A BICATEGORICAL APPROACH TO STATIC MODULES
Dediated to J. Lambek on the oasion of his 75th birthday
RENATO BETTI
ABSTRACT. Thepurposeof thispaperis toindiate somebiategorialproperties
ofringtheory. Inthisinteration, statimodulesareanalyzed.
Introdution
Categorialgeneralizationsofringtheoryusuallystartfromtheobservationthatringsare
(one-objet)ategoriesenrihedinthemonoidallosedategoryAbofabeliangroups. In
this setting, R -S-bimodules are profuntors R +
-S, i.e. funtors R op
S
-Ab,
and their omposition R P
+
-S Q
+
-T is provided by the tensor produt Q
S
P over
the ring S. So, one has a distributive biategory Mod, whose objets are the
(non-ommutative) rings, whose arrows are the profuntors and whose 2-ells are the module
morphisms.
To say that Mod is a distributive biategory is to say that it admits loal olimits
whih distribute over omposition on both sides. Moreover, Mod is bilosed (see e.g.
[6℄), in the sense that it admits right Kan extensions and right liftings: in other words,
ompositionswith anarrow P :R +
-S are funtors:
S Mod
S P
-R Mod
Mod
R P
R
-Mod
S
having right adjoints (here
S
Mod denotes the ategory of left S-modules and similarly
Mod
S
is the ategory of rightS-modules).
In partiularthe rightKan extensionoftheleftE-moduleN alongtheE-R -bimodule
P is the R -module hom E
(P;N) of E-module morphisms P
-N. Analogously, the
right lifting of the right R -module M along the E-R -bimodule P is the right E-module
hom
R
(P;M)of R -module morphisms P
-M.
The main observation whih relates Mod with a relevant property of modules was
rst formulated by Lawvere [6℄: nitely generated projetive modules are exatly those
profuntors whih,when regarded as arrows in Mod, have a right adjoint. To prove this
Researhpartiallysupported bytheItalianMURST.
Reeivedbytheeditors1999February15and,inrevisedform,1999September8.
Publishedon1999November30.
1991MathematisSubjetClassiation: 18D05,18E10,13E05.
Keywordsandphrases: Biategory,module, Cliordtheory.
fat one should onsider the dual module P =hom
R
(P;R ) of P. It is easy to see that
the adjointness P j P
has the evaluation xf
-f(x)as ounit P
Z P
-R
and the assignment of aso-alled dual basis Z
-P
R
P as unit.
In this paper we want to onentrate on the fat that ategorial notions often are
diretly onneted to the notions of ring theory, in general allowing one to formulate
problemsandproveresultsinamoreoneptualway. Here weonsider propertiesrelative
to stati and e-stati modules (with respet to a given module P) i.e. relative to those
modules V for whih the evaluation P
E hom
R (P;V)
-V is an isomorphism (or
an epimorphism). The alulusof adjointsin a biategory and its generalproperties are
put in use tostudy properties of stati and e-statimodules inluding aspets of Morita
theory and an extension of the Dade-Cline version of Cliord theory to one in general
ring theory (see e.g. Cline [3℄, Dade[4℄, Alperin [1℄, Nauman [8℄).
1. Stati and e-stati modules
Notations and the formulation of the problemare from Lambek [5℄ whih regards stati
modules with respet toa projetive module P.
Given a bimodule P :E +
-R , onsider the adjoint pair of funtors:
Mod
R hom
R (P; )
P
E
Mod
E
(1)
(P
E
j hom
R
(P; )).
Bygeneralpropertiesof adjointpairs,theserestrittoanequivalenebetweenthe full
subategory (Fix ) of Mod
R
onsisting of the modulesU whose ounit
U :U
E hom
R (P;U)
-U
isanisomorphismandthe fullsubategory(Fix)ofMod
E
onsisting ofthosemodules
V whose unit
V :V
-hom
R
(P;P
E
V) isan isomorphism.
In ase P is a right R -module and E = hom
R
(P;P) is its ring of endomorphisms,
the modules in(Fix ) are said to be stati with respet toP. If
U
is an epimorphism,
U is said to be e-statiand (Epi ) denotes the full subategory of e-stati modules. In
[7℄, MMaster proves that e-stati modules relative to a projetive P are exatly those
modules whihare otorsionfree withrespet toP.
Observethat the restrition of the adjuntion(1)tothe subategory(Epi )givesan
adjuntion:
(Epi)
(Mono )
where (Mono ) is the full subategory of Mod
E
onstituted by the modules U with a
monomorphiunit :U
When P isnitely generated and projetive then hom
R (P; )
=
P
R
,hene V is
a stati module if and onlyif
P
E P
R V
= V
i.e. V equies the trae ideal :P
E P
!R .
1.1. Theorem. IfP isnitely generated andprojetive,thenitistheequier ofits trae
ideal.
