Let R R R ring
and M M M R-module
.
An injective resolution of M M M cochain complex
0 → M → η I 0 → d 0 I 1 → d 1 I 2 → d 2 ⋯
0 \to M \xrightarrow{\eta} I^0 \xrightarrow{d^0} I^1 \xrightarrow{d^1} I^2 \xrightarrow{d^2} \cdots
0 → M η I 0 d 0 I 1 d 1 I 2 d 2 ⋯ such that:
Each I j I^j I j injective R-module
. The sequence is exact (i.e. an exact complex
). Existence in module categories is guaranteed by injective resolutions exist
.
What resolutions are for If F F F left exact
functor, its right derived functors are computed by applying F F F cohomology
; see derived functor
.
In particular, for any R R R N N N
E x t R n ( N , M ) ≅ H n ( Hom R ( N , I ∙ ) ) ,
\mathrm{Ext}_R^n(N,M)
\cong H^n\bigl(\operatorname{Hom}_R(N, I^\bullet)\bigr),
Ext R n ( N , M ) ≅ H n ( Hom R ( N , I ∙ ) ) , where Hom R ( N , I ∙ ) \operatorname{Hom}_R(N,I^\bullet) Hom R ( N , I ∙ ) Ext
).
Examples Example 1: Over a field, injective resolutions are trivial If R = k R=k R = k k k k k k k V V V
0 → V → = V → 0 → 0 → ⋯ ,
0\to V \xrightarrow{=} V \to 0 \to 0 \to \cdots,
0 → V = V → 0 → 0 → ⋯ , and therefore E x t k n ( W , V ) = 0 \mathrm{Ext}_k^n(W,V)=0 Ext k n ( W , V ) = 0 n > 0 n>0 n > 0 Ext
).
Example 2: An injective resolution of Z \mathbb Z Z In the category of abelian groups (R = Z R=\mathbb Z R = Z Q \mathbb Q Q Q / Z \mathbb Q/\mathbb Z Q / Z
0 → Z → Q → Q / Z → 0
0\to \mathbb Z \to \mathbb Q \to \mathbb Q/\mathbb Z \to 0
0 → Z → Q → Q / Z → 0 is exact, hence yields an injective resolution of Z \mathbb Z Z 1 1 1
Using it, one computes
E x t Z 1 ( Z / n Z , Z ) ≅ H 1 ( Hom ( Z / n , [ 0 → Q → Q / Z → 0 ] ) ) .
\mathrm{Ext}^1_{\mathbb Z}(\mathbb Z/n\mathbb Z,\mathbb Z)
\cong H^1\bigl(\operatorname{Hom}(\mathbb Z/n,\, [0\to \mathbb Q\to \mathbb Q/\mathbb Z\to 0])\bigr).
Ext Z 1 ( Z / n Z , Z ) ≅ H 1 ( Hom ( Z / n , [ 0 → Q → Q / Z → 0 ]) ) . Since Hom ( Z / n , Q ) = 0 \operatorname{Hom}(\mathbb Z/n,\mathbb Q)=0 Hom ( Z / n , Q ) = 0 Hom ( Z / n , Q / Z ) ≅ ( Q / Z ) [ n ] ≅ Z / n Z \operatorname{Hom}(\mathbb Z/n,\mathbb Q/\mathbb Z)\cong (\mathbb Q/\mathbb Z)[n]\cong \mathbb Z/n\mathbb Z Hom ( Z / n , Q / Z ) ≅ ( Q / Z ) [ n ] ≅ Z / n Z
E x t Z 1 ( Z / n , Z ) ≅ Z / n Z .
\mathrm{Ext}^1_{\mathbb Z}(\mathbb Z/n,\mathbb Z)\cong \mathbb Z/n\mathbb Z.
Ext Z 1 ( Z / n , Z ) ≅ Z / n Z . (See Ext
.)
Example 3: An injective resolution of Z / n Z \mathbb Z/n\mathbb Z Z / n Z E x t 1 \mathrm{Ext}^1 Ext 1 Embed Z / n Z \mathbb Z/n\mathbb Z Z / n Z Q / Z \mathbb Q/\mathbb Z Q / Z n n n ( Q / Z ) [ n ] (\mathbb Q/\mathbb Z)[n] ( Q / Z ) [ n ] n n n Q / Z \mathbb Q/\mathbb Z Q / Z ( Q / Z ) [ n ] (\mathbb Q/\mathbb Z)[n] ( Q / Z ) [ n ]
0 → Z / n Z → Q / Z → × n Q / Z → 0
0\to \mathbb Z/n\mathbb Z \to \mathbb Q/\mathbb Z \xrightarrow{\times n} \mathbb Q/\mathbb Z \to 0
0 → Z / n Z → Q / Z × n Q / Z → 0 is exact, with injective terms, hence an injective resolution of Z / n Z \mathbb Z/n\mathbb Z Z / n Z
Applying Hom ( Z / m , − ) \operatorname{Hom}(\mathbb Z/m,-) Hom ( Z / m , − )
0 → Hom ( Z / m , Q / Z ) → × n Hom ( Z / m , Q / Z ) → 0 ,
0\to \operatorname{Hom}(\mathbb Z/m,\mathbb Q/\mathbb Z)\xrightarrow{\times n}
\operatorname{Hom}(\mathbb Z/m,\mathbb Q/\mathbb Z)\to 0,
0 → Hom ( Z / m , Q / Z ) × n Hom ( Z / m , Q / Z ) → 0 , and Hom ( Z / m , Q / Z ) ≅ ( Q / Z ) [ m ] ≅ Z / m Z \operatorname{Hom}(\mathbb Z/m,\mathbb Q/\mathbb Z)\cong (\mathbb Q/\mathbb Z)[m]\cong \mathbb Z/m\mathbb Z Hom ( Z / m , Q / Z ) ≅ ( Q / Z ) [ m ] ≅ Z / m Z
E x t Z 1 ( Z / m , Z / n ) ≅ coker ( × n : Z / m → Z / m ) ≅ Z / gcd ( m , n ) Z .
\mathrm{Ext}^1_{\mathbb Z}(\mathbb Z/m,\mathbb Z/n)
\cong \operatorname{coker}(\times n:\mathbb Z/m\to\mathbb Z/m)
\cong \mathbb Z/\gcd(m,n)\mathbb Z.
Ext Z 1 ( Z / m , Z / n ) ≅ coker ( × n : Z / m → Z / m ) ≅ Z / g cd( m , n ) Z . (See Ext
.)