La structure de la catégorie des algèbres de Hopf en caractéristique p

p
H(k)
H(k)
H1(k)
H(k)
kH(k)H
k H
N
H0
H(k)

k
f:H→KH(k)f
k K
f¯ H:= L
n>0
HnH
K
f
0→H∆ −→ H⊗Hf⊗H −−−→ K⊗H¯ K⊗H
H
H(k)
H(k)k
k
H(k)
k
H(k)H
Hnk n ∈N
P:H(k)→ Egr(k)k
H
H∆−1⊗Id −Id ⊗1 : H→H⊗H
Q:H(k)→ Egr(k)Q(H) =
¯
H/ ¯ H2
P Q
P(H∗)'Q(H)∗Q(H∗)'P(H)∗H P (H)
Q(H)
H
f:H→KH(k)
f
P(f) : P(H)→P(K)Egr(k)
fEgr(k)
f
Q(f) : Q(H)→Q(K)Egr(k)

fEgr(k)
P:H(k)→ Egr (k)Q
P Q
i > 0HH(k)
H(i)i
H(i)n:= Hn/i i|n0
H k p > 0
H(p)→ϕHH(k)ϕH k
H
ϕ:k→k
ϕH H k
k
αH βH H
λ:P→Q
HH(k)
P(H)→¯ HQ(H).
k λ
k p > 0
0→P(αH)→P(H)λH −−→ Q(H)→Q(βH)→0
αH →H
HβH
k P ('Q) : H(k)→
Egr(k)H(k)
H(k)
HH(k)
HH(k)
H
Q(H)k Qn(H)k

H(k)
H(k)
H
Q(H)
k
H(k)H
H(k)K Q(K)
V
Q(H)V
x1, . . . , xrH
Hn→Hi1⊗ · · · ⊗ Hir
i1, . . . , ir>0P
t
it=n
n
H
K H K →H∆ −→
H⊗H K ⊗K∆
L K
H Q(L)K
H(k)
H(k)
H(k)
H(k)
H+HH
Hn= 0 nH−H Q(H)n= 0
n
S(n)n∈N∗
xnn
n n
2S(n)H+
nH−2n
k p ¯ S(n)S(n)
xp
nS(n)
¯
S(n)H+n¯ S(n) = S(n)n
p6= 2
HHH
x n > 0

x H
n k 2H
x S(n)
n k H
x S(n)
k p > 0p n H
k[x]k[x]/(xpr)r > 0
H Hi=
0 0 < i < n H
H x
x
I k[x]
k[x]k[x]/I (xpr)k
p p
Ci
n0< i < n n
p
k S(n)n∈N
H(k)
k p > 0
H(k)
S(n)
¯
S(n)
¯
S(n)n∈N
H(k)
H(k) 1
car(k) = p > 0k[x]/(xpr)
x n n p ¯
S(n.pi)i∈ {0,1, . . . , r−1}
k[x]
(xpr)r∈N
1H
¯
S(npr)r∈N
A
T A
A
T
IAExt1
A(T, I)=0 T∈ T