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Theory and Appli
ations of Categories,

Vol. 6, No. 2, 1999, pp. 25{32.

A BICATEGORICAL APPROACH TO STATIC MODULES
Dedi
ated to J. Lambek on the o

asion of his 75th birthday

RENATO BETTI
ABSTRACT. The purpose of this paper is to indi
ate some bi
ategori
al properties
of ring theory. In this intera
tion, stati
modules are analyzed.

Introdu

tion
Categori
al generalizations of ring theory usually start from the observation that rings are
(one-obje
t)
ategories enri
hed in the monoidal
losed
ategory Ab of abelian groups. In
- Ab,
this setting, R-S -bimodules are profun
tors R +- S , i.e. fun
tors Rop  S
P
Q
and their
omposition R +- S +- T is provided by the tensor produ
t Q
S P over
the ring S . So, one has a distributive bi

ategory Mod, whose obje
ts are the (non
ommutative) rings, whose arrows are the profun
tors and whose 2-
ells are the module
morphisms.
To say that Mod is a distributive bi
ategory is to say that it admits lo
al
olimits
whi
h distribute over
omposition on both sides. Moreover, Mod is bi
losed (see e.g.
[6℄), in the sense that it admits right Kan extensions and right liftings: in other words,
ompositions with an arrow P : R +- S are fun
tors:

Mod
S - RMod

ModR
R - ModS
having right adjoints (here S Mod denotes the
ategory of left
ModS is the
ategory of right -modules).
P

S

P

S

-modules and similarly

S

In parti
ular the right Kan extension of the left E -module N along the E -R-bimodule

- N . Analogously, the
is the R-module homE (P; N ) of E -module morphisms P
right lifting of the right R-module M along the E -R-bimodule P is the right E -module
- M.
homR (P; M ) of R-module morphisms P
The main observation whi
h relates Mod with a relevant property of modules was
rst formulated by Lawvere [6℄: nitely generated proje
tive modules are exa
tly those
profun
tors whi
h, when regarded as arrows in Mod, have a right adjoint. To prove this
P

Resear
h partially supported by the Italian MURST.
Re
eived by the editors 1999 February 15 and, in revised form, 1999 September 8.
Published on 1999 November 30.

1991 Mathemati
s Subje
t Classi
ation: 18D05, 18E10, 13E05.
Key words and phrases: Bi
ategory, module, Cliord theory.
Renato Betti 1999. Permission to
opy for private use granted.



25

Theory and Appli
ations of Categories, Vol. 6, No. 2

26

fa
t one should

onsider the dual module P  = homR (P; R) of P . It is easy to see that
the adjointness P j P  has the evaluation x
f - f (x) as
ounit P
Z P  - R
- P 
R P as unit.
and the assignment of a so-
alled dual basis
In this paper we want to
on
entrate on the fa
t that
ategori
al notions often are
dire
tly
onne
ted to the notions of ring theory, in general allowing one to formulate
problems and prove results in a more

on
eptual way. Here we
onsider properties relative
to stati
and e-stati
modules (with respe
t to a given module P ) i.e. relative to those
- V is an isomorphism (or
modules V for whi
h the evaluation P
E homR (P; V )
an epimorphism). The
al
ulus of adjoints in a bi
ategory and its general properties are
put in use to study properties of stati
and e-stati
modules in
luding aspe
ts 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℄).

