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STAR
O
PERAT
IO
NS
AND
PULLBACKS
MarcoFontana
MiHeePark
AMS
Meeting,Tallahassee,March12-13,2004
RobertGilmerandJoeMott:
FortyYearsofCommutativeRingTheoryatFloridaStateUniversity
InthistalkIwillstudythestaroperationsonapullbackofintegral
domains.
In
particular,Iwillcharacterizethestaroperationsofa
domain
arising
from
a
pullback
of�a
generaltype�by
introducing
new
techniquesfor�projecting�and
�lifting�
staroperationsunder
surjectivehomomorphismsofintegraldomains.
Iwillapplypartofthetheorydevelopedheretogiveacompleteposi-
tiveanswertoaproblem
posedbyD.F.Andersonin1992concerning
thestaroperationsonthe�D+
M�constructions.
1
NOTATION
LetD
beanintegraldomainwithquotient�eldL.
LetF(D)denotethesetofallnonzeroD-submodulesofL,
F(D)thesetofallnonzerofractionalidealsofD,
f(D)bethesetofallnonzero�nitelygeneratedD-submodulesofL.
Obviously,f(D)�F(D)�F(D):
2
InthistalkIwillmainlyconsiderthefollowingsituations:
(þ)T
representsan
integraldom
ain,M
an
idealof
T,kthe
factor
ring
T=M,D
an
integraldom
ain
subring
ofkand
':T!
T=M
=:k
the
canonicalprojection.
SetR:=
'�
1(D)=:T�kD
thepullback
ofD
inside
T
with
respectto
',
hence
R
is
an
integraldom
ain
(subring
ofT).LetK
denote
the
�eld
ofquotientsofR.
(þ+)LetL
be
the
�eld
ofquotients
ofD.
In
the
situation
(þ),we
assum
e,m
oreover,thatL�k,and
denotebyS:=
'�
1(L)=:T�kL
thepullbackofLinsideTwithrespectto'.Then
Sisan
integral
dom
ain
with
�eld
ofquotients
equalto
K.
In
this
situation,M,
which
is
a
prim
e
idealin
R,is
a
m
axim
alidealin
S.
M
oreover,if
M
6=
(0)and
D
(
k,
then
M
is
a
divisorialidealof
R,
actually,
M
=
(R:T).
3
R:=
'�
1(D)
'jR
�!
D
(�L:=
qf(D))
???y
???y
T
'�!
k:=
T=M
(�qf(k))
\
K
:=
qf(R)=
qf(T)
(þ)
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
R:=
'�
1(D)
'jR
�!
D
???y
???y
S:=
'�
1(L)
'jS
�!
L:=
qf(D)
???y
???y
T
'�!
k:=
T=M
\
K
:=
qf(R)=
qf(T)
(þ+)
Recallthatamapping?:F(D)!
F(D);E7!
E?,iscalleda
sem
istar
operation
on
D
forall06=
x2LandE;F2F(D):
(?1)
(xE)?=
xE?;
(?2)
E�F)
E?�F?;
(?3)
E�E?andE?=
(E?)?=:E??:
A
staroperation
on
D
isamap?:F(D)!
F(D),
E
7!
E?;that
satis�estheproperties(?2);(?3)
forallE;F
2F(D);moreover,for
each06=
x2L
andE2F(D):
(??1)(xD)?=
xD;
(xE)?=
xE?:
4
Let?D
[respectively,?T]beastaroperationontheintegraldomain
D
[respectively,T].Our�rstgoalistode�neinanaturalwayastar
operationonR,whichwewilldenoteby3,associatedtothegiven
staroperationsonD
andT.Moreprecisely,ifwedenotebyStar(A)
thesetofallthestaroperationsonanintegraldomainA,thenwe
wanttode�neamap
�
:Star(D)�
Star(T)!
Star(R),
(?D;?T)7!
3
.
ForeachnonzerofractionalidealIofR,set
I3
:=
\ (x�
1'�
1 �xI+
M
M
�?D !
jx2I�
1;x6=
0 )\(IT)?T
;
whereifx
I+M
M
isthezeroidealofD
(i.e.,ifxI�
M
),thenweset'�
1 ��x
I+M
M �?D �:=
M
.
