Homologie de Mac Lane, homologie des foncteurs et quelques

K
(C,⊗, k)A
CηA:k→A
µA:A⊗A→A
B(A)Cn A⊗n+2
A⊗n+2 =A⊗i⊗A⊗2⊗A⊗n−iid⊗µA⊗id −−−−−−−→ A⊗i⊗A⊗A⊗n−i=A⊗n+1
A⊗n+1 =A⊗i⊗A⊗A⊗n−iid⊗ηA⊗id −−−−−−→ A⊗i⊗A⊗2⊗A⊗n−i=A⊗n+2.
M A C C
A⊗M→M M ⊗A→M
B(A;M)n A⊗n⊗M
B(B, A, C) := B(A;BC)
B A C A B C
B⊗C B
CB(A) = B(A, A, A)

A
CA⊗ −
Ck
⊗k A k
M A k B(A;M)
HH∗(A;M)A M
B(A;M)
nB(A;M)n
B(A)B(A, X, Y )X Y A
B(A;M)'
B(A)⊗Aop⊗AM HH∗(A;M)
HomAop⊗A(B(A), M )
k HH∗(A;M)'
TorAop⊗A
∗(A, M)HH∗(A;M)'Ext∗ Aop⊗A(A, M )
k
k
Ck
k
A k M A
kB(A, M)
k
k
B(A;M)
A A−Mod AP(A)
A
Ae:= Aop ⊗A Ae−Mod
AF(A)P(A)→A−Mod
t:Ae−Mod → F(A)M7→ − ⊗
AM
h:Ae−Mod →Fct(P(A)op, A −Mod)M7→ HomA(−, M)
I : P(A)→A−Mod tA
C•:P(A)→Ch+(Ab)

n∈NCn:P(A)→Ab
C0Z[−]¯ Z[−]
C0IH0(C•)' −→ I
n∈NHn(C•) : P(A)→Ab
n > 0F:P(A)op →Ab F⊗
P(A)
Cn= 0 Hom (Cn, A)=0 A:P(A)→
Ab
C•(A)
MP(A)C•(M)
C•(A)
A'H0(C•(A)) C•(A)
B(A, C•(A), A)
A
H∗B(A, C•(A), A)⊗AeM'TorP(A)
∗(hM,I)
H∗HomAe(B(A, C•(A), A), M)'Ext∗
F(A)(I, tM)
A M
VP(A)
B(A, C•(A), C•(V)) →B(A, A, V )→V
V=A V =A⊕i
Hn(C•)
n∈NBn(A, C•(A), C•(−))
F(A)Ci
A⊗Ci1(A)⊗ · · · ⊗ Cir(A)⊗Cj
B∗(A, C•(A), C•(−))
IF(A)
hM
hM⊗
P(A)Bn(A, C•(A), C•(−)) 'hM⊗ P(A)A⊗Cn(A)⊗C0'(A⊗Cn(A)⊗A)⊗
AeM
C0=Z[−]
Q
Q•:Ab →Ch+(Ab)
n∈NQnV7→
Z[Vr]r= 2n
Q0=¯ Z[−]
H0(Q•)'Id
n > 0Qn
n∈NHn(Q•) : Ab →Ab

Qi(A)⊗Qj(B)→Qi+j(A⊗B)
Z→Q∗(Z)
A Q∗(A)
Q∗(A)⊗Q∗(A)→Q∗(A⊗A)→Q∗(A)
A
Z→Q0(Z)→Q∗(Z)→Q∗(A)
A
M A Q∗(M)
Q∗(A)
Q•
A M A
A M
HML∗(A;M) := H∗B(A, Q•(A), A)⊗AeM
A M
HML∗(A;M) := H∗HomAe(B(A, Q•(A), A), M).
HML∗(A;M)'TorP(A)
∗(hM,I)
HML∗(A;M)'Ext∗
F(A)(I, tM)
A M
F(A)
Q•(A)→A
HML∗(A;M)→HH∗(A;M)
HH∗(A;M)→HML∗(A;M).
AQQ•(A)→A
A=M=Z/p p
Z/p

HH2HML2(A;M)
A M
f:BA
A B
A2= 0 M HH2(A;M)
f
HH2(A;M)→HML2(A;M)
Q A n ∈N
Cn:= {0,1}nn Q0
n(A) :=
Z[ACn]CnA
ri, si:Cn−1→Cn1≤i≤n ri(x)
si(x)x0 1 i Q0(A)
δn([t]) =
n
X
i=1
(−1)i[Pi(t)] −[Ri(t)] −[Si(t)]
t∈ACnRi(t) = t◦riSi(t) = t◦siPi(t) = Ri(t) + Si(t)
δnδn+1 = 0 RjRi=
RiRj−1i < j Q0
Q∗(A)Q0
∗(A)
Q0(A)Q(A)
Qn(A)Q0
n(A)
[t]t∈ACnt= 0 n= 0
i1≤i≤n n > 0t(a)=0
a ai= 0 a∈Cnai= 1 i
[t]t∈ACn1≤i≤n−1t(a)=0 a∈Cn
ai6=ai+1
δ
Q0Q Q
Q
A B t ∈ACiu∈BCj[t]
[u] [t.u]t.u t ⊗u
ACi⊗BCj' −→ (A⊗B)Ci×Cj' −→ (A⊗B)Ci+j.
Q0
i(A)⊗Q0 j(B)→Q0 i+j(A⊗B)
Q0Q
Q[1] Q0
0(Z)
Q0(Z)