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
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Giáotrìnhhìnhhọc
đại số
Ngô Bảo Châu
Tháng 8 năm 2003
class="bi x7 ya w1 h6"
class="bi x8 yb w2 h7"
class="bi x9 yc w2 h8"
class="bi xa yd w3 h9"
class="bi x9 ye w2 ha"
Z
0 1
R (+, 0, ×, 1)
R + 0 ∈ R
+
R × 1 ∈ R .
+
x × (y + z) = x × y + x × z.
Z Q R
Z[x]
Q[x] C[x]
0 = 1
R R

φ : R → R

(+, 0, ×, 1) R R

k 0
k p Q
F
p
p
x ∈ R 0
y ∈ R 0 xy = 0 x ∈ R
n ∈ N x
n
= 0
R R
R R
R M
R ×M → M (α, x) → αx
(α + β)x = αx + βx α(x + y) = αx + αy
(αβ)x = α(βx) 1.x = x
R R
R R
R M
1
, M
2
M
1
× M
2
R α(x
1
, x
2

1
+ ··· + α
n
x
n
M R M

M ⊕M

R M N ⊂ M
N M
M/N R
R
R I R
I R R/I
R I I
x, y ∈ R
x y I I = R
I R/I
I R/I
R
Spec(R) R
Spec(R)
R = Z Spec(Z)
{0}
p {0}
Spec(Z)
C[x]
{0}
x −α α


R/p
R → R

Spec(R

) →
Spec(R) R

R
M N R M ⊗
R
N
{x
i
|i ∈ I} M {y
j
| j ∈ J} N M N
I, J
R V x
i
⊗y
j
I × J R W

i∈I
α
i
x
i

j
= 0
M ⊗
R
N = V/W
x ∈ M, y ∈ N x =

i∈I
α
i
x
i
y =

j∈J
β
j
y
j
α
i
, β
j
i, j

i,j
α
i
β
j

N → L ψ = ψ

◦ φ
(M ⊗
R
N, φ)
{x
i
} {y
j
}
M = R
I
N = R
J
M ⊗
R
N = R
I×J
M R
I
M ⊗
R
N ⊗
i∈I
N
i
N
i
N


R M ⊗
R
R

R

β(m ⊗α) = m ⊗(αβ) m ∈ M α, β ∈ R

M
R M ⊗
R
R

R M R
R

R

R

(M ⊗
R
R

) ⊗
R

R


→ R

⊗
R
R

φ

(x

) = x

⊗ 1 φ

(x

) = 1 ⊗ x

(R

⊗
R
, R

; φ

, φ

)
R S R ψ


R

R
R

= R[x
1
, . . . , x
n
]/f
1
, . . . , f
m

n
R

⊗
R
R

= R

[x
1
, . . . , x
n
]/f
1

1
/s
1
+ x
2
/s
2
= (x
1
s
2
+ x
2
s
1
)/s
1
s
2
(x
1
/s
1
)(x
2
/s
2
) = x
1
x

S
−1
R
R
φ : R → S
−1
R
Spec(S
−1
R) → Spec(R)
R S
p R p

S
−1
R
x/s x ∈ p s ∈ S S
−1
R
S
−1
R
p ∩ S = ∅ p

φ
−1
(p

) = p
f ∈ R

R {0}
Spec(R) R −{0}
R
0 R
R K(R)
p R
R/p K(R/p)
R/p (p) R
p
R
p
/(p) = K(R/p).
p R R
R
p
/(p) φ : R → K R
K {0} K
R
p
/(p)
R Spec(R)
p ∈ Spec(R) f ∈ R f R f
R → R
p
/(p)
p
f R
f φ : R → K K 0
R
M

