Let R R R ring
and write $\mathrm{Hom}_R(-,-)$
for the Hom functor.
Statement (covariant in the second variable) Fix a left R R R M M M
0 → A ′ → u A → v A ′ ′
0 \to A' \xrightarrow{u} A \xrightarrow{v} A''
0 → A ′ u A v A ′′ is an exact sequence of left R R R
0 → H o m R ( M , A ′ ) → u ∗ H o m R ( M , A ) → v ∗ H o m R ( M , A ′ ′ )
0 \to \mathrm{Hom}_R(M,A') \xrightarrow{u_*} \mathrm{Hom}_R(M,A) \xrightarrow{v_*} \mathrm{Hom}_R(M,A'')
0 → Hom R ( M , A ′ ) u ∗ Hom R ( M , A ) v ∗ Hom R ( M , A ′′ ) is exact.
Equivalently: H o m R ( M , − ) \mathrm{Hom}_R(M,-) Hom R ( M , − )
Statement (contravariant in the first variable) Fix a left R R R N N N
A ′ → u A → v A ′ ′ → 0
A' \xrightarrow{u} A \xrightarrow{v} A'' \to 0
A ′ u A v A ′′ → 0 is exact, then applying H o m R ( − , N ) \mathrm{Hom}_R(-,N) Hom R ( − , N )
0 → H o m R ( A ′ ′ , N ) → v ∗ H o m R ( A , N ) → u ∗ H o m R ( A ′ , N ) .
0 \to \mathrm{Hom}_R(A'',N) \xrightarrow{v^*} \mathrm{Hom}_R(A,N) \xrightarrow{u^*} \mathrm{Hom}_R(A',N).
0 → Hom R ( A ′′ , N ) v ∗ Hom R ( A , N ) u ∗ Hom R ( A ′ , N ) . Failure of right exactness and Ext The failure of H o m \mathrm{Hom} Hom Ext
: E x t R 1 ( − , N ) \mathrm{Ext}^1_R(-,N) Ext R 1 ( − , N ) E x t n \mathrm{Ext}^n Ext n right derived functors
of H o m R ( − , N ) \mathrm{Hom}_R(-,N) Hom R ( − , N ) short exact sequence
gives a long exact sequence in Ext
(a special case of the long exact sequence for derived functors
).
If M M M projective
, then H o m R ( M , − ) \mathrm{Hom}_R(M,-) Hom R ( M , − ) N N N injective
, then H o m R ( − , N ) \mathrm{Hom}_R(-,N) Hom R ( − , N )
Examples Example 1: Hom need not be right exact (over Z \mathbb Z Z Start from the short exact sequence
0 → Z → × n Z → Z / n → 0.
0 \to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathbb Z/n \to 0.
0 → Z × n Z → Z / n → 0. Apply H o m Z ( − , Z ) \mathrm{Hom}_{\mathbb Z}(-,\mathbb Z) Hom Z ( − , Z ) H o m ( Z / n , Z ) = 0 \mathrm{Hom}(\mathbb Z/n,\mathbb Z)=0 Hom ( Z / n , Z ) = 0 H o m ( Z , Z ) ≅ Z \mathrm{Hom}(\mathbb Z,\mathbb Z)\cong \mathbb Z Hom ( Z , Z ) ≅ Z
0 → 0 → Z → × n Z → E x t Z 1 ( Z / n , Z ) → 0.
0 \to 0 \to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathrm{Ext}^1_{\mathbb Z}(\mathbb Z/n,\mathbb Z)\to 0.
0 → 0 → Z × n Z → Ext Z 1 ( Z / n , Z ) → 0. The map Z → × n Z \mathbb Z\xrightarrow{\times n}\mathbb Z Z × n Z ∣ n ∣ > 1 |n|>1 ∣ n ∣ > 1 H o m ( − , Z ) \mathrm{Hom}(-,\mathbb Z) Hom ( − , Z )
E x t Z 1 ( Z / n , Z ) ≅ Z / n .
\mathrm{Ext}^1_{\mathbb Z}(\mathbb Z/n,\mathbb Z)\cong \mathbb Z/n.
Ext Z 1 ( Z / n , Z ) ≅ Z / n . Example 2: If M M M If M ≅ R ⊕ r M\cong R^{\oplus r} M ≅ R ⊕ r
H o m R ( M , A ) ≅ A ⊕ r .
\mathrm{Hom}_R(M,A)\cong A^{\oplus r}.
Hom R ( M , A ) ≅ A ⊕ r . Thus H o m R ( M , − ) \mathrm{Hom}_R(M,-) Hom R ( M , − )
Example 3: Vector spaces over a field If k k k V V V k k k V V V H o m k ( V , − ) \mathrm{Hom}_k(V,-) Hom k ( V , − )
E x t k n ( V , W ) = 0 ( n > 0 )
\mathrm{Ext}^n_k(V,W)=0 \quad (n>0)
Ext k n ( V , W ) = 0 ( n > 0 ) for all k k k W W W