Download The Functor Category in Relation to the Model Theory of Modules

The Functor Category in Relation to the Model
Theory of Modules
Example
Example 1
Consider the pp-formula θ:
x3 = 0
in the language of Mod(Z).
Define
Fθ (Z/6Z) = {x ∈ Z/6Z | x3 = 0}
Solution
Z/6Z = {0, 1, 2, 3, 4, 5}
One easily checks that
0·3=0
1·3=3
2·3=6=0
3·3=9=3
4 · 3 = 12 = 0
5 · 3 = 15 = 3
Hence
Fθ (Z/6Z) = {0, 2, 4}
Changing the Module Z/6Z
Let M ∈ Mod(Z). Using the same pp-formula θ as above, define
Fθ (M ) = {x ∈ M | x3 = 0}
How Does This Assignment Behave With Respect To
Morphisms?
Start with a ∈ Fθ (M ).
Then a satisfies a3 = 0.
Take f : M −→ N
Then
f (a)3 = f (a3) = f (0) = 0
Hence f (a) ∈ Fθ (N ) and f restricts to a morphism
Fθ (f ) : Fθ (M ) → Fθ (N )
Fθ Is a Functor
For every f : M −→ N , there is a commutative diagram of abelian
groups
/M
Fθ (M )
Fθ (f )
Fθ (N )
f
/N
where the horizontal arrows are inclusions and Fθ (f ) is the
restriction and co-restriction of f . The following properties are
satisfied:
1
Fθ (1M ) = 1Fθ (M )
2
Fθ (g ◦ f ) = Fθ (g) ◦ Fθ (f )
3
Fθ (g + f ) = Fθ (f ) + Fθ (g)
The Functor Corresponding to θ
The pp-formula θ gives rise to an additive covariant functor
Fθ : Mod(Z) −→ Ab
Moreover, from the commutative diagram
Fθ (M )
Fθ (f )
Fθ (N )
/M
f
/N
Fθ is a subfunctor of the identity functor.
A Closer Look at Fθ
From the exact sequence
[3]
Z −→ Z −→ Z/3Z −→ 0
we have the exact sequence of functors
0 −→ (Z/3Z,
) −→ (Z,
(Z/3Z,
([3],
)
) −→ (Z,
)∼
= Fθ
)
The Functor Corresponding to a pp-formula
Let φ be any pp-formula with n-free variables. Define
Fφ : mod(R) −→ Ab
by
Fφ (M ) := {x ∈ M n : φ(x)}
This is easily seen to be a functor.
Another “Example”
Example 2
Let φ denote the quantifier free formula
xA = 0
for some n × m matrix

a11
 a21

A= .
 ..
a12
a22
···
···
a1m−1
a2m−1
an1
an2
···
anm−1
with entries from a ring R.

a1m
a2m 



anm
The Functor Fφ
For the quantifier free pp-formula φ:
xA = 0
define Fφ : Mod(R) −→ Ab by
Fφ (M ) = {m ∈ M n | mA = 0}
Then Fφ is again an additive covariant functor and again is a
subfunctor of the functor ( )n .
Example
This matrix A gives rise to a morphism of free right modules
A
Rm −→ Rn
where it is viewed as left multiplication of column vectors. In
particular if we write d1 , . . . , dm for the basis of Rm


a11 a12 · · · a1m−1 a1m
 a21 a22 · · · a2m−1 a2m 


A= .

.
 .

an1
an2
···
anm−1
↑
↑
···
↑
↑
Ad1
Ad2
···
Adm−1
Adm
anm
Denote by
e1 , . . . , e n
d1 , . . . , dm
the standard basis vectors in Rn and Rm respectively.
There Exists a Commutative Diagram
(Rn ,
(
)
(A, )
/ (Rm ,
α
)n
τ
/(
)
β
)m
where
n
αM (f : R −→ M ) := f (e1 ), . . . , f (en )
m
βM (g : R −→ M ) := g(d1 ), . . . , g(dm )
Computing τ
The natural transformations α and β are both natural
isomorphisms. This will allow us to compute the natural
transformation τ between the forgetful functors.
Computing τ
f
fA
(Rn , M )
(A, M )
αM
Mn
f (e1 ), . . . , f (en )
(Rm , M )
βM
τM
Mm
f (A(d1 )), . . . f (A(dm ))
Computing τ
Since A(dk ) = Ak is the k-th column of A we have


