4
Equivalent Categories.
Suppose that R and S are two rings. We want to determine what it would mean for them to have the
same representation theories — or for these theories to be equivalent in some meaningful way. What we
want should be pretty clear — the two rings have equivalent representation theories when their module
(= representation) categories RMod and SMod are “the same”. But that just trades one problem
for another — what does it mean for two categories “to be the same”? This is a bit dicey. We’d
probably like to say that they are the same if they are isomorphic. But that’s too strong. For example,
the category of all finite sets, an enormous sucker, has exactly the same categorical structure as the
category of all finite subsets of N, but there is no possible way to build a bijective function between
these two categories. As we shall see, natural transformations will come to the rescue, and we will be
able to determine an appropriate notion of equivalent categories and hence of equivalent representation
theories for rings.
Let C and D be two (additive) categories. A covariant (additive) functor F : C −→ D is an
equivalence in case there exists a covariant (additive) functor G : D −→ C such that GF and F G are
naturally isomorphic, respectively, to the identity functors 1C and 1D . If such an equivalence exists,
then we say that the categories are equivalent. This relation is symmetric. Indeed, the two categories
are equivalent iff there exist (additive) covariant functors
F : C −→ D
and G : D −→ C
and natural isomorphisms
η : GF −→ 1C
and µ : F G −→ 1D .
This means that η assigns to each object C in C an isomorphism ηC : GF (C) −→ C in C and µ
assigns to each object D in D an isomorphism µD : F G(D) −→ D in D such that for all C-morphisms
f : C −→ C 0 , and all D-morphisms g : D −→ D0 , the diagrams
GF (C)
GF (f )
- GF (C 0 )
ηC 0
ηC
?
C
F G(D)
f
F G(g)
- F G(D0 )
µD0
µD
?
- C0
?
D
g
?
- D0
commute. When this happens, we call F and G inverse equivalences for C and D.
When such inverse equivalences exist, it means that all “categorical” information available in one
of the categories carries over unchanged to the other. In other words, using only categorical machinery
we have no way to distinguish equivalent categories. So now let’s see how to characterize category
equivalences.
31
32
Section 4
A covariant functor F : C −→ D is faithful in case it is one-to-one on the class of morphisms of C.
The functor F is full in case for each pair C, C 0 in C the map F : MorC (C, C 0 ) −→ MorD (F (C), F (C 0 ))
is surjective. Finally, the functor F is dense in case for each object D in D there is an object C in C
with F (C) ∼
= D. It turns out that these three properties characterize category equivalences.
4.1. Theorem. Let C and D be (additive) categories and let F : C −→ D be a covariant (additive)
functor. Then F is an equivalence iff F is full, faithful, and dense.
Proof. (=⇒) Suppose that F is an equivalence and that G, η, and µ are as above. Then the
map f 7−→ GF (f ) is a bijection (isomorphism) from MorC (C, C 0 ) onto MorC (GF (C), GF (C 0 )). Thus,
the map f 7−→ F (f ) is injective (monic) from MorC (C, C 0 ) to MorD (F (C), F (C 0 ), and the map G :
F (f ) 7−→ GF (f ) from MorD (F (C), F (C 0 )) to MorC (GF (C), GF (C 0 )) is surjective. Using the natural
isomorphism µ gives that f 7−→ F (f ) is surjective (epic). Thus, F is full and faithful. That it is dense
is trivial.
(⇐=) This time suppose that F is full, faithful, and dense. We must build a functor G : D −→ C
and natural isomorphisms ϕ : 1D −→ F G and ψ : 1C −→ GF .
Since F is dense, for each object D in D we can choose an object C in C and an isomorphism
ϕD : D −→ F (C). Set G(D) = C. Then ϕD is an isomorphism D −→ F G(D). Now for each morphism
g : D −→ D0 in D, since F is full and faithful, there is a unique morphism f : C = G(D) −→ C 0 = G(D0 )
for which the diagram
g
D
- D0
ϕD 0
ϕD
?
F (f )
- F G(D0 )
?
