1. Généralités

A
n Sln(A)
A M ∈Sln(A)
M
n
n= 1 n−1
M= (mi,j )1≤i,j≤n∈Sln(A)
M=
m1,10· · · · · · 0
0m−1
1,1
1 0
0
0· · · · · · 0 1
1 0 · · · · · · 0
0m2,2m1,1
m3,30
0
0· · · · · · 0mn,n
α0
0α1
α0
0α1∼α1
0α1∼α1
1 0∼0 1 1 0∼0 1 1 1∼1 1 0 1∼1 0 0 1

A
n M ∈Sln(A)
B1, ..., BpB0
1, ..., B0
q
B1...BpMB0
1...B0
q
A(2)
A(ii)
Sln(A)
n n = 1
n−1M= (mi,j )1≤i,j≤n∈Sln(A)
m1,1
1≤i≤n i mi,1
m1,1
im1,i
m1,1
M1
M1=
m1,10. . . 0
0
M0
0
M2=
1 0 . . . 0
0
M00
0
M3=
m1,10 0 . . . 0
0m−1
1,10. . . 0
0 0
Idn−2
0 0
M1=M2M3M00 ∈Sln−1(A)
M00
M2
M3
A ν M = (mi,j )1≤i,j≤n∈Sln(A)
1≤i0≤n
ν(mi0,1) = min{ν(mi,1, i ≤i≤n)}1≤i≤n i 6=i0

mi,1=mi0,1qi+riν(qi)< ν(mi0,1)
qii0
A
1
i16= 1 1
i1M0
1.3
M= (mi,j )1≤i,j≤n∈Sln(A)I=xA
A
1
i0mi0,1∈A×
m1,1
m1,1m−1
i0,1+ 1 i0
1.3
M= (mi,j )1≤i,j≤n∈Sln(A)I
A φ :A→A/I
φ:Sln(A)→Sln(A/I)
I A/I
M=Y
k≤p
Bik,jk(λk), k ≤p λk∈
k≤p,λ0
kφ(λ0 k) = λk

M0(m0
i,j )1≤i,j≤n=MY
k≤p
Bik,jk(−λ0
k)
φ(M0) = Id
i6=j m0
i,j ∈I m0 i,i ∈A×
m0
1,11.3
A I A A/I
A1A2A1×A2
s:A→A/I
˜s:Sln(A)→Sln(A/I)
Sln(A)Sln(A/I)
A A/I 
φ:A→A1×A2φ:Sln(A)→Sln(A1)×Sln(A2)
M∈Sln(A) (M1, M2) = φ(M)
A1A2M1M2
M1=Y
k≤n1
Bik,jk(λk)M2=Y
k≤n2
Bi0
k,j0 k(λ0
k)
M1=Y
k≤n1
Bik,jk(λk)Y
k≤n1
Bi0
k,j0 k(0) M2=Y
k≤n1
Bik,jk(0) Y
k≤n2
Bi0
k,j0 k(λ0
k)
M1=Y
k≤n1+n2
Bik,jk(λk)M2=Y
k≤n1+n2
Bik,jk(λ0
k)

M=φ−1(M1, M2) = φ−1
Y
k≤n1+n2
Bik,jk(λk),Y
k≤n1+n2
Bik,jk(λ0
k)
=Y
k≤n1+n2
φ−1(Bik,jk(λk), Bik,jk(λ0
k))
=Y
k≤n1+n2
(Bik,jk(φ−1(λk, λ0
k))
A
A
A
A A
Ar×
k
Y
i=1
A/niA n1|n2...|nk
A/niA
A
A A
Ar×
k
Y
i=1
A/niA n1|n2...|nk
A/niA
A/pαA p
Ar×Qk
i=1 A/niA