5
Proposition1Keeping
the
notation
and
hypotheses
introduced
in
(þ),
then
3
de�nes
a
star
operation
on
the
integraldom
ain
R
(=
T�kD).
Thepreviousconstructionofthestaroperation3
givestheideafor
�liftingastaroperation�withrespecttoasurjectiveringhomomor-
phim
betweentwointegraldomains.
Corollary2LetRbean
integraldom
ain
with
�eld
ofquotientsK,M
a
prim
eidealofR,D
:=
R=M
and
':R!
D
thecanonicalprojection.
Assum
e
that?isa
staroperation
on
D.Foreach
I2F(R),set:
I?'
:=
\ nx�
1'�
1 ��xI+M
M
�? �jx2I�
1;x6=
0 o
=
\ �x'�
1 ��x�1
I+M
M
�? �
jx2K;I�xR �;
Then
?'
isa
staroperation
on
R.
6
Let�:R
,!
T
beanembeddingofintegraldomainswith
thesame
�eldofquotientsK
andlet�beasemistaroperationonR.De�ne
��:F(T)!
F(T)bysetting:
E��:=
E�
;
foreach
E2F(T)(�F(R)):
Thenitiseasytoseethat:
(a)If�isnottheidentitym
ap,then��isasem
istar,possiblynon�star,
operation
on
T,even
if�isa
staroperation
on
R.
Notethat,when
�isastaroperation
on
R
and
(R
:K
T)=
(0),a
fractionalidealE
ofTisnotnecessarilyafractionalidealofR,hence
��isnotde�nedasastaroperationonT.
(b)W
hen
T:=
R�,then
��de�nesa
staroperation
on
R�
:
7
Conversely,let?be
a
semistaroperation
on
the
overring
T
ofR.
De�ne
?�:F(R)!
F(R)bysetting
E?�
:=
(ET)?
foreach
E2F(R):
Thenitiseasytoseethat
(c)?�isa
sem
istaroperation
on
R.
(d)Foreach
sem
istaroperation
?on
T,we
have
(?�)�=
?.
(e)For
each
sem
istar
operation
�on
R,
we
have
(�� )��
�(since
E(�� )�
=
(ET)��=
(ET)�
�E�
foreachE2F(R)).
8
Using
thenotation
introduced
above,weimmediatelyhavethefol-
lowing:
Corollary3W
ith
thenotation
and
hypothesesintroduced
in
(þ)and
Proposition
1,ifwe
use
the
de�nition
given
in
Corollary
2,we
have
3
=
(?D)'^(?T)�:
9
Wenextexaminetheproblem
of�projecting
astaroperation�with
respecttoasurjectivehomomorphism
ofintegraldomains.
Proposition4LetR,K,M,D,'be
asin
Corollary
2
and
letLbe
the
�eld
ofquotients
ofD.
Let
�be
a
given
staroperation
on
the
integraldom
ain
R.Foreach
nonzero
fractionalidealF
ofD,set
F�'
:=
\ ny�
1' ��'�
1(yF) �� �jy2F�
1=
(D
:LF);y6=
0 o:
Then
�'
isa
staroperation
on
D.
10
Incaseofapullbackoftype(þ+)thede�nitionofthestaroperation
�'
givenaboveissimpli�edasfollows:
Proposition5Let
T,K,M,k,D,',L,S
and
R
be
as
in
(þ+).
Let�be
a
given
staroperation
on
the
integraldom
ain
R.
Foreach
nonzero
fractionalidealF
ofD,we
have
F�'
=
' ��'�
1(F) �� �
=
�'�
1(F) ��
M
:
11
Proposition6LetT,K,M,k,D,',L,Sand
R
be
asin
(þ+).Let
?be
a
given
staroperation
on
the
integraldom
ain
D,let�:=
?'
be
the
staroperation
on
R
associated
to
?(which
isde�ned
in
Corollary
2)
and
let
�'
(=
(?')')be
the
staroperation
on
D
associated
to
�
(which
isde�ned
in
Proposition
4).Then
?=
�'
(=
(?')').