F G(D)
commutes. Set G(g) = f . By uniqueness it is clear that for each D we have G(1D ) = 1G(D) and that
for every pair of morphisms g and g 0 in D, if g ◦ g 0 is defined, then G(g ◦ g 0 ) = G(g) ◦ G(g 0 ). Thus,
G : D −→ C is a covariant functor, and ϕ : 1D −→ F G is a natural isomorphism. In the additive case,
if g, g 0 ∈ M orD (D, D0 ), then since F (G(g) + G(g 0 )) = F G(g) + F G(g 0 ), uniqueness again gives that
G(g + g 0 ) = G(g) + G(g 0 ), so that G is additive.
Finally, we claim that there is a natural isomorphism ψ : 1C −→ GF . Let C be an object of C. Then
there is an isomorphism ϕF (C) : F (C) −→ F GF (C). But F is full and faithful, so there is a unique
morphism ψC : C −→ GF (C) with F (ψC ) = ϕF (C) . Since ϕF (C) is an isomorphism, there exists a
morphism g : F GF (C) −→ F (C) with
gϕF (C) = 1F (C)
and ϕF (C) g = 1F GF (C) .
Again, since F is full and faithful there is a unique morphism h : GF (C) −→ C with F (h) = g. Thus,
F (hψC ) = 1F (C) = F (1C ) and F (ψC h) = 1F GF (C) = F (1GF (C) . Since F is faithful, this means that
Non-Commutative Rings
Section 4
33
hψC = 1C and ψC h = 1GF (C) so that ψC : C −→ GF (C) is an isomorphism. We finish by showing that
it is natural. Let f : C −→ C 0 be a morphism and consider the diagram
C
f
- C0
ψC
ψC 0
?
? GF (f )
- GF (C 0 )
GF (C)
Hit it with F and it commutes. But F is faithful, so the diagram itself commutes.
Although an equivalence of two categories need not be an isomorphism, under which virtually
eveything of mathematical interest is preserved, it does preserve the really important categorical stuff.
For example,
4.2. Corollary. Let F : C −→ D be a categorical equivalence. Then a morphism f : C −→ C 0 in
C is a monomorphism, epimorphism, or isomorphism, respectively, iff F (f ) : F (C) −→ F (C 0 ) is
a monomorphism, epimorphism, or isomorphism in D. If C, D, and F are additive, then for each
pair C, C 0 of objects in C,
F : MorC (C, C 0 ) −→ MorD (F (C), F (C 0 ))
is a group isomorphism, and for each object C in C
F : MorC (C, C) −→ MorD (F (C), F (C))
is a ring isomorphism.
Proof. Suppose that f : C −→ C 0 is a monomorphism. Let g, g 0 : D −→ F (C) be morphisms in D
with F (f )g = F (f )g 0 . Since F is dense, there is some C 00 and some isomorphism g 00 : F (C 00 ) −→ D.
So gg 0 and g 0 g 00 are morphisms F (C 00 ) −→ F (C). Since F is full, there exist morphisms h, h0 in C with
F (h) = gg 00 and F (h0 ) = g 0 g 00 . So F (f h) = F (f )F (h) = F (f )gg 00 = F (f )g 0 g 00 = F (f )F (h0 ) = F (f h0 ).
But F is faithful, so f h = f h0 and since f is monic, h = h0 . So gg 00 = g 0 g 00 . But g 00 is an isomorphism,
so g = g 0 . Thus, F (f ) is monic. We’ll leave the rest of this to the reader.
We want to apply this to the representation theory of rings. Two rings R and S are (Morita)
equivalent in case the additive categories RMod and SMod are equivalent. Not at all surprisingly,
if F : RMod −→ SMod is an additive equivalence, then F is an exact functor.
