Déterminant des matrices à coefficients dans un anneau

Z/nZ
A1
M n A f :An→An
A
f f det(f)A
fdet(f)A
A=Zf|det(f)|
A=k[X]f k deg(det(f))
coker(f) =
An/f(An)
fcoker(f) = 0
e1, . . . , enAn
f i if(i) = eig(ei) = i
i g :An→Anf◦g= Id
eidet(f) det(g)=1
det(f)
Mt˜ M=t˜ MM = det(M) Id
f
d= det(f)d f(x)=0
Mx = 0 dx = 0
x d x = 0 f
d f u ∈A
ud = 0
µ M uµ = 0
1M f(ue1) = 0 ue16= 0 f
N M u det(N)6= 0
r r < n ud = 0

M
N r
i∈ {1, . . . , n}r M
i Pir+ 1
Pi=
m1,1. . . m1,r+1
mr,1. . . mr,r+1
mi,1. . . mi,r+1
.
i≤r Pii≥r+1
M r + 1 u
r u det(Pi)=0
uPr+1
j=1 (−1)jmi,j µj= 0 µj(r+ 1, j)
i M(ux)=0 x
(−µ1,...,(−1)r+1µr+1,0,...,0) uµr+1 =udet(N)6= 0 ux 6= 0 f
R, S ZD:= RMS
d1, . . . , dndi|di+1
i
coker(f)'Zn/D(Zn)'Z/d1Z× · · · × Z/dnZ
|coker(f)|=d1. . . dn=|det(f)|
R, S
k[X]D:= RMS
P1, . . . , PnPi|Pi+1 i
coker(f)'k[X]
(P1)× · · · × k[X] (Pn)
dimk(coker(f)) = deg(P1) + · · · + deg(Pn) = deg(P1. . . Pn) = deg(det(f)) 