Let R R R M , N M,N M , N R R R
N ↦ Hom R ( M , N ) N \mapsto \operatorname{Hom}_R(M,N) N ↦ Hom R ( M , N ) N N N left exact
,M ↦ M ⊗ R N M \mapsto M\otimes_R N M ↦ M ⊗ R N right exact
.Their failure to be exact is measured by derived functors, giving rise to Ext
and Tor
.
Definitions (via resolutions) Ext as a right derived functor Fix M M M injective resolution
0 → N → I 0 → I 1 → ⋯ 0\to N\to I^0\to I^1\to \cdots 0 → N → I 0 → I 1 → ⋯ Hom R ( M , − ) \operatorname{Hom}_R(M,-) Hom R ( M , − ) Hom R ( M , I ∙ ) \operatorname{Hom}_R(M,I^\bullet) Hom R ( M , I ∙ )
Ext R n ( M , N ) : = H n ( Hom R ( M , I ∙ ) ) .
\operatorname{Ext}^n_R(M,N) := H^n(\operatorname{Hom}_R(M,I^\bullet)).
Ext R n ( M , N ) := H n ( Hom R ( M , I ∙ )) . Equivalently (often more computationally), choose a projective resolution
⋯ → P 1 → P 0 → M → 0 \cdots \to P_1\to P_0\to M\to 0 ⋯ → P 1 → P 0 → M → 0
Ext R n ( M , N ) : = H n ( Hom R ( P ∙ , N ) ) .
\operatorname{Ext}^n_R(M,N) := H^n(\operatorname{Hom}_R(P_\bullet,N)).
Ext R n ( M , N ) := H n ( Hom R ( P ∙ , N )) . These agree and are well-defined up to canonical isomorphism because projective/injective resolutions exist (see projective resolutions exist
and injective resolutions exist
).
Tor as a left derived functor Fix N N N ⋯ → P 1 → P 0 → M → 0 \cdots \to P_1\to P_0\to M\to 0 ⋯ → P 1 → P 0 → M → 0 − ⊗ R N -\otimes_R N − ⊗ R N P ∙ ⊗ R N P_\bullet\otimes_R N P ∙ ⊗ R N
Tor n R ( M , N ) : = H n ( P ∙ ⊗ R N ) .
\operatorname{Tor}^R_n(M,N) := H_n(P_\bullet\otimes_R N).
Tor n R ( M , N ) := H n ( P ∙ ⊗ R N ) . Functorial consequences Examples Example 1 (Tor over Z \mathbb Z Z Tor 1 Z ( Z / n , Z / m ) \operatorname{Tor}_1^\mathbb Z(\mathbb Z/n,\mathbb Z/m) Tor 1 Z ( Z / n , Z / m ) Use the standard projective resolution
0 → Z → ⋅ n Z → Z / n → 0.
0\to \mathbb Z \xrightarrow{\cdot n} \mathbb Z \to \mathbb Z/n\to 0.
0 → Z ⋅ n Z → Z / n → 0. Tensor with Z / m \mathbb Z/m Z / m
0 → Z / m → ⋅ n Z / m → ( Z / n ) ⊗ ( Z / m ) → 0.
0\to \mathbb Z/m \xrightarrow{\cdot n} \mathbb Z/m \to (\mathbb Z/n)\otimes(\mathbb Z/m)\to 0.
0 → Z / m ⋅ n Z / m → ( Z / n ) ⊗ ( Z / m ) → 0. Then
Tor 1 Z ( Z / n , Z / m ) = ker ( ⋅ n : Z / m → Z / m ) ≅ Z / gcd ( n , m ) ,
\operatorname{Tor}_1^\mathbb Z(\mathbb Z/n,\mathbb Z/m)=\ker(\cdot n:\mathbb Z/m\to \mathbb Z/m)\cong \mathbb Z/\gcd(n,m),
Tor 1 Z ( Z / n , Z / m ) = ker ( ⋅ n : Z / m → Z / m ) ≅ Z / g cd( n , m ) , and Tor i = 0 \operatorname{Tor}_i=0 Tor i = 0 i ≥ 2 i\ge 2 i ≥ 2 1 1 1
Example 2 (Ext over Z \mathbb Z Z Ext Z 1 ( Z / n , Z / m ) \operatorname{Ext}^1_\mathbb Z(\mathbb Z/n,\mathbb Z/m) Ext Z 1 ( Z / n , Z / m ) Apply Hom Z ( − , Z / m ) \operatorname{Hom}_\mathbb Z(-,\mathbb Z/m) Hom Z ( − , Z / m )
0 → Z → ⋅ n Z → Z / n → 0
0\to \mathbb Z \xrightarrow{\cdot n} \mathbb Z \to \mathbb Z/n\to 0
0 → Z ⋅ n Z → Z / n → 0 to obtain
0 → Hom ( Z , Z / m ) → ⋅ n Hom ( Z , Z / m ) → Ext 1 ( Z / n , Z / m ) → 0.
0\to \operatorname{Hom}(\mathbb Z,\mathbb Z/m)\xrightarrow{\cdot n}\operatorname{Hom}(\mathbb Z,\mathbb Z/m)\to \operatorname{Ext}^1(\mathbb Z/n,\mathbb Z/m)\to 0.
0 → Hom ( Z , Z / m ) ⋅ n Hom ( Z , Z / m ) → Ext 1 ( Z / n , Z / m ) → 0. Since Hom ( Z , Z / m ) ≅ Z / m \operatorname{Hom}(\mathbb Z,\mathbb Z/m)\cong \mathbb Z/m Hom ( Z , Z / m ) ≅ Z / m
Ext Z 1 ( Z / n , Z / m ) ≅ ( Z / m ) / n ( Z / m ) ≅ Z / gcd ( n , m ) ,
\operatorname{Ext}^1_\mathbb Z(\mathbb Z/n,\mathbb Z/m)\cong (\mathbb Z/m)/n(\mathbb Z/m)\cong \mathbb Z/\gcd(n,m),
Ext Z 1 ( Z / n , Z / m ) ≅ ( Z / m ) / n ( Z / m ) ≅ Z / g cd( n , m ) , and Ext i = 0 \operatorname{Ext}^i=0 Ext i = 0 i ≥ 2 i\ge 2 i ≥ 2
If M M M projective
, then Ext R n ( M , N ) = 0 \operatorname{Ext}^n_R(M,N)=0 Ext R n ( M , N ) = 0 n ≥ 1 n\ge 1 n ≥ 1 N N N N N N flat
, then Tor n R ( M , N ) = 0 \operatorname{Tor}^R_n(M,N)=0 Tor n R ( M , N ) = 0 n ≥ 1 n\ge 1 n ≥ 1 M M M 0 0 0 Hom/tensor exactness
).