Bài tập Lý thuyết phạm trù - Thầy Đông

υo.



†
†
‡
§
 id

G S
A G
G [Z
2
, Z
2
] = {0, id}
S [Z
2
, Z
2
] = {0, id, e} e(0) =
1, e(1) = 1
A G
(A
i
)
i∈I
C C
S
C

P
Ob(C
S
) = {(α
i
: A
i
−→ X)
I
|X ∈ (C)},
[(α
i
: A
i
−→ X)
I
, (β
i
: A
i
−→ Y )
I
]
C
S
= {δ : X −→
Y ∈ [X, Y ]
C
|
β

i
= δα
i
, ∀i ∈ I}
C
S
C
Ob(C
P
) = {(α
i
: X −→ A
i
)
I
|X ∈ (C)},
[(α
i
: X −→ A
i
)
I
, (β
i
: Y −→ A
i
)
I
]
C

P
= {γ : X −→
Y ∈ [X, Y ]
C
|
β
i
= γα
i
, ∀i ∈ I}
C
P
C
∗ [(α
i
: A
i
−→ X)
I
, (β
i
: A
i
−→ Y )
I
]
C
S
⊂ [X, Y ]
C

∗ 1
(α
i
:A
i
−→X)
I
= 1
X
∗ C
S
C 1
(α
i
:A
i
−→X)
I
= 1
X
A B C
O
B
U
A
Ob(O
B
) = {(X −→ B)|X ∈ (C)},
[X
α

−→ B, Y
β
−→ B]
O
B
= {γ : X −→ Y ∈ [X, Y ]
C
|βγ =
α}
Ob(U
A
) = {(A −→ Y )|Y ∈ (C)},
[A
α
−→ Y, A
β
−→ X]
U
A
= {δ : X −→ Y ∈ [X, Y ]
C
|δβ =
α}
α β βα βα
αβ α β
α C [α]
C
f g βαf = βαg β
αf = αg α f = g βα
f g αf = αg βαf = βαg

βα f = g α
β
N
α
−→ Z
β
−→ N
n −→ n
z −→ |z|
βα = id
N
β
|z
1
| = |z
2
| z
1
= z
2
N
Mor(N)
a b ∈ Mor(N) a ∼ b ⇐⇒ a b 2
2
Mor(N) [0], [1]
N
Ob(N) = {N}, Mor(N) = {[0], [1]}
[a], [b]
[a][b] = [ab] =




[1] a b
[0]
N [2]
N
Mon )
j : N −→ Z
g
1
g
2
Z M
n ∈ Z g
1
(n) = g
2
(n)
g
1
(−n) = g
2
(−n) n −n /∈ N g
1
j = g
2
j
j
Div
q : Q −→ Q/Z

qf = qg f g : G −→ Q G
qh = 0
h = f − g
h(x) x ∈ G h(x) = 0
h

x
2008h(x)

=
1
2008
qh

x
2008h(x)

= 0
qh = 0 h(x) = 0 q
A
O −→ A
A −→ O
1
A
a ⇐⇒ b a ⇐⇒ d
a =⇒ b) A f g : A −→
X f.0
OA
= g.0
OA

f = g [A, X]
O −→ A
(b =⇒ a) O −→ A
O −→ A
A
0
AO
//
O
0
OA
//
A
∗ 0
OA
.0
AO
= 0
AA
∈ [A, A] 1
AA
∈ [A, A]
0
AA
0
OA
= 1
AA
0
OA

= 0
OA
0
OA
0
AA
= 1
AA
∗ 0
AO
.0
OA
= 0
OO
= 1
OO
a ⇐⇒ b
a =⇒ d)
(d =⇒ a) 1
A
A
0
XA
∈ [X, A] f ∈ [X, A] 1
A
f =
0
XA
= 1
A