a11
 a21 


f (A(d1 )) = f  . 
 .. 
an1
= f e1 a11 + e2 a21 + · · · + en an1
= f (e1 )a11 + f (e2 )a21 + · · · + f (en )an1
= f (e1 ), . . . , f (en ) A1
Computing τ
A similar calculation will show that for all k
f A(dk ) = f (e1 ), . . . , f (en ) Ak
so in fact
f A(d1 ), . . . f A(dm ) = f (e1 ), . . . f (en ) A
In other words τM is multiplication on the right by the matrix A.
There is an Exact Sequence of Modules
A
p
Rm −→ Rn −→ C −→ 0
where C is a finitely presented R-module.
Moving to the Functor Category
Embed via the Left Exact Yoneda functor:
Exact Sequence of Functors
0
/ (C,
∼
=
0
)
/ Fφ
/ (Rn ,
∼
=
/(
)
)n
(A,
A
)
/ (Rm ,
/(
)
∼
=
)m
Functorial Explanation of Quantifier Free pp-formulas
Theorem
quantifier free formulas = representable functors:
If φ is a quantifier free pp-formula, then Fφ is representable in
(mod(R), Ab).
If F is a representable functor in (mod(R), Ab), then F ∼
= Fφ
for some quantifier free formula φ
Correspondence
φ 7→ Fφ
Important Observation
Proposition
If we have an R-module homomorphism
A
Rm −→ Rn
where A is an n × m matrix with coefficients from R interpreted as
multiplication of column vectors on the left by A, then the induced
natural transformation in the functor category
(
A
)a −→ (
)b
is multiplication of row vectors on the right by the matrix A.
Finding General pp-formulas
Suppose that F is a finitely generated subfunctor of a representable
functor. Then we have a diagram with exact rows and columns
(D,
0
)
/F
0
%
/ (C,
)
We have seen that representable functors arise from quantifier free
pp-formulas:
) = Fφ
(D,
0
/F
0
(
/ (C,
) = Fψ
Apply Functorial Methods
Yoneda ensures that the natural transformation Fφ −→ Fψ comes
via some morphism C −→ D. Move to mod(R) to get a
commutative diagram with exact rows.
Rb
Rq
A
/ Ra
L
B
/
Rp
/C
/D
/0
/0
where A, B, L are all matrices with coefficients from R and these
maps are multiplication on the left of column vectors.
Move Back to the Functor Category
0
0
/ Fφ
0
F
0
/(
)p
B
)a
A
/(
)q
~
Lr
/ Fψ
L
/(
/(
)b
1
A, B, L, Lr are all matrices and the natural transformations
are multiplication on the right.
2
Lr is the restriction of L
New Description of F
Given M ∈ mod(R), we have x ∈ F (M ) ⊆ M a if and only if
1
xA = 0
2
x = vLr for some v ∈ M p with vB = 0
This establishes that
F (M ) = x ∈ M a :
∃v ∈ M
p
(x v)
I
−Lr
A
0
0
B
=0
F = Fϕ
The finitely generated subfunctor F comes via the general
pp-formula ϕ
ϕ := ∃v (x v)T = 0
and
T =
I
−Lr
A
0
0
B
Functorial Explanation of pp-formulas 1
Theorem
pp-formulas = finitely generated subfunctors of representables:
If φ is a pp-formula, Fφ is a finitely generated subfunctor of a
representable functor in fp(mod(R), Ab).
If F is a finitely generated subfunctor of a representable
functor, then there exists pp-formula φ such that F ∼
= Fφ .
Particular Example
)
(D,
0
/F
0
%
/ (Rn ,
)
0
/ Fφ
0
F
|
Lr
"
0
0
/(
/(
)p
B
)n
0
/(
)q
L
)n
(
/0
1
A, B, L, Lr are all matrices and the natural transformations
are multiplication on the right.
2
Lr is the restriction of L
F = Fϕ
The finitely generated subfunctor F comes via the general
pp-formula ϕ
ϕ := ∃v (x v)T = 0
and
T =
I
−Lr
0
B
The Tuple Natural Transformation
Proposition
For any pp-formula φ there exists a natural transformation
(c, ) −→ Fφ such that
(C,
) −→ Fφ −→ (Rn ,
is induced by c ∈ C n if and only if c ∈ Fφ (C).
)
Connections With Free Realisation
The epimorphism p corresponds to the notion of a free realisation
of ϕ:
0
Ker(p)
(D,
(c,
0
{
/ Fψ
)
)
p
M m Fϕ

