luận án tiến sĩ toán học hệ nhân tử trog nhóm phạm trù phân bậc

Γ
H
3
ab
(M, N)
Γ
Γ
c
X,X
= id
X
H
3
Shu
(R, M)
G
k k
k K
K
K = Z
Γ Γ
Γ
Γ
(ϕ, f)
(Π, A)
Γ Γ
Γ
Γ
(G, ⊗, I, a, l, r)

X Y
X ⊗ Y  I  Y ⊗ X
X Y X ⊗ Y = I = Y ⊗ X
a l, r
G
G Π
Π A k ∈ Z
3
(Π, A)
S
G
G
S
G
G S
G
(Π, A, k) (Π, A)
Γ G = (G, gr, ⊗, I, a, l, r)
Γ (G, gr) Γ ⊗ :
G ×
Γ
G → G I : Γ → G
a
X,Y,Z
: (X ⊗ Y ) ⊗ Z
∼
→ X ⊗ (Y ⊗ Z), l
X
:
I ⊗ X

∼
→ X, r
X
: X ⊗ I
∼
→ X
G
G
Γ (G, gr)
Γ
A
⊕, ⊗ : A × A → A
⊕, ⊗
F F
⊕
⊗
R R M h ∈ Z
3
MacL
(R, M).
S
A
= (R, M, h)
A (R, M)
(ϕ, f)
(ϕ, f)
(F,

F ) : G → G


(F,

F )
F
0
: π
0
G → π
0
G

, [X] → [F X],
F
1
: π
1
G → π
1
G

, u → γ
−1
F I
(F u),
F
1
(su) = F
0
(s)F
1

(u) γ
X
(u)
γ
X
(u) = l
X
◦ (u ⊗ id) ◦ l
−1
X
.
(F,

F ) : G → G

S
G
→ S
G

.
⊗ G, G

(I, l, r)
(I

, l

, r


) (F,

F , F
∗
) : G → G

⊗
γ
−1
F I
(F u) = F
−1
∗
F (u)F
∗
.
F γ
X
(u) = γ
F X
(γ
−1
F I
F u).
S, S

(Π, A, h) (Π, A, h

)
F : S → S


(ϕ, f)
F (x) = ϕ(x), F (x, a) = (ϕ(x), f(a)),
ϕ : Π → Π

f : A → A

f(xa) =
ϕ(x)f(a) x ∈ Π, a ∈ A.
(F,

F ) : G → G

S
F
: S
G
→ S
G

(ϕ, f) ϕ = F
0
, f = F
1
S
F
= G

F H, H, G


(F,

F ) : S → S

(ϕ, f).
(ϕ,f)
[S, S

]
(ϕ, f) S = (Π, A, h) S

= (Π, A

, h

) k = ϕ
∗
h

− f
∗
h
F (ϕ, f)
F : S → S

(ϕ, f)
k H
3
(Π, A


)
i) Hom
(ϕ,f)
[S, S

] ↔ H
2
(Π, A

),
ii) Aut(F) ↔ Z
1
(Π, A

).
CG
H
3
Gr
(Π, A, h)
h ∈ H
3
(Π, A) (ϕ, f) : (Π, A, h) → (Π

, A

, h

) (ϕ, f)
g : Π

2
→ A

(ϕ, f, g) (Π, A, h) →
(Π

, A

, h

)
d : CG → H
3
Gr
G → (π
0
G, π
1
G, h
G
)
(F,

F ) → (F
0
, F
1
)
i) dF F
ii) d

iii) d (ϕ, f) : dG → dG

d :
(ϕ,f)
[G, G

] → H
2
(π
0
G, π
1
G

).
Π Π A G
(Π, A) p : Π → π
0
G, q : A → π
1
G
q(su) = p(s)q(u), s ∈ Π, u ∈ A
CG[Π, A]
(Π, A)
Γ : CG[Π, A] → H
3
(Π, A),
[G] → q
−1
∗