0
XA
f = 0
XA
1
A
a ⇐⇒ d
C
P
p
1
//
p
2

B
1
β
1

D
2
β
2
//
B
P
β
1
: B

1
−→ B β
2
: B
2
−→ B C
Ob(Pull) = {(p
1
: P −→ B
1
, p
2
: P −→ B
2
)|p
1
, p
2
∈
Mor(C), β
1
p
1
= β
2
p
2
} [(p
1
, p

2
), (p

1
, p

2
)]
P
= {γ : P

−→
P ∈ Mor(C)| p

1
= p
1
γ, p

2
= p
2
γ}
C 1
(p
1
,p
2
)
= 1

C
P
∗ P β
1
β
2
(P, p
1
, p
2
) (p
1
, p
2
)
∗ P β
1
β
2
(p
1
, p
2
) P
(P, p
1
, p
2
)
α Kerα = 0

β Ker(α) = Ker(βα)
u : K −→ A α : A −→ B
p : A −→ K
∗
u u
p
X
0
XA
−→ A
α
−→ B
∗ α.0
XA
= 0
∗ u

: K

−→ A
αu

= 0 α u

= 0
XA
λ = id
X
0
XA

.λ = u

Kerα = 0
R − Smod
Λ
3
= {0, 1, a} a 0 1
Λ
3
N = {0, 1, 2, . . .}
ϕ : N × Λ
3
−→ Λ
3
(n, x) −→ nx = x + x + . . . x
  
∀m, n ∈ N ∀x, y ∈ Λ
3
n(x + y) = (x + y) + . . . (x + y)
  
= x + x + . . . x
  
+ y + y + . . . y
  
= nx + ny
(m + n)x = nx + mx
(mn)x = m(nx)
1x = x
Λ
3

N −
f : Λ
3
−→ Λ
3
0 −→ 0
1 −→ 1
a −→ 1
f(0 + 1) = f(1) = 1 = 0 + 1 = f(0) + f(1)
f(0 + a) = f(0) + f(a)
f(1 + a) = f(1) + f(a)
f(m1) = mf(1)
f(m0) = mf(0)
f(ma) = mf(a)
f N−
K = {x ∈ Λ
3
|f(x) = 0} = {0}
R − Smod
K

0
K

K
=λ
~~
g


B
B
B
B
B
B
B
B
K
g=0
//
Λ
3
f
//
Λ
3
g : K −→ Λ
3
0 −→ 0
fg = f.0
KΛ
3
= 0
KΛ
3
g

: K


−→ Λ
3
fg

= 0
K

Λ
3
g

= 0
λ : K

−→ K = {0}
x −→ 0
gλ(x) = g(0) = 0 = g

(x) ∀x ∈ K

K = Kerf f
Ker(α) Ker(βα)
Ker(α) = Ker(βα)
(=⇒)
X
kerα
−→ A
α
−→ B
β

−→ C
(βα)kerα = β(αkerα) = β0
XB
= 0
XC
K

λ
~~
u


A
A
A
A
A
A
A
A
X
kerα
//
A
α
//
B
β
//
C

u

: K

−→ A βαu

= 0
K

C
β
αu

= 0
XB
αkerα = 0XB
λ : K

−→ K ker(α).λ = u

Ker(α) = Ker(βα)
(⇐=)
K

λ
~~
u


A

A
A
A
A
A
A
A
K
u
//
A
p
//
α

@
@
@
@
@
@
@
K
∗
γ
}}
B
∗ pu = 0 p = cokeru
αu = 0 cokeru
γ : K

∗
−→ B
γp = α
∗ u

: K

−→ A pu

= 0
γpu

= 0 = αu

u = kerα
λ : K

−→ K uλ = u

u p
(β
i
:
B
i
−→ B)
i∈I
) C (β
i
:

B
i
−→ B)
i∈I
) O
B
B
β
i
(β
i
:
B
i
−→ B)
i∈I
) B
i

i∈I
B
i
C B
O
B
B
(B
i
)
i∈I

(β
i
:
B
i
−→ B)
i∈I
)
C
Ov
B
≡ C
Ob(C)
1−1
←→ Ob(Ov
B
)
[X
f
−→ B, Y
g
−→ B] = [X, Y ]
C
Ob(Ov
B
) 1 − 1
Ob(C)
(X −→ B) ∈ Ob(Ov
B
)

1−1
←→ X ∈ Ob(C)
∀γ ∈ [X
f
−→ B, Y
g
−→ B]
Ov
B
γ : X −→
Y : gγ = f
γ ∈ [X, Y ]
C
[X
α
−→ B, Y
β
−→ B] ⊂
[X, Y ]
C
γ ∈ [X, Y ]
C
γ : X −→ Y
gγ : X −→ B ∈ [X, B]
f : X −→ B ∈ [X, B] B
gγ = f γ ∈ [X
f
−→ B, Y
g
−→

B]
Ov
B
[X, Y ]
C
⊂ [X
f
−→ B, Y
g
−→ B]
Ov
B
[X
f
−→ B, Y
g
−→ B] = [X, Y ]
C
(P, P
p
i
−→ B
i
)
i∈I
(B
i
)
i∈I
C (P, P

p
i
−→ B
i
)
i∈I
(β
i
: B
i
−→ B)
i∈I
β
i
p
i
: P −→ B, β
j
p
j
: P −→ B
B β
i
p
i
= β
j
p
j
∀i, j ∈ I

β
i
p
i
∈ Ob(Ov
B
)
∀X ∈ Ob(C) (X, X
α
i
−→ B
i
)
i∈I
β
i
α
i
: X −→ B, β
j
α
j
: X −→ B
B β
i
α
i
= β
j
α

j
∀i, j ∈ I
β
i
α
i
∈ Ob(Ov
B
)
P γ : X −→ P
p
i
γ = α
i
∀i ∈ I
P (β
i
: B
i
−→ B)
i∈I
α Kerα = 0
α :
A −→ B α = uv u = ker(cokerα) v =
coker(kerα)
α α = ker(cokerα)
α kerα = 0
XA
v = coker(kerα) =
coker0

XA
= 1
A
α = u1
A
= ker(cokerα)
kerα = 0 =⇒ α
f, g : X −→ A αf = αg
uvf = uvg u vf = vg
coker(kerα)f = coker(kerα)g 1
A
f = 1
A
g
f = g
α ⇐⇒ Cokerα = 0
Coequ(α, β) = Coker(α − β)
α : A −→ B cokerα : B −→ Y
Cokerα = 0 =⇒ α
f, g : B −→ Y
∗
fα = gα
fα − gα = 0 (f − g)α = 0
A
α
//
B
Cokerα
//
f−g

B
B
B
B
B
B
B
B
Y
γ
~~
Y
∗
γ : B −→ Y
∗
f −g =
γcokerα = 0 f = g α
(C, h) = Coker(α − β)
Coequ(α, β) = Coker(α − β)
Coequ(α, β)
h(α − β) = hα − hβ = 0 hα = hβ
u : B −→ Z uα = uβ u(α−β) = 0
γ : Y −→ Z γh = u
A
α−β
//
B
h
//

u

@
@
@
@
@
@
@
C
γ

Z
Coequ(α, β) = Coker(α − β)
H
A
: C −→ S
X −→ [A, X]
C
X
α
−→ Y −→ H
A
(X)
[A,α]=H
A
(α)
−→ H
A
(Y )

f −→ αf
C A
C
A ⊗
R
− : R − −→ A
X −→ A ⊗
R
X
X
α
−→ Y −→ A ⊗
R
X
1⊗α
−→ A ⊗
R
X
A R − R −
R −
Hom
R
(A, −) : R − −→ A
X −→ Hom
R
(A, X)
X
α
−→ Y −→ Hom
R

