultiplications by
appro-priate
elements from either ,S or T.COROLLARY 2.2. A valuation domain R is maximal
if
andonly if
all uniserial R-modules U zvith
U ll U
a arequasi-projective.
PROOF. Theorem 3.5 of Herrmann
[8]
states that is maximal if all submodulesof Q
arequasi-projective.
This provessufficiency
of thecondition.
Conversely,
if R ismaximal,
then R and all its factors arelinearly compact
in the discretetopology.
Thus all uniserial R-modules are stan- dard.Hence,
for every uniserial moduleU = I/J, J I ~ Q,
withthe
endomorphism ring EndR ( U)
is the localization(R/Ann
U) , which is also maximal.Therefore, endomorphisms
U and its factors areinduced
by multiplications by appropriate
elementsof RUa . Thus,
everysuch U is
weakly quasi-projective.
The statement follows from Theo-rem 1.5.
Corollary
2.2 for ideals is aparticular
case of a moregeneral
resultby
Fuchs
[4].
He proves that for a cardinal x a valuation domain R is K-ma-ximal if and
only
if everyK-generated
ideal of R isquasi-projective.
We are
going
to show thatquasi-projectivity
of a uniserial module iscompletely
determinedby
itstype.
THEOREM 2.3. Let R be a vaLuation domain and U be a uniserials R-module.
Suppose
thatt( U) = [I/J] (J
I ~Q).
Then U isquasi-pro- jective if
andonly if
IIJ is.PROOF.
By
Theorem1.5,
thequasi-projectivity
of(non-)
standard uniserial Usatisfying Up (UU
a = isequivalent
to its weakquasi- projectivity.
The«sharp»
ideals associated to U and IIJ arenecessarily
the same.
Thus,
we have to show that the two modules areweakly quasi- projective
at the same time.Suppose
that V is a(proper)
submodule ofU,
V ~ U. We can choose I such thatt(V) = [I1/J]
and=
[I/I1 ].
On the otherhand,
for anygiven
idealI, (J ~ I)
there is a submodule V of U withtype
as above. Since uniserial modules of the sa- metype
havenaturally isomorphic endomorphism rings,
the endomor-phisms
of UIV lift to U if andonly
if those ofI/h
lift to I/J . Theproof
isfinished.
COROLLARY 2.4. Let U be a uniserial R-rrzodule.
If Tor1 (U, KIL) (L
K ~Q)
isquasi-projective,
thenTor1(U, V)
isquasi-projective for
each V
of type [K/L].
PROOF. It is known
(see [2]) that,
for uniserial R-modules U and V withtypes [7/J]
and[K/L] respectively,
theTorR product
of U and V is uniserial andThus, TorR ( U, KlL)
andTorR (U, V)
have the sametype.
A reference toTheorem 2.3 finishes the
proof.
Bazzoni, Fuchs,
and Salce prove in[2]
that theisomorphy
classes oftorsion uniserial R-modules form a commutative
semigroup
under opera-tion
TorR. Corollary
2implies
that the orbit of a module U under theTorR operation
with uniserials of the sametype
consistsentirely
of qua-si-projective
modules if it contains at least onequasi-projective.
3. Ideals of valuation domains.
In this section we use Theorems 1.5 and 2.1 to
study quasi-projectivi- ty
of ideals of a valuation domain R. We will show thatquasi-projectivity
of an archimedean ideal
depends
on thecompleteness
of certain localiza- tions offactor-rings
of R .THEOREM 3.1. Let R be a valuation domain. An ideal I
of R is
qua-si-projective if
andonly if for
each ideal J I with Ann IIJ =(J : I)+
the
ring
is
complete in
itsS-topology.
PROOF. This is a direct consequence of the Theorem 1 and Theorem 2. Since for an ideal
I ,
0 = I a ~I always,
thequasi-projectivity
of I isequivalent
tolifting
of eachendomorphism
of every factor IIJ to I. When Ann IIJ =(J : 1)+
and thering
is notcomplete,
thering EndR I/J = S
contains elements which are not induced
by
elements ofThe absence of such elements is
equivalent
to thecompleteness
of S(for
every such
J),
and thequasi-projectivity
of I.It is
possible
togive
a moreexplicit
characterization ofquasi-projec- tivity
of archimedean ideals. There are two cases,according
as P isprin- cipal
or not. If P isprincipal,
thenonly
theprincipal
ideals are archime-dean.
Thus, they
all areprojective and, therefore, quasi-projective.
Toconsider the second
alternative,
we need aspecial
case of Lemma 1.4from
[2].
Theproof
isprovided
for the sake ofcompleteness.
LEMMA 3.2. Let J I be archimedean ideals
of
a valuation do- main R . Then Ann IIJ =(J : 1)+ if
andonly if
I is notprincipal.
PROOF. The
«only
if»part
is trivial. To prove the converse, let I bean
infinitely generated
archimedean ideal. Assume that there is an ele- ment i E I such that Ann iR/J = Ann I/J . Since I is notprincipal,
we canchoose i ’ E I such that iR i ’ I~ I . That
is,
i = si ’ for a non-unit s of I~ .Then Ann iRIJ = Ann
i ’ R/J,
hencei -1 J
= andi ,
i ’ are associa-tes. A contradiction.