p
∗
h
G
.
B = (B, ⊗, I, a, l, r, c)
S
B
= (M, N, h, η) η
(F,

F ) : S → S

(ϕ, f, g)
ϕ
∗
(h

, η

) − f
∗
(h, η) = ∂
ab
(g).
H
3
BGr
(M, N, (h, η))
(h, η) ∈ H

3
ab
(M, N) BCG
d : BCG → H
3
BGr
B → (π
0
B, π
1
B, (h, η)
B
)
(F,

F ) → (F
0
, F
1
)
i) dF F
ii) d
iii) d (ϕ, f) : dB → dB

Br
(ϕ,f)
[B, B

]
∼

=
H
2
ab
(π
0
B, π
1
B

),
Br
(ϕ,f)
[B, B

]
B B

(ϕ, f)
BCG[M, N]
(M, N)
Γ : BCG[M, N] → H
3
ab
(M, N),
[B] → q
−1
∗
p
∗

(h, η)
B
.
Γ C Psd(Γ, C)
Γ C
Γ
BCG
Γ
Γ
Γ
BCG  Psd(Γ, BCG).
(Π, G, ψ) ψ : Π → G/ G
(Π, G, ψ) k ∈ H
3
(Π, ZG)
Aut
G
G
(α, β) = {c ∈ G|α = µ
c
◦ β}.
Aut G/InG, ZG ψ
∗
h ψ
∗
h k.
G
M = (B, D, d, θ) B
d
→ D

B → D d : B → D, θ : D → B
C
1
. θd = µ
C
2
. d(θ
x
(b)) = µ
x
(d(b)), x ∈ D, b ∈ B
µ
x
x
M = (B, D, d, θ)
i) Kerd ⊂ Z(B)
ii) Imd D
iii) θ ϕ : D → Aut(Kerd)
ϕ
x
= θ
x
|
Kerd
,
iv) Kerd Cokerd
sa = ϕ
x
(a), a ∈ Kerd, x ∈ s ∈ Cokerd.
(B, D, d, θ)

P
B→D
:= P
(f
1
, f
0
) : (B, D, d, θ) →
(B

, D

, d

, θ

) P, P

(B, D, d, θ) (B

, D

, d

, θ

)
i) F : P → P

F (x) = f

0
(x), F (b) = f
1
(b),
x ∈ ObP b ∈ MorP
ii)

F
x,y
: F (x)F (y) → F (xy) F

F
x,y
= ϕ(x, y) ϕ ∈ Z
2
(Coker d, Ker d

)
Cross
(f
1
, f
0
, ϕ) (f
1
, f
0
) ϕ ∈
Z
2

(Cokerd, Kerd

).
(F,

F ) : P → P

S
1
. F (x) ⊗ F(y) = F (x ⊗ y), x, y ∈ ObP
S
2
. F (b) ⊗ F(c) = F (b ⊗ c), b, c ∈ MorP
P P

(B, D, d, θ) (B

, D

, d

, θ

) (F,

F ) : P → P

(f
1
, f

0
, ϕ)
f
1
(b) = F (b), f
0
(x) = F (x), ϕ(x, y) =

F
x,y
,
b ∈ B, x ∈ D, x ∈ Coker d, Cross
Grstr
Φ : Cross → Grstr,
(B → D) → P
B→D
(f
1
, f
0
, ϕ) → (F,

F )
F (x) = f
0
(x), F (b) = f
1
(b),

F

x,y
= ϕ(x, y) x, y ∈ D, b ∈ B
M = (B
d
→ D) Q
B Q M
E : 0
//
B
j
//
E
p
//
ε

Q
//
1,
B
d
//
D
(B, E, j, θ
0
) θ
0
(id
B
, ε)

ψ : Q → Coker d
Ext
B→D
(Q, B, ψ)
B Q B → D ψ : Q → d
Dis Q (Q, 0, 0)
(0, Q, 0, 0)
Dis Q → P
B → D ψ : Q → Coker d
(F,

F ) : Dis Q → P F (1) = 1
(ψ, 0) : (Q, 0) → (Cokerd, Kerd) E
F
B Q
B → D ψ
Ω : Hom
(ψ,0)
[DisQ, P
B→D
] → Ext
B→D
(Q, B, ψ).
P = P
B→D
B → D π
0
P = Coker d π
1
P = Ker d S

P
S
P
= (Cokerd, Kerd, k), k ∈ H
3
(Cokerd, Kerd).
ψ : Q → Cokerd ψ
∗
k ∈ Z
3
(Q, Kerd).
(B, D, d, θ) ψ : Q → Cokerd
ψ
∗
k H
3
(Q, Kerd)
B Q B → D ψ
ψ
∗
k
H
2
(Q, Kerd)
Γ
Γ Γ
Γ
Γ (ϕ, f)
Γ H
i

Γ
(Π, A)
i = 1, 2, 3.
(ϕ, f)
Γ (ϕ, f)
Γ G
Γ

Γ
(Π, A, h)
G
Γ (H
Γ
,

H
Γ
, id) :