Proof. Themodule P equies the traeideal :P
E P
-R beauseP
R P
= 1
E .
Moreover, any stati module U fators uniquely up to isomorphism in the form U
=
P
E hom
R
(P;U) beause hom
R
(P;R )
R U
= hom
R
(P;U).
If moreover P is a generator of Mod
R
, then the trae ideal is anisomorphism and
any module inMod
R
is stati. This is one of the main results of Morita theory, namely
the assertion that inthis ase Mod
R
and Mod
E
are equivalent ategories (i.e. R and E
are Morita equivalent rings).
In general, fora bimodule P :E +
-R with E =hom
R
(P;P), one has:
1.2. Theorem. Any nitely generated projetive E-module V is in (Fix ).
Proof. It is enough to remind the reader that, in any biategory with right liftings, the
funtor hom
R
(P; )preserves omposition with rightadjoints.
IntheaseofMod,ifV hasarightadjoint,foranyW inMod
E
onehasthefollowing
sequene of bijetions,natural inW:
W
-hom
R
(P;P)
E V
P
E W
-P
E V
W
-hom
R
(P;P
E V)
From the previous theorem, one has that the subategory of nitely generated E
-modulesisontainedin(Fix). Nauman[8℄partiularlyonsiders thissubategoryand,
through his result ([8℄, theorem 3.7) one onludes:
1.3. Corollary. A right R -module U weakly divides P in Mod
R
(i.e. it is a diret
summand of nitely many opies of P) if and only if it is of the form P
E
V for a
nitely generated and projetive E-module V.
Let ( ) 0
denote the funtor whih assoiates to any R -module V the image of the
evaluation:
V :P
E hom
R (P;V)
e
V
-V 0
m
V
-V (2)
1.4. Theorem. The funtor ( ) lands in (Epi ) and it is the right adjoint to the
em-bedding (Epi )
-Mod
R .
Proof. First one proves that V 0
is in (Epi ). Consider the diagram:
P
E hom
R (P;V
0
)
V 0
-V 0
P
E hom
R (P;V) m
0
?
V
-V m
V
?
Here,m 0
denotesP
E hom
R (P;m
V
). Now,hom
R (P;m
V
)ismonobeausehom
R (P; )
is a left exat funtor. Moreover, under the bijetion of P
E
j hom
R
(P; ), the
evaluation(2) orresponds tothe omposite:
hom
R (P;V)
f
e
V
-hom
R (P;V
0
) hom
R (P;m
V )
-hom
R (P;V)
whih is the identity of hom
R
(P;V). Hene hom
R (P;m
V
) is also epi. As a onsequene
m 0
is an isomorphismand from
e
V
(P
E hom
R (P;m
V ))=
V 0
one has that also
V
0 isepi.
For the adjointness, it iseasy toprove that, if U isin (Epi), the universal property
of images provides a naturalbijetion:
U
-V 0
U
-V
given by ompositionwith m
V .
A known haraterization of stati modules is provided by Auslander equivalene [2℄
(see also Alperin [1℄): a module V is stati with respet to P if and only if there is a
presentation (i.e. an exat sequene):
Y P
X P
-V
-0
suh that the appliation of the funtor hom
R
(P; ) gives another exat sequene (here
X and Y are sets and denotes oprodut). In ase P is projetive, the presentation is
enough, as proved diretly by Lambek[5℄.
BysubstitutingthepresentationofV with aondition onitsgenerationbyP,e-stati
modulesanbeharaterized. SaythattheR -moduleV isgeneratedbyP ifthereisan
epi-morphism
X P
-V. Moreover,saythatthegeneratorsarepreservediftheappliation
of the funtor hom (P; ) gives anotherepimorphism hom (P; P)
1.5. Theorem. The R -module V is in (Epi ) if and only if it is generated by P and
the generators are preserved by hom
R
(P; ).
Proof. First, observe that (Epi ) is a oreetive subategory of Mod
R
and thus it is
losed under olimits. Moreover the module P isstati beauseE =hom
R
(P;P).
NowsupposethatV isgeneratedbyP andthatgeneratorsarepreservedbyhom
R
(P; ).
Tensoring the epimorphism hom
R (P;
X P)
-hom
R
(P;V) with P gives another
epi-morphism, as tensoring isa rightexat funtor. So, one has the ommutativediagram:
P
E hom
R (P;
X P)
-P
E hom
R (P;V)
X P
X P
?
-V
V
?
where the horizontal arrows are epimorphism and also
X P
is an isomorphism beause
(Epi )is losed under oproduts. Hene
V
isan epimorphism.