Remark7W
ith
the
notation
and
hypothesesofProposition
6,
for
each
nonzero
fractionalidealF
ofD,we
have
F?=
' �'�
1(F)?' �
:
Asamatteroffact,bythepreviousproofandProposition5,wehave
thatF?=
F�'
=
'�
1(F)?'
=M.
12
Corollary8LetT,K,M,k,D,',L,Sand
R
be
asin
(þ+).
(a)Them
ap
(�)':Star(R)!
Star(D),�7!
�',isorder�preserving
and
surjective.
(b)Them
ap
(�)':Star(D)!
Star(R),?7!
?',isorder�preserving
and
injective.
(c)Let?be
a
staroperation
on
D.Then
foreach
nonzero
idealIof
R
with
M
�I�R,
I?'
=
'�
1 �('(I))? �:
13
Thenextresultshowshow
thecompositionmap
(�)'Æ
(�)':Star(R)!
Star(R)
compareswiththeidentitymap.
Theorem
9LetT,K,M,k,D,',L,Sand
Rbeasin
(þ+).Assum
e
thatD
(k.Then
foreach
staroperation
�on
R,
��((�)')'
:
Wewillshow
thatingeneral��((�)')'.However,insomerelevant
cases,theinequalityis,infact,anequality:
Corollary10LetT,K,M,k,D,',L,Sand
R
be
asin
Theorem
9.
Then
vR
=
((vR)')';
(vD)'=
vR;
(vR)'=
vD
:
14
Ournextgoalisto
applythepreviousresultsforgiving
a
compo-
nentwisedescriptionofthe�pullback�staroperation�consideredin
Proposition1.
Proposition11Let
T,K,M,k,D,',L,S
and
R
be
as
in
(þ+).
Assum
e
thatM
6=
(0)and
D
(k.Let
�
:Star(D)�Star(T)!
Star(R),
(?D;?T)7!
�:=
(?D)'^(?T)�,
bethem
apconsideredin
Proposition
1andCorollary3.Thefollowing
propertieshold:
(a)�'=
?D.
(b)��=
(vR)�^?T
(2Star(T)).
(c)�=
(�')'^(�� )�.
15
Example12W
ith
the
sam
e
notation
and
hypothesesofProposition
11,we
show
that,in
general,��6=
?T
(even
ifL=
k).
LetD
be
any
integraldomain
(nota
�eld)with
quotient�eld
L.
LetT
:=
L[X;Y](X;Y)
and
letM
:=
(X;Y)T.
NotethatT
isa
2-
dimensionallocalUFD,thusMvT
=
T.Set�:=
(vD)'^(vT)�(thus
?T
=
vT).ThenM��=
M�
=
M(vD)'
\M(vT)�
=
MvR
\M(vT)�
=
M,
becauseMvR
=
M
andM(vT)�
=
(MT)vT
=
MvT
=
T.
16
Remark13(a)Notethat,withthesamenotationandhypotheses
ofProposition11,the
m
ap
�
isnotone-to-one
in
general.
Thisfactimmediatelyfollowsfrom
Example12andProposition
11
(b)and(c),since
(?D)'^(?T)�=
�=
(�')'^(�� )�.
(b)Inthesamesettingasabove,the
m
ap
�
isnotonto
in
general.
Anexample,evenincaseL=
k,isgivennext.
17
Example14LetD
bea1-dim
ensionaldiscretevaluationdom
ainwith
quotient�eld
L.
SetT:=
L[X2;X3],M
:=
(X2;X3)T=
XL[X]\T
and
K
:=
L(X).Let'and
R
be
asin
(þ+).Then
vR
=2Im(�).
Notethat,foreach�2Im(�),��(vD)'^(vT)��vR.Inordertoshow
thatvR
62
Im(�),itsu�cesto
provethat(vD)'^(vT)�6=
vR.