(c,
)
%
/ (Rn ,
)
/ (Rn ,
)
pp-pairs
1
From the functorial perspective, it is now clear that the
collection of pp-formulas may not tell the whole story.
2
We can recover finitely generated subfunctors of
representables by passing from a pp-formula φ to the functor
Fφ .
3
This process misses certain functors in fp(mod(R), Ab).
4
This is interesting because the category fp(mod(R), Ab) tells
a different story about mod(R), namely that we can embed
this category into a larger category and the extra objects give
us more information.
pp-pairs
Definition 1
A pp-pair is a pair of pp-formulas φ/ψ such that ψ implies φ.
From the functorial point of view, a pp-pair is a pair of
pp-formulas φ/ψ such that the there is an exact sequence in the
functor category
0 −→ Fψ −→ Fφ
The Category
eq +
LR
Definition 2 (Herzog)
One can turn the pp-pairs into a category denoted by
The objects of
eq +
LR
eq +
LR :
are pp-pairs.
A morphism from φ/ψ (both with free variable x) to σ/τ
(both with free variable y) is a pp-formula ρ(x y) satisfying:
1 ρ(x y) ∧ φ(x) −→ σ(y)
2 ρ(x y) ∧ ψ(x) −→ τ (y)
3
φ(x) −→ ∃y ρ(x y)
The Functorial Look
Each pp-pair determines a functor in fp mod(R), Ab . The
morphisms between the pp-pairs are precisely the data needed to
produce a natural transformation.
0
/ Fψ
/ Fφ
/ Fφ/ψ
/0
[ρ]
0
/ Fσ
/ Fτ
/ Fσ/τ
/0
Structure of
eq +
LR
Theorem (Herzog)
eq +
LR
The category
abelian category.
of pp-pairs with morphisms between them is an
Theorem (Burke)
The category Leq+
is equivalent to the category fp(mod(R), Ab)
R
Restricting The Yoneda Embedding
The Yoneda Embedding
Y : Mod(R) −→ (Mod(R), Ab)
sends every R-module to a left exact functor in the functor
category (Mod(R), Ab).
This assignment is a contravariant embedding.
The functor category (Mod(R), Ab) is quite large because
Mod(R) is not skeletally small.
If we “restrict” the Yoneda embedding to
Y : Mod(R) −→ (mod(R), Ab) it is no longer an embedding.
Tensor Embedding
Theorem
The covariant functor t : Mod(R) −→ (mod(Rop ), Ab) given by
t(M ) = M ⊗
is an embedding, i.e fully faithful. Moreover, the functor t is the
left adjoint to evaluation at the ring. In particular
1 There is a natural isomorphism Nat(
⊗ M, F ) ∼
= (M, F (R))
2
t is right exact, commutes with direct limits, and commutes
with products.
Purity in the Functorial Language
Theorem
A monomorphism f : A −→ B is pure if and only if the morphism
A⊗
is a monomorphism.
f⊗
/B⊗
Theorem
An exact sequence
0 −→ A −→ B −→ C −→ 0
is pure exact if and only if the sequence
0 −→ A ⊗
−→ B ⊗
op
is exact in mod(R ), Ab .
−→ C ⊗
−→ 0
Pure Injectivity
Theorem
A module M ∈ Mod(R) is pure injective if and only if M
⊗
an injective object of the functor category mod(R), Ab
is
Duality
Theorem (Auslander-Gruson-Jensen, Prest, Herzog, Burke)
There is an exact duality
D : fp(mod(R), Ab) −→ fp(mod(Rop ), Ab)
which is completely determined by its exactness and the property
that
D(X, ) ∼
⊗X
=
D(
⊗ X) ∼
= (X,
)