Comparaison des modules d`extensions dans des catégories de

∗
p
Ext∗(Id, F )
Fp
pP
pExt
Fp
Ext P
Ext Id
pF
Fp
FpId F
SnΛnΓn
F
U
Ext∗
F(Id, Sn)
∗

P
Ext∗
F(Id, Sn)
Id
Ext∗(Id, F )
Ext∗
F(Id,Id)
P
F
P
F P

pE
FpEf
F EfE
Id EfE F n
SnV∈ EfV⊗n
P
(V, W ) Hompol(V, W )
S∗(V∗)⊗W
V W
f V W d Sd(V∗)⊗W
V W
V W
P
EfEfV7→ P(V)
(V, W )Ef
PV,W Hom(V, W ) Hom(P(V), P (W))
PV,V (idV) = idP(V)
PV,W
d
PV,W
d
P
PdP
d
U:P −→ F .
Id
PTw
Id UTw
p
p
p
F F (1) =F◦Tw

F(n)=F(n−1) ◦Tw n
F F (n)n>0
F U
P F
Si1⊗ · · · ⊗ Sisi1+· · · +is=d
Pd
P
UF
FPdS F
PF
d S F
FU S P
Ext∗(Id, F )F
PFP P
U F
F
F G F P
F(0) = 0 = G(0) A A = Id
FA= Id(n)P
Ext∗(A, G ⊗F) = 0.
F P
Ext∗
F(Id,Id) e`, ` > 0
2p`−1

ep
`= 0 Ext∗
F(Id, Sn)n
pExt∗
F(Id, Sph)
Ext∗
F(Id,Id) e1, . . . , eh
Ext∗
F(Id, Sph)
Ext∗
P(Id(n),Id(n))
e(n−`)
`Ext2p`−1
P(Id(n),Id(n)) 0 < ` 6n
³e(n−`)
`´p
= 0, h
06h6nExt∗
P(Id(n), Sph(n−h))
Ext∗
P(Id(n),Id(n))e(n−1)
1, . . . , e(n−h)
h
Ext∗
P(Id(n), Sph(n−h))
F P S
F
d
2d−2
F P
FPn
n p Ext∗
F(Id, F ) = 0
n=ph
Ext∗
P(Id(h+k), F (k))∼
=Ext∗
P(Id(h), F )⊗FpExt∗ P(Id(h+k), Sph(k)),
Ext∗
F(Id, F )∼
=Ext∗
P(Id(h), F )⊗FpExt∗ F(Id, Sph).