Zum maximalen Ideal
(
x
,
y
,
z
,
w
)
⊆
Q
[
x
,
y
,
z
,
w
]
/
(
x
3
+
y
3
+
z
3
+
w
3
)
{\displaystyle {}(x,y,z,w)\subseteq \mathbb {Q} [x,y,z,w]/(x^{3}+y^{3}+z^{3}+w^{3})}
H
2
{\displaystyle {}H^{2}}
9
,
375
{\displaystyle {}9,375}
q
∑
h
2
(
S
q
(
S
y
z
2
)
(
m
)
)
{\displaystyle {}\sum h^{2}(S^{q}(Syz_{2})(m))}
durch Rang
durch Rang
q
3
{\displaystyle {}q^{3}}
∑
h
2
(
F
e
(
S
y
z
2
)
(
m
)
)
{\displaystyle {}\sum h^{2}(F^{e}(Syz_{2})(m))}
durch
4
{\displaystyle {}4}
durch
q
3
{\displaystyle {}q^{3}}
1
42
10,5
10,5
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
2
801
80,1
10,0125
311
77,75
9,7187
3
5401
270,05
10,0018
4
22442
641,2
10,0187
2430
607,5
9,4921
5
70244
1254,3571
10,0348
4714
1178,5
9,428
6
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
7
12898
3224,5
9,4008
8
19316
4829
9,4316
9
10
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
11
49970
12492,5
9,3858
12
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
13
82454
20613,5
9,3825
14
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
15
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
16
17
18
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
19
20
−
{\displaystyle {}-}
−
{\displaystyle {}-}
−
{\displaystyle {}-}
In positiver Charakteristik kann man die "divided powers"
Γ
n
(
S
y
z
2
)
=
S
n
(
S
y
z
2
∨
)
∨
{\displaystyle {}\Gamma ^{n}(Syz_{2})=S^{n}(Syz_{2}^{\vee })^{\vee }}
S
q
(
F
2
)
→
S
q
−
1
(
F
2
)
⊗
F
1
,
e
ν
↦
∑
i
,
j
e
ν
−
1
i
⊗
M
e
i
,
{\displaystyle {}S^{q}(F_{2})\rightarrow S^{q-1}(F_{2})\otimes F_{1},e_{\nu }\mapsto \sum _{i,j}e_{\nu -1_{i}}\otimes Me_{i},}
M
{\displaystyle {}M}
F
2
→
F
1
{\displaystyle {}F_{2}\rightarrow F_{1}}
Damit kann man dann
∑
m
h
2
(
Γ
n
(
S
y
z
2
)
{\displaystyle {}\sum _{m}h^{2}(\Gamma ^{n}(Syz_{2})}
∑
m
h
2
(
S
n
(
S
y
z
2
∨
(
m
)
)
{\displaystyle {}\sum _{m}h^{2}(S^{n}(Syz_{2}^{\vee }(m))}
Man erhält ganz ähnliche Werte wie in Charakteristik Null, nämlich für
F
2
[
x
,
y
,
z
,
w
]
/
(
x
3
+
…
+
w
3
)
{\displaystyle {}\mathbb {F} _{2}[x,y,z,w]/(x^{3}+\ldots +w^{3})}
F
5
[
x
,
y
,
z
,
w
]
/
(
x
3
+
…
+
w
3
)
{\displaystyle {}\mathbb {F} _{5}[x,y,z,w]/(x^{3}+\ldots +w^{3})}
![{\displaystyle {}(x,y,z,w)\subseteq \mathbb {Q} [x,y,z,w]/(x^{3}+y^{3}+z^{3}+w^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47adfdfb1ee8441ecf17942b45e2015881cdadb8)
![{\displaystyle {}\mathbb {F} _{2}[x,y,z,w]/(x^{3}+\ldots +w^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc62f4381ab4e44fda208801036f841d0beb1939)
![{\displaystyle {}\mathbb {F} _{5}[x,y,z,w]/(x^{3}+\ldots +w^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbd2c21e3c026ce387892669425bf708ca20cdc)