Z

1. Stati
and e-stati
modules

Notations and the formulation of the problem are from Lambek [5℄ whi
h regards stati
modules with respe
t to a proje
tive module P .
Given a bimodule P : E +- R,
onsider the adjoint pair of fun
tors:
homR (P;

ModR 

P


E

-) ModE

(1)

(P
E
j homR (P; )).
By general properties of adjoint pairs, these restri
t to an equivalen
e between the full
sub
ategory (F ix ) of
R
onsisting of the modules U whose

ounit

Mod

U

: U
E homR (P; U )

-U

Mod

is an isomorphism and the full sub
ategory (F ix  ) of
E
onsisting of those modules
homR (P; P
E V ) is an isomorphism.
V whose unit V : V

In
ase P is a right R-module and E = homR (P; P ) is its ring of endomorphisms,
the modules in (F ix ) are said to be stati
with respe
t to P . If U is an epimorphism,
U is said to be e-stati
and (Epi ) denotes the full sub
ategory of e-stati
modules. In
[7℄, M
Master proves that e-stati
modules relative to a proje
tive P are exa
tly those
modules whi
h are
otorsionfree with respe
t to P .
Observe that the restri
tion of the adjun
tion (1) to the sub
ategory (Epi ) gives an
adjun
tion:
(Epi ) 

- (M ono )

Mod

where (M ono  ) is the full sub
ategory of
E
onstituted by the modules
monomorphi
unit U : U - homR (P; P
E U ).

U

with a

Theory and Appli
ations of Categories, Vol. 6, No. 2

27

When P is nitely generated and proje
tive then homR (P; ) 
= P 
R , hen
e V is
a stati
module if and only if


E P 
R V = V
equies the tra
e ideal  : P
E P  ! R.
P

i.e.

V

1.1. Theorem.

ideal.

If P is nitely generated and proje
tive, then it is the equier of its tra
e

roof. The module P equies the tra
e ideal  : P
E P  - R be
ause P 
R P 
= 1E .
Moreover, any stati
module U fa
tors uniquely up to isomorphism in the form U 
=

P
E homR (P; U ) be
ause homR (P; R)
R U = homR (P; U ).

P

Mod
Mod Mod

If moreover P is a generator of
R , then the tra
e ideal  is an isomorphism and
any module in
is
stati
.
This
is
one
of the main results of Morita theory, namely
R
the assertion that in this
ase
R and
E are equivalent
ategories (i.e. R and E
are Morita equivalent rings).

Mod

In general, for a bimodule P : E +-

1.2. Theorem.

R

with E = homR (P; P ), one has:

Any nitely generated proje
tive

E -module V

is in (F ix  ).

roof. It is enough to remind the reader that, in any bi
ategory with right liftings, the
fun
tor homR (P; ) preserves
omposition with right adjoints.
In the
ase of
, if V has a right adjoint, for any W in
E one has the following
sequen
e of bije
tions, natural in W :
P

Mod

Mod

- homR (P; P )
E V
- P
E V
P
E W
- homR (P; P
E V )
W

W

From the previous theorem, one has that the sub
ategory of nitely generated E modules is
ontained in (F ix  ). Nauman [8℄ parti
ularly
onsiders this sub
ategory and,
through his result ([8℄, theorem 3.7) one
on
ludes:

Mod

1.3. Corollary. A right R-module U weakly divides P in
R (i.e. it is a dire
t
summand of nitely many
opies of P ) if and only if it is of the form P
E V for a
nitely generated and proje
tive E -module V .

Let ( )0 denote the fun
tor whi
h asso
iates to any
evaluation:
V :P

(here eV is epi and

mV


E homR (P; V ) eV - V 0

is mono).

R-module V

-V

mV

the image of the
(2)

Theory and Appli
ations of Categories, Vol. 6, No. 2

()

28

lands in (Epi ) and it is the right adjoint to the em(Epi )  - ModR.
Proof. First one proves that V is in (Epi ). Consider the diagram:

1.4. Theorem. The fun
tor

0

bedding

0

P


E homR (P; V ) V 0 - V
0

m0
P

0

mV

?


E homR (P; V )

V

- V?