4.3. Theorem. If R and S are equivalent rings with F : RMod −→ SMod an additive equivalence,
then F is exact.
f
g
Proof. Let 0 → M 0 −→ M −→ M 00 → 0 be exact over R. Then by Corollary 4.2, F (f ) is monic
34
Section 4
and F (g) is epic. Also, F (g) ◦ F (f ) = F (g ◦ f ) = F (0) = 0, and thus, Im F (f ) ≤ Ker F (g). So, it will
suffice to show that Ker F (g) ≤ Im F (f ). Let K = Ker F (g). Since F is dense, there is an H in RMod
with F (H) ∼
= K. Since F is also full and faithful, there exist R-homomorphisms for which the diagram
0
?
- M0
0
h
?
- H
0
f-
M
g-
M 00
=
=
?
? g
- M
- M 00
- 0
- 0
commutes and has exact rows and columns. Thus, h is an isomorphism, so that F (h) : F (M 0 ) −→ K
is an isomorphism, so that Ker F (g) = Im F (f ).
A property of modules is said to be a Morita invariant if it is preserved by additive equivalences.
Our next result is the key to identifying a whole lot of important Morita invariants.
4.4. Theorem. If R and S are equivalent rings with F : RMod −→ SMod an additive equivalence,
then for each module
RM
the lattice of submodules of
RM
and the lattice of submodules of S F (M )
are isomorphic.
Proof. Let
RM
be a left R-module. For each N ≤ M , let ιN : N ,→ M be the inclusion
monomorphism, and let Φ(N ) = Im F (ιN ) ≤ F (M ). We claim that the map Φ : N 7−→ Φ(N ) from the
submodule lattice of M to that of F (M ) is a lattice isomorphism.
To begin with Φ is injective. For if Φ(N ) = Φ(N 0 ), then there is an S-isomorphism h : F (N ) −→
F (N 0 ) with h ◦ F (ιN 0 ) = F (ιN ). But since F is full, there must be an R-homomorphism, necessarily
an isomorphism by Corollary 4.2, g : N −→ N 0 with F (g) = h. Then since F is faithful, g ◦ ιN 0 = ιN ,
and so N = N 0 .
g pp
N p p p p p p p p p p pN0
@
ιN @
ιN 0
R
M
h = F (g)
p p F (N 0 )
F (N ) p p p p p p p p p p p p p p pQ
F (ι 0 )
+
s
Q
F (ιN ) Q
N
F (M )
We claim next that Φ is surjective. Indeed, let K be a submodule of F (M ) with, say, ιK : K ,→ F (M )
the inclusion mono. Since F is dense, there is some
RH
and some S-isomorphism f : F (H) −→ K.
Since F is full, there is an R-monomorphism g : H −→ M with F (g) = ιK ◦ f : F (H) −→ F (M ). Thus,
Φ(Im g) = K.
Finally, for submodules N, N 0 of M we have N ≤ N 0 iff there is some monomorphism j : N −→ N 0
with ιN = ιN 0 ◦ j iff there is some monomorphism g : F (N ) −→ F (N 0 ) with F (ιN ) = F (ιN 0 ) ◦ g iff
Non-Commutative Rings
Section 4
35
Φ(N ) ≤ Φ(N 0 ).
4.5. Corollary.
Each of the following properties of a module is a Morita invariant.
That
is, if F is an equivalence from the category RMod of left R-modules to the category
SMod of left S-modules, then one of the following applies to
RM
iff it applies to
(a)
simple;
(d)
artinian;
(g)
finitely generated;
(b)
semisimple;
(e)
noetherian;
(h)
projective;
(c)
indecomposable
(f )
composition length c;
(i)
injective.
S F (M ):
Proof. By Corollary 4.2, epimorphisms and monomorphisms are preserved, and splitting is functorial, so projective and injective are Morita invariants. All the rest of these are characterized in the
lattice of submodules.
So what isn’t a Morita invariant? By and large properties that deal with the elements or with the
arithmetic of the ring are not invariants. Once we have found another characterization of equivalence,
some of these will become quite clear. But some properties that are not invariants are commutativity,
division ring, existence of nilpotent elements — all dealing with the arithmetic of the ring. A possibly
surprising non-invariant is that of being a free module. Projectivity is, but freeness is not.
