Cover Page The handle http://hdl.handle.net/21633 holds

Cover Page
The handle http://hdl.handle.net/1887/21633 holds various files of this Leiden University
dissertation.
Author: Gioia, Alberto
Title: On the Galois closure of commutative algebras
Issue Date: 2013-09-04

R f n
R[Z]A R R[Z]/(f)i= 0, . . . , n
Fi(A)fiFi(A)[Z]
F0(A) = R f0=f Fi(A)fi
Fi+1(A) = Fi(A)[xi+1]/(fi(xi+1)), fi+1(Z) = fi(Z)
Z−xi+1 ∈Fi+1(A)[Z].
i= 0, . . . , n fi
R f
SnFn(A)
A/R Fn(A)
Sn
R
α:R→K R π =
π(R, α)R α
R n
n
π π
X π n i = 0, . . . , n
Inj({1, . . . , i}, X){1, . . . , i}X
π
π X R A n
π X R Gi(A)
n(n−1) ···(n−i+ 1) πInj({1, . . . , i}, X)
R A R[Z]/(f)f
i= 0, . . . , n Gi(A)Fi(A)
R→A
n SnGn(A)

A Gn(A)
Gn(A)
i= 0, . . . , n Fi(A)Gi(A)
R A R
n
A/R
A R n R
A(n)i= 1, . . . , n R
αi:A→A(n)A a
An
Y
i=1 Z−αi(a)∈A(n)[Z]
Pa(Z)a A(n)[Z]
R[Z]→A(n)[Z]A(n),(αi)i
A(n)
1A(n)
3 4
1R
R A n A(i)
i= 0, . . . , n Fi(A)Gi(A)
A(n)
4
A(i)
A
R A(i)
6i= 1, . . . , m AiR
niA AiR
n=PniA
A(n)∼
= m
O
i=1
A(ni)
i!n!
n1!··· nm!
.

A(n)A(i)
K f
n K[Z]L K[Z]/(f)M
L/K M K
Sni= 0, . . . , n Fi(L)
K→Fi(L)M
MSn−i
A(n)Sn
A(n)3
A(i)(A(n))Sn−i
RF2(X2)A
F2(X)R A(2)
A S2R→(A(2))S2
R→A R →R0
Spec A⊗RR0→Spec R0
A R
n i {0, . . . , n}A(i)→
A(n)Sn−i
2
X→S G X/G
S
X→S G
X→S
X/G X/G →S
X→S K
S G X(K)→S(K)

4AnA(n)
K
2f
n K[Z]M K[Z]/(f)
M/K Sn
∆ff M
K→K[p∆f]M MAnK→K[p∆f]
M/K f
RZ[1/2] A R
nVnA R 1
VnA⊗VnA→R R
R⊕VnA R ∆1/2(A/R)
A R
A=R[Z]/(f)R
A Sn∆1/2(A/R)R[p∆f]
RZ[1/2] A R
n
R λ: ∆1/2(A/R)→A(n)∆1/2(A/R)→
(A(n))Anλ
λ
R
n R