Here, m denotes P
E homR (P; mV ). Now, homR (P; mV ) is mono be
ause homR (P; )
is a left exa
t fun
tor. Moreover, under the bije
tion of P
E
j homR (P; ), the
evaluation (2)
orresponds to the
omposite:
0

fV hom (P; V ) homR (P; mV-) hom (P; V )
( ) eR
R
whi
h is the identity of homR (P; V ). Hen
e homR (P; mV ) is also epi. As a
onsequen
e
m is an isomorphism and from
homR P; V

0

0

eV


(P
E homR (P; mV )) = V 0

one has that also V 0 is epi.
For the adjointness, it is easy to prove that, if U is in (Epi ), the universal property
of images provides a natural bije
tion:
U
U

-V
-V

0

given by
omposition with mV .
A known
hara
terization of stati
modules is provided by Auslander equivalen
e [2℄
(see also Alperin [1℄): a module V is stati
with respe
t to P if and only if there is a
presentation (i.e. an exa
t sequen
e):

-0
Y P -  X P - V
su
h that the appli
ation of the fun
tor homR (P; ) gives another exa
t sequen
e (here
X and Y are sets and  denotes
oprodu
t). In
ase P is proje
tive, the presentation is
enough, as proved dire
tly by Lambek [5℄.
By substituting the presentation of V with a
ondition on its generation by P , e-stati
modules
an be
hara
terized. Say that the R-module V is generated by P if there is an epimorphism X P - V . Moreover, say that the generators are preserved if the appli
ation
of the fun
tor homR (P; ) gives another epimorphism homR (P; X P ) - homR (P; V ).

Theory and Appli
ations of Categories, Vol. 6, No. 2

1.5. Theorem. The

29

R-module V is in (Epi )
hom (P; ).

the generators are preserved by

if and only if it is generated by

P

and

R

roof. First, observe that (Epi ) is a
ore
e
tive sub
ategory of Mod and thus it is
losed under
olimits. Moreover the module P is stati
be
ause E = hom (P; P ).
Now suppose that V is generated by P and that generators are preserved by hom (P; ).
Tensoring the epimorphism hom (P;  P ) - hom (P; V ) with P gives another epimorphism, as tensoring is a right exa
t fun
tor. So, one has the
ommutative diagram:
P

R

R

R

R

R

X

-P


P
hom (P;  P )
E

R

X

X

E

hom (P; V )
R



P

V

?

?

-V
 P
where the horizontal arrows are epimorphism and also X is an isomorphism be
ause
(Epi ) is
losed under
oprodu
ts. Hen
e  is an epimorphism.
Conversely, suppose that  : P
hom (P; V ) - V is epi. Now, hom (P; V )
as a right E -module is a quotient of a free E -module, i.e. it is generated in the form
 E ! hom (P; V ). By tensoring and
omposing with  , one has that V is generated
by P :
X

P

V

V

X

E

R

R

R

V


 P
= P
 E - P
hom (P; V )
X

E

E

X

-V

V

R

Applying hom (P; ) we get the
ommutative diagram:
R



 E

- hom (P; V )

X

X

E

R



?

hom (P; P
 E )
R

E

X

-

R (P;V )

hom

?

hom (P; P
hom (P; V )))
R

E

R

where the arrow  is epi. Taking into a

ount the triangular identity:
id = hom (P;  )

R

V

R (P;V )

hom

one has that:
 = hom (P;  )

X
R

V

E

hen
e hom (P;  ) is epi and the generators of V by P are preserved.
R

V

Theory and Appli
ations of Categories, Vol. 6, No. 2

30

2. Appli
ation to Cliord theory
We shall now prove a general result about modules obtained by indu
tion and restri
tion
along a ring homomorphism.
In [3℄, Cline extended the
lassi
al Cliord theory of proje
tive representations and
stable modules by utilising rings and modules whi
h are graded by a group. His main result
an be stated as an equivalen
e of
ategories and was reobtained by Dade ([4℄, theorem
7.4) by using only the fun
tors H om and
in a natural way. Restri
ting it to the essential
ase of group algebras, the equivalen
e regards the homomorphism kH - kG indu
ed
by a normal subgroup H of G (here G is a nite group, k is a eld and kG is the group
algebra). In this
ase, Dade's result [4℄
hara
terizes those kG-modules whi
h are stati
when restri
ted to kH . Furthemore, Alperin [1℄ gives a generalization to non-normal
subgroups H .
In general, the equivalen
e 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
. Pre
isely: f j f  .
Indu
tion and restri
tion fun
tors along f are given by the
ompositions with f and

f
respe
tively. These are adjoint fun
tors (f
R
j f 
S ):