Conversely, suppose that
V
: P
E hom
R (P;V)
-V is epi. Now, hom
R (P;V)
as a right E-module is a quotient of a free E-module, i.e. it is generated in the form
X
E !hom
R
(P;V). By tensoring and omposing with
V
, one has that V is generated
by P:
X P
=P
E
X E
-P
E hom
R (P;V)
V
-V
Applying hom
R
(P; ) we get the ommutativediagram:
X E
-hom
R (P;V)
hom
R
(P;P
E
X E)
X E
?
-hom
R
(P;P
E hom
R
(P;V)))
hom
R (P;V)
?
where the arrow is epi. Taking intoaount the triangularidentity:
id=hom
R (P;
V )
hom
R (P;V)
one has that:
=hom
R (P;
V
)
X E
2. Appliation to Cliord theory
Weshall nowprove ageneralresult about modulesobtained by indutionand restrition
along aring homomorphism.
In [3℄, Cline extended the lassial Cliord theory of projetive representations and
stablemodulesbyutilisingringsandmoduleswhiharegradedbyagroup. Hismainresult
an be stated as an equivalene of ategories and was reobtained by Dade ([4℄, theorem
7.4)byusingonlythefuntorsHomandinanaturalway. Restritingittotheessential
ase of group algebras,the equivalene regards the homomorphismkH
-kG indued
by a normalsubgroup H of G (here G is anite group, k is a eld and kG is the group
algebra). In this ase, Dade's result [4℄haraterizes those kG-modules whih are stati
when restrited to kH. Furthemore, Alperin [1℄ gives a generalization to non-normal
subgroups H.
In general, the equivalene is relative to a given ring homomorphism f : R
-S.
This gives rise to a bimodule f
: R +
-S and to a bimodule f
: S +
-R . These
bimodules are given by S itself, and it is easy to prove that they are adjoint arrows in
Mod. Preisely: f
j f
.
Indution and restrition funtors along f are given by the ompositionswith f
and
f
respetively. Theseare adjointfuntors (f
R
j f
S ):
Mod
R f
R
f
S
Mod
S
If P : E +
-R is suh that E = hom
R
(P;P), onsider the module P 0
= f
R P
indued by P through f as a module F +
-S, where F = hom
S (P
0
;P 0
). One has
another ring homomorphism i : E
-F, easily desribed in Mod by the anonial
morphismof right liftings
hom
R (P;P)
-hom
S (P
0
;P 0
)
The generalization of Cline's [3℄ and Dade's [4℄ works on Cliord theory referred to
above regards those S-modules U whose restrition f
S
U is stati with respet to P.
In [8℄, Nauman expresses it ina purely ring theoretial way:
2.1. Theorem. [Nauman [8℄, theorem 5.5℄ The restritions of the additive funtors
hom
S (P
0
; ) and P 0
F
form an equivalene between the full additive subategory of
Mod
S
having as objets all S-modules U suh that f
S
U weakly divides P in Mod
R
and the full additive subategory of Mod
F
whose objets are the F-modules V suh that
i
V is nitely generated and projetive.
By the universal property of adjoint pairs one proves that:
P 0
F i
= f
R
2.2. Theorem. The restrition of the right S-module U along f is in (Fix ) if and
only if the restrition of hom
S (P
0
;U) along i is in (Fix ). In symbols:
f
S
U 2(Fix )()i
F hom
S (P
0
;U)2(Fix )
Proof. Forany right S-module U:
f
S
U 2(Fix )()hom
R (P;f
S
U)2(Fix)
Consider the following diagram. By (3) the square of left adjoints ommutes up to
isomorphisms, and thusso doesthe square of rightadjoints:
Mod
R
hom
R (P; )
P
E
Mod
E
Mod
S f
R
? f
S 6
hom
S (P
0
; )
P 0
F
Mod
F i
E
? i
F 6
Hene:
hom
R (P;f
S U)
= i
F hom
S (P
0
;U)
Aknowledgment.Iwouldliketothanktheanonymousrefereeforhelpfulsuggestionsandorretions.
Referenes
[1℄ J.L. Alperin, Stati modules and non-normal Cliord theory, J.Austral. Math. So. (Series A) 49
(1990),347-353.
[2℄ M.Auslander,Representation of Artinalgebras, Comm.inAlgebra1(1974),177-268.
[3℄ E.Cline, StableCliordtheory,J.ofAlgebra22(1972),350-364.
[4℄ E.C.Dade,Group-gradedringsandmodules,Math. Z.(1980),241-262.
[5℄ J.Lambek,Remarks onoloalization andequivalene, Comm.inAlgebra10(1983),1145-1153.
[6℄ F.W.Lawvere,Metrispaes,generalizedlogi,andlosedategories,Rend.Sem.Mat.eFis.Milano
[7℄ R.J.MMaster,Cotorsiontheories andoloalization, Can.J.Math. 27(1975),618-628.
[8℄ S.K.Nauman, Statimodules andstable Cliordtheory,J.ofAlgebra128(1990),497-509.
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