Exercises 4.
4.1. Finish the proof of Corollary 4.2 by proving that if F : C −→ D is an equivalence and if f : C −→ C 0
is a surjective, then so is F (f ) : F (C) −→ F (C 0 ).
4.2. Prove that the category sets of all finite sets is equivalent to its full subcategory C with object
class the collection of all finite initial subsets of N.
4.3. Let C be a category. Choose an object from each isomorphism class of objects of C. (Yes, we are
using a monster version of the Axiom of Choice here.) For each object A denote the representative
of its class by Â. So
 = B̂ ⇐⇒ A ∼
= B.
Let Cˆ be the full subcategory of C with object class the representatives  of the objects of C. We
call this the skeleton of C. Using the word “the” is premature, but in Exercise 4.4 we will prove
that there is essentially one skeleton.
(a) Find a skeleton for sets whose objects are all subsets of N;
36
Section 4
(b) Find a skeleton for ab the category of finitely generated abelian groups.
(c) Find a skeleton for the category of finite abelian 2-groups.
4.4. Two categories C and D are isomorphic in case there exist covariant functors, S : C −→ D and
T : D −→ C such that T S = 1C and ST = 1D .
(a) Show that there exists a category with two objects that is not isomorphic to any of its
skeletons.
(b) Show that if Cˆ and C are two skeletons of the category C, then they are isomorphic.
4.5. Prove that two categories C and D are equivalent iff they have isomorphic skeletons.
4.6. Let C and D be two (additive) categories. A contravariant (additive) functor F : C −→ D is a
duality in case there exists a contravariant (additive) functor G : D −→ C such that GF and F G
are naturally isomorphic, respectively, to the identity functors 1C and 1D . Prove that for a pair C
and D of categories the following statements are equivalent:
(a) There is a duality F : C −→ D;
(b) There is an equivalence H : C op −→ D;
(c) There is a full, faithful, and dense contravariant functor F : C −→ D.
4.7. Let F : Top −→ Sets be the forgetful functor from the category of topological spaces to the
category of sets. On the other hand, let T : Sets −→ Top be the functor that assigns to each
set S the discrete space T (S) on S, and to each f : S −→ S 0 in Sets, the continuous function
T (f ) = f : T (S) −→ T (S 0 ) in Top.
(a) Show that the functor F : Top −→ Sets is faithful and dense but not full.
(b) Show that the functor T : Sets −→ Top is faithful and full but not dense.
4.8. Let F : C −→ D and G : D −→ C be two functors. These two functors then determine two
bi-functors
MorD (F (−), −) : C op × D −→ Sets
and
MorC (−, G(−)) : C op × D −→ Sets.
We say that (F, G) is an adjoint pair in case there is a natural isomorphism
η : MorC (F (−), −) −→ MorD (−, G(−)).
Alternatively, we say that F is a left adjoint of G and that G is a right adjoint of F . Of course,
Non-Commutative Rings
Section 4
37
this means that for each f : C 0 −→ C in C and each g : D −→ D0 in D, the diagram
MorD (F (C), D)
Mor(F (f ), g)
- MorD (F (C 0 ), D0 )
η(C,D)
η(C 0 ,D0 )
?
MorC (C, G(D))
Mor(f, G(g))
?
- MorC (C 0 , G(D0 ))
commutes. That is, for each ϕ ∈ MorD (F (C), D),
G(g) ◦ η(C,D) (ϕ) ◦ f = η(C 0 ,D0 ) (g ◦ ϕ ◦ F (f )).
(a) Let F : C −→ D and G : D −→ C be inverse equivalences. (See Exercise 0.16.) Prove that
both (F, G) and (G, F ) are adjoint pairs.
(b) Let K be a field and let V be a K-vector space. Consider the additive functors
F = HomK (V, −) : KMod −→ KMod
and G = (−) ⊗K V : KMod −→ KMod .
Show that (F, G) is an adjoint pair.
(c) Show that if F and T are the functors of Exercise 4.7, then (T, F ) is an adjoint pair.