(A, α)
A R − R −
R −
X −→ [A, X]
C
X
α
−→ Y −→ H
A
(X)
H
A
(α)
−→ H
A
(Y )
f −→ αf
∗ H
A
(1
X
) = 1
H
A
(X)
φ ∈ [A, X]
C
H
A
(1

X
)(φ) = 1
X
(φ) = φ
1
H
A
(X)
(φ) = φ.
∗ φ ∈ [A, X]
C
f ∈ [X, Y ]
C
g ∈ [Y, Z]
C
H
A
(gf)(φ) = (gf)φ = gfφ
H
A
(g)H
A
(f)(φ) = H
A
(g)(fφ) = gfφ,
H
A
(gf) = H
A
(g)H

A
(f)
H
A
α : A −→ B Mor(C)
H
A
H
B
X ∈ Ob(C)
H
α
: H
A
(X) −→ H
B
(X)
f −→ αf
H
α
H
A
H
B
H
A
H
B
H
α

φ : X −→ Y ∈ Mor(C)
H
A
(Y )
H
A
(φ)

H
α
(Y )
//
H
B
(Y )
H
B
(φ)

H
A
(X)
H
α
(X)
//
H
B
(X)
[Y, A]

C
H
A
(φ)

H
α
(Y )
//
[Y, B]
C
H
B
(φ)

[X, A]
C
H
α
(X)
//
[X, B]
C
f ∈ [Y, A]
H
B
(φ)H
α
(Y )(f) = H
B

(φ)(αf) = αfφ
H
α
(X)H
A
(φ)(f) = H
α
(X)(fφ) = αfφ.
H
B
(φ)H
α
(Y ) = H
α
(X)H
A
(φ).
F : C −→ D
F (0) = 0
F (−α) = −F (α) ∀α ∈ Mor(C)
F (id
0
+ id
0
) = F (id
0
) + F(id
0
) =⇒ F (id
0

) = 0
=⇒ id
F (0)
= 0 =⇒ F (0) = 0
F (α − α) = F (0) = F (α) + F (−α) = 0
F (−α) = −F (α)
F
C D
X
α
−→ Y
β
−→ Z −→ 0
F (X)
F (α)
−→ F (Y )
F (β)
−→ F (Z) −→ 0
(=⇒) F
X
α
−→ Y
β
−→ Z −→ 0
F (X)
F (α)
−→ F (Y )
F (β)
−→ F (Z) −→ 0
F

C X
α
−→ Y
β
−→ Z β =
Cokerα β β = Cokerα = Coimβ
Imα = Kerβ
X
α
−→ Y
β
−→ Z −→ 0
F (X)
F (α)
−→ F (Y )
F (β)
−→ F (Z) −→ 0
ImF (α) = KerF(β) F (β)
CokerF (α) = CoimF (β) = F (β)
(⇐=) F C
X
α
−→ Y
β
−→ Z β = Cokerα
F (X)
F (α)
−→ F (Y )
F (β)
−→ F (Z) −→ 0

CokerF (α) = F (β)
X
α
−→ Y
β
−→ Z −→ 0
Imα = Kerβ β
Cokerα = Coimβ = β
F (X)
F (α)
−→ F (Y )
F (β)
−→ F (Z)
CokerF (α) = CoimF (β) = F (β)
ImF (α) = KerF (β) F (β)
H
A
H
B
Hom
R
(A, −) Hom
R
(−, B) :
R − −→ A
A ⊗
R
− : R − −→ A
− ⊗
R

B : R − −→ A
H
A
A ⊗
R
− C
R −
H
A
∗ H
A
∗ H
A
f, g ∈ [A, X]
C
[A, α](f+g) = α(f+g) = αf+αg = [A, α]f+[A, α]g
∗ H
A
C
X
α
−→ Y
β
−→ Z α = Kerβ A
[A, X]
C
[A,α]
−→ [A, Y ]
C
[A,β]

−→ [A, Z]
C