Giáo trình hình học đại số pot

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Giáotrìnhhìnhhọc
đại số
Ngô Bảo Châu
Tháng 8 năm 2003
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


R
φ
R
: Z → R
n φ
R
n 1+···+1
n R n n −φ
R
(−n)
x ∈ R y ∈ R
xy = 1
R
×
R R
R
×
= R −{0} Q
R, C Z
p
F
p
F
p
p Q
Z
k Q
F
p

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

) = (αx
1
, αx
2
)
M
1
M
2
M
1
⊕ M
2
R n
R
n
= R ⊕···⊕R n
M
R
n
→ M M
M
x
1
, . . . , x
n
∈ M x ∈ M
x = α
1
x

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

C[x] C
R {0}
R
φ : R → R

p
p

R

p
p

R

p p

R/p → R

/p

φ
R

/p

R/p
R

/p


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

⊗ y
j
j ∈ J
α
i
i

i∈I
α
i
x
i
= 0

j∈J
α
j
x
i
⊗ y
j
i ∈ I
α
j
j

j∈J
α
j
y

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

x
i
y
j
∈ V V/W x =

i∈I
α
i
x
i
y =

j∈J
β
j
y
j
x y
φ : M ×N → M ⊗
R
N
(M ⊗
R
N, φ)
ψ : M × N → L
ψ

: M ⊗
R

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

M = R/p p R M ⊗ N = N/pN pN
N αy α ∈ p y ∈ N
R R
R

φ : R → R

R (φ
1
, R
1
) (φ
2
, R
2
) ψ : R
1
→ R
2
ψ ◦ φ
1
= φ
2
R

R M R M ⊗
R
R

R


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


= M ⊗
R
R

.
R

R

R R
R

⊗
R
R

R φ

: R

→ R

⊗
R
R

φ

: R


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


: R

→ S
ψ

: R

→ S R ψ : R

⊗
R
R

→
S ψ

= ψ ◦ φ

ψ

= ψ ◦ φ

R

⊗
R
R

R

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

, . . . , f
m
.
R S R
1 ∈ S x, y ∈ S xy ∈ S
R S
S
−1
R = {(x, s) ∈ R × S}/ ∼
(x
1
, s
1
) ∼ (x
2
, s
2
) s ∈ S s(x
1
s
2
− x
2
s
1
) = 0
(x, s) x/s S
−1
R
x

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

2
/s
1
s
2
S
−1
R
φ : R → S
−1
R
φ(x) = x/1 ∈ S
−1
R φ φ(s)
s ∈ S S
−1
R
φ

: R → R

φ(s) ∈ R

s ∈ S
φ

= ψ ◦ φ ψ : S
−1
R → R


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 S = {1, f, f
2
, . . .}
f Spec(S
−1
R)
Spec(R) R f
p S
S = R − p p S
S
−1
R R S
Spec(S
−1
R) R p p
R S
−1
R S
−1
R
S
p
p p
R R −p
R − p R
R p
R
p
R
R − p

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

R S R
S
−1
M M S
S
−1
M = M ⊗
R
S
−1
R.
S
−1
M (m, s) ∈
M × S (x
1
, s
1
) ∼ (x
2
, s
2
) s ∈ S s(x
1
s
2
−
x
2
s

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

k R/m
M R
¯x
1
, . . . , ¯x
m
k
¯
M ⊗
R
k
x
1
, . . . , x
m
M x
i
¯
M ¯x
i
φ : R
m
→ M
x
1
, . . . , x
n
φ M
φ
N φ : 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

y
i
¯
M ¯x
1
, . . . , ¯x
n
¯
M
¯y
i
=
m

j=1
¯α
ij
¯x
j
.
α
ij
∈ R ¯α
ij
∈ k
y
i
−
m


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

∈ m
Id
n
¯y = β ¯y β β
ij
¯y
(¯
y
1
, . . . ,
¯
y
n
) Id
n
−
β
R 1 k
R
Id
n
− β ¯y N