Γ
(Π, A, h) → G










H
Γ
(s) = X
s
H
Γ
(r
(a,σ)
→ s) = (X
r
ˆγ
X
s
(a)◦Υ
(r,σ)
−−−−−−−→ X
s
)
(

H
Γ
)
r,s
= i
−1
X
r
⊗X
s

,
σr = s Γ
Γ
G Γ ∆F
G Γ ∆F

Γ
(Π, A, h)
(ϕ, f)
(ϕ, f)
G, G

S = (Π, A, h), S

= (Π

, A

, h

)
Γ
i) Γ (F,

F ) : G → G

Γ
S
F
: S

G
→ S
G

(ϕ, f) ϕ = F
0
, f = F
1
F
0
: π
0
G → π
0
G

, [X] → [F X],
F
1
: π
1
G → π
1
G

, u → ˆγ
−1
F I
(F u).
S

F
= G

Γ
F H
Γ
, H
Γ
, G

Γ
Γ
ii) Γ (F,

F ) : S → S

Γ (ϕ, f).
iii) Γ F : S → S

(ϕ, f) Γ
ξ H
3
Γ
(Π, A

)
Hom
(ϕ,f)
[S, S


] ↔ H
2
Γ
(Π, A

).
Γ
B, D Γ Γ
M = (B, D, d, θ) d : B → D, θ : D → B Γ
C
1
. θd = µ,
C
2
. d(θ
x
(b)) = µ
x
(d(b)),
C
3
. σ(θ
x
(b)) = θ
σx
(σb),
σ ∈ Γ, x ∈ D, b ∈ B, µ
x
x
Γ

Γ
F = (G, F
σ
, η
σ,τ
) Γ
G η
σ,τ
= id F
σ
σ, τ ∈ Γ
(P, gr)
i) Ker P
ii) P F Γ
Ker P
Γ M
Γ P
M
:= P
Γ
Γ
Γ
(f
1
, f
0
) : M → M

Γ
Γ (F,


F ) : P
M
→ P
M

F (x) =
f
0
(x), F (b, 1) = (f
1
(b), 1) f = p
∗
ϕ, ϕ ∈ Z
2
Γ
(Coker d,
Ker d

) p : D → Coker d
Γ
Cross Γ
(f
1
, f
0
, ϕ) (f
1
, f
0

) : M → M

Γ
ϕ ∈ Z
2
Γ
(Coker d, Ker d

)
Γ (F,

F ) : P → P

Γ
S
1
. F (x ⊗ y) = F(x) ⊗ F(y),
S
2
. F (b ⊗ c) = F (b) ⊗ F (c),
S
3
. F (σb) = σF (b)
S
4
. F (σx) = σF (x)
x, y ∈ Ob P, b, c P
p : D → Coker d
P P


Γ
Γ M M

(F,

F ) : P → P

Γ (f
1
, f
0
, ϕ)
i) f
0
(x) = F (x), (f
1
(b), 1) = F (b, 1), σ ∈ Γ, b ∈ B, x, y ∈ D,
ii) p
∗
ϕ = f
Γ
Cross
Γ
Γ
Grstr
Φ :
Γ
Cross →
Γ
Grstr,

(B → D) → P
B→D
(f
1
, f
0
, ϕ) → (F,

F )
F (x) = f
0
(x), F (b, 1) = (f
1
(b), 1),
F (x
(0,σ)
→ σx) = (ϕ(px, σ), σ),

F
x,y
= (ϕ(px, py), 1),
x, y ∈ D, b ∈ B, σ ∈ Γ
Γ
Γ
Γ B
d
→ D Γ Q
B Q Γ B
d
−→ D

Γ
E 0
//
B
j
//
E
p
//
ε

Q
//
1,
B
d
//
D
(B, E, j, θ
0
) Γ θ
0
(id, ε) Γ
Γ ψ : Q → Cokerd
Ext
Γ
B→D
(Q, B, ψ)
B Q Γ B → D
ψ : Q → d

Dis
Γ
Q Γ Γ
(0, Q, 0, 0) Γ
Dis
Γ
Q → P
B→D
B
d
→ D Γ ψ : Q → Coker d
Γ Γ (F,

F ) : Dis
Γ
Q → P
B→D
F (1) = 1 Γ (ψ, 0) : (Q, 0) → (Coker d, Ker d)
E
F
B Q Γ B → D
ψ
Γ
Ω : Hom
(ψ,0)
[Dis
Γ
Q, P
B→D
] → Ext