1
) = 0
S
−1
M
S
−1
R
p M
p
M
R − p
M
(p)
= M
p
⊗
R
p
(R
p
/(p))
(R
p
/(p))
M M
(p)
p ∈ Spec(R)
M R
R m R

n
→ M N

= M/N
N

= 0 M
N

y

1
, . . . , y

n
N

y
1
, . . . , y
n
M y
i
N

y

i
¯y
i

j=1
α
ij
x
j
∈ mM.
N

¯y
i
x
j
∈ N ¯y
i
mN

¯y
i
¯y
i
=
n

j=1
β
ij
¯y
j
β
ij

r
¯
M
¯
M r = n
φ : R
n
→ M x
1
, . . . , x
n
φ
ij
x
i
=
n

j=1
φ
ij
v
j
.
¯
φ : k
n
→
¯
M

k
¯
M ⊕
¯
M

= k
n
x
1
, . . . , x
m
∈ M ¯x
1
, . . . , ¯x
m
∈
¯
M
¯
M x

1
, . . . , x

m

∈ M

¯x


φ
R
R
I
1
⊂ I
2
⊂ ··· ⊂ I
n
⊂ ···
M R
M

⊂ M
M = R I R R
M = R I
I R
M
R
n
→ M
M = R
n
M = R
n
M = R
{0} Z
R R[x
1

−1
(
¯
I
i
) φ
¯
I
i
= φ(φ
−1
(I
i
)) R I
i
¯
I
i
R

= S
−1
R φ : R → R

I

1
⊂ I

2

1
, . . . , x
n
] → R

R

R
R[x
1
, . . . , x
n
] R

= R[x
1
, . . . , x
n
]/I
I
R

R

= R[x
1
, . . . , x
n
]/f
1

A
∈ Hom
C
(A, A)
A A, B, C ∈ Ob(C)
Hom
C
(A, B) × Hom
C
(B, C) → Hom
C
(A, C)
(φ, ψ) → ψ ◦φ
A
i
∈ Ob(C) i = 0, 1, 2, 3 φ
i
∈ Hom
C
(A
i−1
, A
i
)
(φ
2
◦ φ
1
) ◦ φ
0

C
(A, B) id
A
C C
Set
Ring Set
F C C

Ob(C) → Ob(C

) A → F A
A, B ∈ Ob(C)
Hom
C
(A, B) → Hom
C

(F A, FB)
φ → F (φ)
F (φ ◦ψ) = F(φ) ◦ F (ψ)
F (id
A
) = id
F A
id
C
C → C
C C
Ring → Set R
R


f : F → F

f(A) : F (A) → F

(A)
A ∈ Ob(C) φ ∈ Hom
C
(A, B)
F A
f(A)
−−−→ F

A
F (φ)






F

(φ)
F B −−−→
f(B)
F

B
F = F


id
C

F

◦F
id
C
F ◦F

= id
C
F ◦F

id
C
C C
opp
C
C
Hom
C
opp
(A, B) = Hom
C
(B, A).
C
opp
C

C
(A, B)
φ : A → A

h
A

→ h
A
B ∈ ObC
Hom
C
(A

, B) → Hom
C
(A, B).
ψ → ψ ◦φ
Hom
C
(A

, B) −−−→ Hom
C
(A, B)







, h
A
)
φ, φ

∈
Hom
C
(A, A

) B = A

φ = id
A

∈ Hom
C
(A

, A

) h(φ)(id
A

) = φ
h(φ

)(id
A

) f
−1
f

Y(A) → Y(A)
h
f
−1
f

: h
A

→ h
A
φ : A → A

h(φ) = f
−1
f

F : C → Set
(A, f)
f F
A A
A
1
: Ring → Set R
R
Z[t] t

α = φ(x) β = φ(y) α β R
αβ = 1 α R
α β G
m
 Spec(Z[x, y]/xy −1)
µ
n
: Ring → Set n ∈ N
R n
µ
n
(R) = {x ∈ R | x
n
= 1}.
µ
n
(R)
Z[x]/x
n
− 1 → R.
µ
n
= Spec(Z[x]/x
n
− 1)
J
J
j ∈ J S
j
i ≤ j J s

jk
= s
ik
i ≤ j ≤ k
J J
Hom
J
(i, j) i ≤ j
i ≤ j Hom
J
(i, j)
J Set
C C
J S J C
C J S J C
opp
C C C
c
j
: S
j
→ C i ≤ j c
i
= c
j
◦ s
ji
(C

; (c

j
i ≤ j
(C, (c
j
)
j∈J
)
Set C
C Set
F
j
: C → Set
F
j
(C) C ∈ ob(C)
(S
j
, s
ji
) Set
C (j, x) j ∈ J
x ∈ S
j
(j, x) ∼ (j

, x

) i
j j


(x
j
) = x
i
i
≤
j C 