Hence,
no such element i E I exists and Ann IIJ == (J : I)+.
+ . 8THEOREM 3.3. Let R be a valuation domain with
infinitely
genera- ted maximal ideal P. Anon-principal
archimedean ideal I isquasi- projective if
andonly if RIK
iscomplete
in theRIK-topology for
each ar-chimedean ideal
K 0
P .PROOF.
By
Theorem3.1,
I isquasi-projective
if andonly
if RIK iscomplete
whenever Ann(IIJ)
= K + for some ideal J. The ideal J has to be archimedeanby
Theorem 2.1. It remains to prove that for differentJ,
Ann
(IIJ)
can be any archimedeanideal 0 P,
butnothing
else.If then Ann I/IK = K + .
If K = P then K = sP for some s E R . Assume that Ann IIJ = K + for
some
(archimedean)
ideal J. We have IK = IsP = sI ~ J. But then(sR)I ~ J,
which means that IIJ has annihilatorlarger
than K. Contra- diction.Hence,
idealsisomorphic
to P cannot be annihilators of IIJ. Thiscompletes
theproof.
The
following corollary
is an immediate consequence of Theorem 3.3.COROLLARY 3.4. Let R be a valuation domain with
infinitely
gene- rated maximal ideal P .If
there exists anon-principal quasi-projective
archimedean
ideal,
then all archimedean idealsof
R arequasi-projec-
tive.
It is difficult to
give
a moreexplicit
characterization of thequasi-pro- jectivity
of non-archimedean idealsbeyond
Theorem 3.1.However,
weshow that one has a way of
«producing»
newquasi-projective
ideals star-ting
with aquasi-projective
ideal I andtaking
tensorproducts
of I witharchimedean ideals. This result is in a way
analogous
to Theorem 3.3 from[1].
First,
we need thefollowing
lemma.LEMMA 3.5. Let R be a valuation domain with maximal ideal P .
Suppose
that I is anon-principal
idealof R . If
0 ~ r ER ,
then the endo-mor~phism rings of IlrR
and IlrP are(naturally) isomorphic.
PROOF. This is a consequence of Theorem 2.1 and the
following
twoobservations.
Firstly,
IlrR and IlrP have the same annihilator A+ ,
where A = rR : I = rP : I.
For,
if sI c rR(0 ~ s E A),
then sI c rP since I is notprincipal
and rP is the maximal submodule of rR .Secondly,
bothmodules have the same
« sharp »
ideals because rR and rP are archime- dean.By
Theorem2.1,
theendomorphism rings
of I/rR and IlrP are iso-morphic
to thecompletion
of R/A in theR/A-topology.
Thisisomorphism
is natural.
We have the
following
theorem.THEOREM 3.6. Let R be a vaLuation domain with
non-principal
maximal ideal P .
Suppose
that I is anon-principal
idealof
R and J isan archimedean ideal. Then I is
quasi-projective if
andonly if I ®R J
is.PROOF. Over valuation domains
I ®R J is naturally isomorphic
to IJ .This allows us to consider submodules of
I ®R J
as ideals contained in IJ and vice-versa. For ideals of valuation domains thequasi-projectivity
isequivalent
to the weakquasi-projectivity.
Thus we need to prove that I isweakly quasi-projective
if andonly if I ®R J
is. Since thearguments
ineach way are
similar,
wegive
theproof
of the «if»part only.
The case ofprincipal
J is trivial.Suppose
thatI ®R J
isweakly quasi-projective,
K I and cp ee
EndR (I/K).
If K is notprincipal, tensoring
with J is«reversible»,
thatis, J) = K ®R P = K.
Theisomorphisms
are natural. Consi-der the
following diagrams.
Taking
the tensorproduct
of the solidpart
of thediagram
on the leftwith
J,
one obtains thediagram
on theright. Here, map f is
alifting
ofcp
OR 1,
which existsby assumption. Maps
x and x0R
1 are canonical pro-jections. Taking
the tensorproduct
with R :J,
one returns to the leftdiagram. Map f ’
is alifting
of cp , which makes the left square commute. It remains to consider thespecial
case ofprincipal
K.If K = rR then
K0RJ0R(R: J)
* rP.By
Lemma3.5,
theendomorphism
cp : I/rR -I/rR is an element of the R/Ann(I/rR )-
completion
of thering
R/Ann(I/rR ).
Since thisring
is also the endomor-phism ring of I/rP ,
cp can be considered as anendomorphism
of I/rP . Assuch,
it can be lifted to Iusing technique
of theprevious paragraph.
Clearly,
thislifting
is also alifting
of theoriginal
cp :IlrR
Theproof
is finished.We conclude with the
following
observation. Sinceisomorphy
classesof archimedean ideals form a group under the tensor
product operation, application
of Theorem 3.6 delivers an alternativeproof
ofCorollary
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Manoscritto pervenuto in redazione il 15