M R v
1
, . . . , v
r
¯v
1
, . . . , ¯v

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

¯
φ det(
¯
φ) ∈ k
×
det(φ) ∈ R
×
φ
φ : R
n
→ M
M R
M M

M ⊕
M

R
n
¯
M = M ⊗
R
k
¯
M

= M

⊗
R

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


1
, . . . , ¯x

m

∈
¯
M

¯
M

φ : R
m
→ M φ

: R
m

→ M

φ ⊕φ

: R
m+m

→ R
n
φ φ


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

, . . . , x
n
]
n R
R I R
¯
R = R/I S R
S
−1
R
φ : R →
¯
R = R/I
¯
I
1
⊂
¯
I
2
⊂ ···
¯
R
I
1
⊂ I
2
⊂ ··· R I
i
= φ

−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

⊂ ···
R

I
1
⊂ I
2
⊂ ··· R
I
i
= φ
−1
(I

i
) I

i
= S
−1
I
i
I
i
I

i
R R

R ψ : R[x

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

, . . . , f
m

f
1
, . . . , f
m
 f
1
, . . . , f
m
∈ R
C
Ob(C) C Hom(C)
C
s × b : Hom(C) → Ob(C) ×Ob(C),
s : Hom( C) → Ob(C)
b
: Hom(
C
)
→
Ob(
C
)
A, B ∈ Ob(C) Hom
C
(A, B) φ ∈
Hom(C) A B
A ∈ Ob(C) id

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

= φ
2
◦ (φ
1
◦ φ
0
),
φ ∈ Hom
C
(A, B) φ ◦ id
A
= id
B
◦ φ = φ
Set
Ring
R
R −Alg R R
C Ob(C

) Ob(C)
C

Ob(C

) Hom
C

(A, B) =
Hom

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

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


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
F
A ∈ Ob(C) id
F
(A) = id
F A
: F A → FA
f : F → F

f

: F

→ F f ◦ f

= id
F

f

◦ f = id
F
F : C → C

F

: C

→ C 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

φ ∈ Hom
C
opp
(A, B) ψ ∈ Hom
C
opp
(B, C)
ψ ◦ φ C
opp
φ ◦ ψ C
Ring
A Spec(A)
A Ring
opp
C
F(C) F : C → Set
F, F

: C → Set F(C) Hom
F(C)
f : F → F

F F

F(C) id
F
A ∈ ObC
h
A
: C → Set B → Hom

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)







Hom
C
(A

, B

) −−−→ Hom
C
(A, B

)
h : C
opp
→ F(C)
h : C
opp
→ F(C) A → h
A
A, A

∈ Ob(C)
h : Hom
C
(A, A

) → Hom
F(C)
(h
A


, 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


) = φ

h(φ) = h(φ

) φ φ

f : h
A

→ h
A
φ ∈ Hom
C
(A, A

) φ = f(id
A

)
h(φ) = f
F : C → Set
A ∈ ObC f : h
A
→ F.
(A, f)
(A

, f


) 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

φ : Z[t] → R
φ(t) ∈ R h
A
→ F A = Z[t]
A
1
Z[t]
Ring
opp
F(Ring)
A
1
 Spec(Z[t])
R F : R − Alg → Set
R R

R

R[t]
R A
1
R
 Spec(R[t])
G
m
: Ring → Set R R
×
R
Z[x, y]/xy −1
φ : Z[x, y]/xy−1 → R

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

ji
: S
i
→ S
j
s
ii
= 1
s
kj
◦ s
ji
= s
ki
i ≤ j ≤ k
J
j ∈ J S
j
i ≤ j J s
ij
: S
j
→ S
i
s
ii
= 1
s
ij
◦ 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
)
j∈J
) (C, (c
j
)
j∈J
)
b : C → C

c

j
= b ◦ c
j
(C, (c
j
)
j∈J )
C C
c
j
: C → S
j
c
i
= s
ij
◦ 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


s
ij
(x) = s
ij

(x

) j ∈ J c
j
: S
j
→ C
x ∈ S
j
(j, x)
(S
j
, s
ij
) Set
C (x
j
)
j∈J
x
j
∈ S
j
s
ij

(x
j
) = x
i
i
≤
j C 