TD7 : Localisation

Z[X]/(2X2+ 3X+ 2)
C
Z[(Xn)n∈N]
A K
A[X]
I A[X]
I∩A A
I∩A= 0
J K[X]I I =J∩A[X]
I
I∩A k =A/(I∩A)
I I ∩A I
I∩A P ∈A[X]k[X]
A[X]
AmA
A[X]m
C[X, Y ]
Z[X]
αC
Z
Z[α]

A I S A
S S A =A/I S−1I S−1A I
S−1A S−1A/S−1I
S−1A
A=ZS=Z\ {0}
A=ZS={−1,1}
A=ZS={10k|k≥0}
A=Z×ZS={(0, y)|y6= 0}∪{(1,1)}
A=Z×ZS={(x, y)|x6= 0}
A=Z×ZS={(x, y)|x6= 0, y 6= 0}
A=C[X]/(X5)S={Xk|k≥0}
A=C[X, Y ]/(X5Y)S={Xk|k≥0}
A=C[X, Y ]/(X5Y)S={P∈A|P6∈ (X)}
A=C[X, Y ]/(X5Y)S={P∈A|P6∈ (Y)}
A=C[X, Y ]/(XY )S={(X+Y)k|k≥0}
A S A
S−1A
A
S−1A A
S
A S−1A
A S 0
S−1A
A S 0
S−1A
A K B
K A
S A B =S−1A
C(X, Y )C[X, Y ]
C[X, Y ]C[X, Y ]

C[0,2] Rm
S=C \ mS−1C
A S A
S−1A= lim
−→
f∈S
A[1/f].
A
A Am
m∈(A)
A K
A=∩m∈AAm
A K
A A
A Amm∈
(A)
A Am
m∈(A)
A=C[X, Y ]/(XY )B=C[X, Y, Z]/(X2(Y+Z2), XY )
Q(A)Q(B)
A A
A A
A
A
A
m
n > 0mn= 0
A
A

S
A M A S−1A S−1M
S−1A
A a, b ∈A(a, b) = A M A
0→M→M1
a⊕M1 b→M1 ab.
(ai)i∈IA
A M N P A
S A
A f :M→N g :N→P M →N→P
A
S−1A
S−1M→S−1N→S−1P.
M→Qm∈AMm
M= 0 Mm= 0 m∈(A)
A f :M→N g :N→
P0→M→N→P→0
0→Mm→Nm→Pm→0
m∈(A)
M Mm
m∈(A)
M
(Mp)p∈AAp