Mod


R ModS


S

ModR 

f

f

If P : E +- R is su
h that E = homR (P; P ),
onsider the module P 0 = f
R P
indu
ed by P through f as a module F +- S , where F = homS (P 0 ; P 0). One has
- F , easily des
ribed in
another ring homomorphism i : E
by the
anoni
al
morphism of right liftings

Mod

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 restri
tion f 
S U is stati
with respe
t to P .
In [8℄, Nauman expresses it in a purely ring theoreti
al way:
2.1. Theorem. [Nauman [8℄, theorem 5.5℄ The restri
tions of the additive fun
tors
0
homS (P ;
) and P 0
F form an equivalen
e between the full additive sub
ategory of

S having as obje
ts all S -modules U su
h that f
S U weakly divides P in
R
and the full additive sub
ategory of
whose
obje
ts
are
the
F
-modules
V
su
h
that
F

i
V is nitely generated and proje
tive.
By the universal property of adjoint pairs one proves that:

Mod

Mod

Mod

P

0


F  = 
R
i

f

P

Adjointness is essential also in proving the following:

(3)

Theory and Appli
ations of Categories,

Vol. 6, No. 2

31

S -module U along f is in (F ix ) if and
0
only if the restri
tion of homS (P ; U ) along i is in (F ix  ). In symbols:

2.2. Theorem. The restri
tion of the right

f




S U 2 (F ix ) () i
F homS (P 0; U ) 2 (F ix )

Proof. For any right

f



S -module U :


S U 2 (F ix ) () homR (P; f 
S U ) 2 (F ix )

Consider the following diagram. By (3) the square of left adjoints
ommutes up to
isomorphisms, and thus so does the square of right adjoints:

ModR 
f


R

6

f

?

homR (P;
P


S


E
i

homS (P 0 ;

ModS 

- ModE
6

)

P0


F

)


E

i

?
- Mod
F


F

Hen
e:

homR (P; f




S U ) = i
F homS (P 0; U )

A
knowledgment. I would like to thank the anonymous referee for helpful suggestions and
orre
tions.

Referen
es
[1℄ J.L. Alperin,

Stati
modules and non-normal Cliord theory,

J. Austral. Math. So
. (Series A) 49

(1990), 347-353.

Representation of Artin algebras, Comm. in Algebra 1 (1974), 177-268.
E. Cline, Stable Cliord theory, J. of Algebra 22 (1972), 350-364.
E.C. Dade, Group-graded rings and modules, Math. Z. (1980), 241-262.
J. Lambek, Remarks on
olo
alization and equivalen
e, Comm. in Algebra 10 (1983), 1145-1153.
F.W. Lawvere, Metri
spa
es, generalized logi
, and
losed
ategories, Rend. Sem. Mat. e Fis. Milano

[2℄ M. Auslander,
[3℄
[4℄
[5℄
[6℄

43 (1973), 135-166.

Theory and Appli
ations of Categories,

Vol. 6, No. 2

32

Cotorsion theories and
olo
alization, Can. J. Math. 27 (1975), 618-628.
S.K. Nauman, Stati
modules and stable Cliord theory, J. of Algebra 128 (1990), 497-509.

[7℄ R.J. M
Master,
[8℄

Dipartimento di Matemati
a
Polite
ni
o di Milano
Piazza L. da Vin
i, 32
20133 Milano (Italy)
Email:

renbetmate.polimi.it

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