Γ
B→D
(Q, B, ψ).
Γ B, Q
Hom
Γ
[Dis
Γ
Q, Hol
Γ
B] → Ext
Γ
(Q, B).
P
B→D
S
P
= (Cokerd, Kerd, h),
h ∈ Z
3
Γ
(Cokerd, Kerd), Γ ψ : Q → Cokerd
ψ
∗
h ∈ Z
3
Γ
(Q, Kerd).
Γ (B, D, d, θ) Γ ψ : Q → Cokerd
ψ

∗
h H
3
Γ
(Q, Kerd)
B Q Γ B → D ψ
ψ
∗
h
H
2
Γ
(Q, Kerd)
H
3
Shu
(R, M)
R Z
H
2
MacL
(R, M)
H
3
MacL
(R, M)
(B, D, d) D
K B D d : B → D
D
d(b)b


= bd(b

), b, b

∈ B.
(k
1
, k
0
) : (B, D, d) → (B

, D

, d

)
k
1
: B → B

k
0
: D → D

k
1
k
0
K x ∈ D, b ∈ B

k
0
d = d

k
1
,
k
1
(xb) = k
0
(x)k
1
(b), k
1
(bx) = k
1
(b)k
0
(x).
K Z
(B, D, d)
E M = (B, D, d, θ) d : B →
D, θ : D → M
B
x ∈ D, b ∈ B
θ ◦ d = µ,
d(θ
x
b) = x.d(b), d(bθ

x
) = d(b).x.
(B, D, d, θ) θ
θ(D)
(f
1
, f
0
) : (B, D, d, θ) → (B

, D

, d

, θ

)
f
1
: B → B

f
0
: D → D

f
0
d = d

f

1
,
f
1
(θ
x
b) = θ

f
0
(x)
f
1
(b), f
1
(bθ
x
) = f
1
(b)θ

f
0
(x)
.
B → D
A
B→D
(f
1

, f
0
) : (B, D, d, θ) → (B

, D

, d

, θ

)
i) F : A
B→D
→ A
B

→D

F (x) = f
0
(x), F (b) = f
1
(b), x ∈ Ob A, b ∈ Mor A.
ii)
˘
F
x,y
: F (x + y) → F x + F y

F

x,y
: F (xy) → F xF y
F
˘
F

F Ker d

x, y ∈ D
θ

F x
(

F ) = (

F )θ

F y
=

F ,
θ

F x
(
˘
F ) = (
˘
F )θ


F y
=
˘
F +

F .
F (f
1
, f
0
)
(F,
˘
F ,

F ) F (0) = 0, F(1) = 1
˘
F ,

F
(F,
˘
F ,

F ) : A
B→D
→ A
B


→D

(f
1
, f
0
) : (B → D) → (B

→ D

)
f
1
(b) = F (b), f
0
(x) = F (x), b ∈ B, x ∈ D
(F,
˘
F ,

F ), (F

,
˘
F

,

F


) : A
B→D
→
A
B

→D

(F,
˘
F ,

F ), (F

,
˘
F

,

F

)
F = F

F, F

: A
B→D
→ A

B

→D

Annstr HoAnnstr Annstr
Hom
HoAnnstr
(A, A

) =
Hom
Annstr
(A, A

)
.
ESyst
Φ : ESyst → HoAnnstr,
(B → D) → A
B→D
(f
1
, f
0
) → [F ]
F (x) = f
0
(x), F (b) = f
1
(b) x ∈ ObA, b ∈ MorA

(B, D, d, θ) B
Q B → D
0
//
B
j
//
E
p
//
ε

Q
//
0,
B
d
//
D
(B, E, j, θ
0
) θ
0
(id, ε)
ψ : Q → Coker d
Ext
B→D
(Q, B, ψ) B Q
B → D ψ
Dis Q → A

B→D
(B, D, d, θ) ψ : Q → Coker d
(F,
˘
F ,

F ) : Dis Q → A (ψ, 0)
E
F
B Q B → D
ψ : Q → Coker d
Ω : Hom
Ann
(ψ,0)
[DisQ, A] → Ext
B→D
(Q, B, ψ).
A
B→D
S
A
=
(Cokerd, Kerd, k), k ∈ H
3
Shu
(Cokerd, Kerd) ψ : Q →
Cokerd ψ
∗
k ∈ H
3

Shu
(Q, Kerd).
(B, D, d, θ) ψ : Q →
Cokerd ψ
∗
k H
3
Shu
(Q, Kerd)
B Q B → D ψ
ψ
∗
k
Ext
B→D
(Q, B, ψ) ↔ H
2
Shu
(Q, Kerd).
Γ
Γ
Γ
Γ (M, N)