Le déterminant sur un anneau

K
K= (A)A
M∈n(A)n(M)∈A
O M
Z[Xi,j ]1≤i,j≤n
Q(Xi,j )1≤i,j≤nn
Detn=n((Xi,j )1≤i,j≤n)∈Z[Xi,j ]1≤i,j≤n
A M ∈n(A)
ϕA,M :Z[Xi,j ]1≤i,j≤n−→ A
ϕA,M (Xi,j ) = Mi,j 1≤i, j ≤n
M
n(M) = ϕA,M (Detn)
f:A−→ B
f(n(M)) = n(f(M))
f◦ϕA,M =ϕB,f(M)
n(MN) = n(M)n(N)
M, N ∈n(A)
O
ϕA,M (Xi,j )1≤i,j≤n
ϕA,M,N :Z[Xi,j , Yi,j ]1≤i,j≤n−→ A
ϕA,M (Xi,j ) = Mi,j ϕA,N (Yi,j ) = Ni,j
(Xi,j )1≤i,j≤n(Yi,j )1≤i,j≤n
M

A A n ≥1
A,n :n(A)−→ A
M= (Mi,j )1≤i,j≤n∈n(A)
A,n(M) = X
σ∈Sn
(σ)Mσ(1),1· · · Mσ(n),n
f:A−→ B
f(A,n(M) = B,n(f(M))
f(M) = (f(Mi,j ))1≤i,j≤n∈n(B)
M∈n(A)
A,n(M) = M1,1· · · Mn,n
OM
i>j⇒Mi,j = 0
σ∈Snσ(i)≤i1≤i≤n σ =
A,n(M) = P
σ∈Sn
(σ)Mσ(1),1· · · Mσ(n),n σ6=j1≤j≤n
σ(j)> j Mσ(j),j = 0 (σ)Mσ(1),1· · · Mσ(n),n = 0
A,n(M) = M1,1· · · Mn,n M
M, P, Q ∈n(A)s6=j Cs(M) = Cs(P) = Cs(Q)
Cj(M) = a Cj(P) + b Cj(Q)
A,n(M) = aA,n(P) + bA,n(Q)
O
A,n(M) = X
σ∈Sn
(σ)Mσ(1),1· · · Mσ(j),j · · · Mσ(n),n
=X
σ∈Sn
(σ)Mσ(1),1· · · (a Pσ(j),j +b Qσ(j),j )· · · Mσ(n),n
=aX
σ∈Sn
(σ)Mσ(1),1· · · Pσ(j),j · · · Mσ(n),n +bX
σ∈Sn
(σ)Mσ(1),1· · · Qσ(j),j · · · Mσ(n),n
=aA,n(P) + bA,n(Q)
M
M∈n(A)
A,n(M)=0

OCi(M) = Cj(M) 1 ≤i < j ≤n τ = (i, j)∈Snσ∈Sn
(σ)Mσ(1),1· · · Mσ(i),i · · · Mσ(j),j · · · Mσ(n),n+ (στ)Mστ(1),1· · · Mστ(i),i · · · Mστ(j),j · · · Mστ (n),n = 0
Sn
Sn={σ∈Sn/σ(i)< σ(j)}∪{σ∈Sn/σ(i)> σ(j)}
{σ∈Sn/σ(i)< σ(j)} −→ {σ∈Sn/σ(i)> σ(j)}
σ−→ στ
A,n(M)=0 M
Y1,· · · Yn∈n,1(A)< Y1|···|Yn>∈n(A)
Cj(< Y1|···|Yn>)) = Yj1≤j≤n
(Y1,· · · Yn)−→ A,n(< Y1|···|Yn>)
σ∈Sn
A,n(< Yσ(1)|···|Yσ(n)>) = (σ)A,n(< Y1|···|Yn>)
M, N ∈n(A)
A,n(MN) = A,n(M)A,n(N)
O
A,n(MN) = X
σ∈Sn
(σ)(MN)σ(1),1· · · (MN)σ(n),n
=X
σ∈Sn
(σ)(X
k1
Mσ(1),k1Nk1,1)· · · (X
kn
Mσ(n),knNkn,n)
=X
k1,··· ,kn
Nk1,1· · · Nkn,n(X
σ∈Sn
(σ)Mσ(1),k1· · · Mσ(n),kn)
=X
k1,··· ,kn
Nk1,1· · · Nkn,n A,n(M(k1,··· ,kn))
M(k1,··· ,kn)= (Mi,kj)1≤i,j≤n
k1,· · · , kn
A,n(M(k1,··· ,kn))=0
κ:i−→ kiSn
A,n(M(k1,··· ,kn)) = (κ)A,n(M)
A,n(MN) = A,n(M)X
κ∈Sn
(κ)Nκ(1),1· · · Nκ(n),n
=A,n(M)A,n(N)
M
M∈n(A) 1 ≤i, j ≤n M (i,j)∈n−1(A)
i j M

M∈n(A)i1≤i≤n Mk,1= 0
k6= 1
A,n(M)=(−1)i+1Mi,1A,n−1M(i,1)
OS={σ∈Sn/σ(1) = i}
A,n(M) = X
σ∈Sn
(σ)Mσ(1),1· · · Mσ(n),n
=Mi,1X
σ∈S
(σ)Mσ(2),2· · · Mσ(n),n
γ= (1,· · · , i)∈SnS0={σ0∈Sn/σ0(1) = 1} ' Sn−1
S0−→ S
σ0−→ σ=γ−1◦σ0
A,n(M) = Mi,1X
σ0∈S0
(γ−1σ0)Mγ−1σ0(2),2· · · Mγ−1σ0(n),n
= (−1)i+1Mi,1X
σ0∈S0
(σ0)Mγ−1σ0(2),2· · · Mγ−1σ0(n),n
= (−1)i+1Mi,1A,n−1M(i,1)
σ0∈S0{2,· · · , n}
M(i,1) σ=γ−1◦σ0
{1,· · · , i −1, i + 1,· · · , n}M(i,1) M
M∈n(A)
A,n(M) =
n
P
i=1
(−1)i+jMi,j A,n−1(M(i,j)) 1 ≤j≤n
A,n(M) =
n
P
j=1
(−1)i+jMi,j A,n−1(M(i,j)) 1 ≤i≤n
O1≤j≤n1≤i≤nc Mij
M e?
i= (δr,i)1≤r≤n
A,n(M) =
n
X
i=1
Mi,j A,n(c Mi)
c
M?
ic
Miσ= (1,· · · , i)∈Sn
(c M?
i)j,1= 1 ((c
M?
i)[j])(j,1) =M(i,j)
A,n(c
M?
j) = (−1)j+1 A,n−1(M(i,j))
= (−1)i+1 A,n(c Mi)
A,n(M) =
n
P
i=1
(−1)i+jMi,j A,n−1(M(i,j))
MM
M∈n(A)f M= ((−1)i+jA,n−1(M(j,i)))1≤i,j≤n

M∈n(A)
M. f M=f M.M =A,n(M)
ON=M. f M1≤i, j ≤n
Ni,j =
n
X
k=1
(−1)k+jMi,k A,n−1(M(j,k))
i=j Ni,i =
n
P
k=1
(−1)k+iMi,k A,n−1(M(i,k)) = A,n(M)
i6=j M
Lk(M) = (Lk(M)k6=j
Li(M)k=j
M j
A,n(M) =
n
X
k=1
(−1)j+kMj,k A,n−1(M(j,k))
=
n
X
k=1
(−1)j+kMi,k A,n−1(M(j,k)) = Ni,j = 0
M
M
M∈n(A)M∈n(A)A,n
A