The
fractionaloverringT
ofR
isnotadivisorialidealofR,sinceTvR
=
(R
:K
(R
:K
T))=
(R
:K
M)�
L[X])
T.Therefore,T(vD)'^
(vT)�
=
TvR^
(vT)�
=
TvR
\T(vT)�
=
TvR
\TvT
=
TvR
\T=
T(TvR.
18
Theorem
15W
ith
the
notation
and
hypotheses
ofProposition
11,
set
Star(T;vR):=
f?T
2Star(T)j?T
�(vR)� g.
Then
(a)Star(T;vR)=
f?T
2Star(T)j(vR^(?T)�)
�=
?T
g
=
f��j�2Star(R)g
\
Star(T)
=
f��j�2Star(R)and
T�
=
Tg:
(b)The
restriction
�0
:=
�jStar(D)�
Star(T;vR)isone-to-one.
(c)Im(�0)=
Star(R;(þ+)):=
f�2
Star(R)jT�
=
Tand
�=
(�')'^(�� )�g.
19
Wenextapplysomeofthetheorydevelopedaboveforansweringa
problem
posedbyD.F.Andersonin1992[A-1992].
Example16(�D
+
M
��constructions).
LetTbeanintegraldomainofthetypek+M,whereM
isamaximal
idealofT
andkisasubringofT
canonicallyisomorphictothe�eld
T=M,andletD
beasubringofkwith�eldofquotientsL(�k).Set
R:=
D+
M.NotethatR
isafaithfully�atD�module.
Given
a
staroperation
�on
R,D.F.Anderson
[A-1988,page835]
de�nedastaroperationonD
inthefollowingway:foreachnonzero
fractionalidealF
ofD,setF
�D
:=
(FR)�
\L:
20
From
[A-1988,Proposition5.4(b)]itisknownthat:
ForeachnonzerofractionalidealF
ofD,
(a)F�D
+
M
=
(F+
M)�
;
(b)F�D
=
(F+
M)�
\L=
(F+
M)�
\k.
DavidF.Andersonin[A-1992]observedthatthepreviousconstruc-
tion
givesrise
to
a
map
�
:Star(D
+
M)!
Star(D),
�7!
�D
,
which
isorder-preserving
butnotinjective.
Heposesthequestion
whether�
maybesurjectiveor,moreprecisely,whether�
mayhave
arightinverse�:Star(D)!
Star(D+
M),whichisan(injective)
order-preserving
map.
Hegavean
answerin
aparticularsituation,
consideringjustthestaroperationsde�nedbyfamiliesofoverrings.
21
Thetheorydevelopedabovegivesacompleteanswertotheseques-
tions.
Westartbycomparingtheoperation�D
de�nedin[A-1988]withthe
�projection�,
�',
consideredaboveinageneralpullbacksetting.
Claim.If':R!
D
is
the
canonicalprojection
and
if�'
is
the
star
operation
de�ned
in
Proposition
4,then
�D
=
�'
(i.e.
the
m
ap
�
coincideswith
the
m
ap
(�)':Star(R)!
Star(D)).
Inparticular,by[A-1992,Proposition2(a),(c)],we
deduce
that
(1)(dR)'=
dD
;
(tR)'=
tD
;
(vR)'=
vD
;
and
(2)(�f)'=
(�')f.
22
ByapplyingProposition6andCorollary8(a)totheparticularcase
ofR=
D+
M
(specialcaseof(þ+)),weknow
thatthemap
(�)':Star(D+
M)!
Star(D),
�7!
�'=
�D
,
issurjectiveandorder-preservingandithastheinjectiveorder-preserving
map
(�)'
:Star(D)!
Star(D+
M),
?7!
?'
,
asarightinverse.
Thisfactgivesacompletepositiveanswertotheproblem
posedby
D.F.Anderson.
23
[A-1988]David
F.Anderson,
A
general
theory
of
class
groups.
Comm.Algebra16(1988),805�847.
[A-1992]David
F.Anderson,
Staroperations
and
the
D+
M
con-
structions.
Rend.Circ.Mat.Palermo41IISerie(1992),221�230.
24
