Derived functors formalize the idea that a non-exact functor can be “corrected” by replacing objects with resolutions and then taking homology/cohomology.
Throughout, let A , B \mathcal A,\mathcal B A , B abelian categories
and F : A → B F:\mathcal A\to\mathcal B F : A → B
Right derived functors (from injective resolutions) Assume F F F left exact
and A \mathcal A A injective resolutions
exist).
For A ∈ A A\in\mathcal A A ∈ A 0 → A → I ∙ 0\to A\to I^\bullet 0 → A → I ∙ F F F F ( I ∙ ) F(I^\bullet) F ( I ∙ ) B \mathcal B B right derived functors of F F F
R n F ( A ) : = H n ( F ( I ∙ ) ) ( n ≥ 0 ) ,
R^nF(A)\;:=\;H^n\bigl(F(I^\bullet)\bigr)\qquad (n\ge 0),
R n F ( A ) := H n ( F ( I ∙ ) ) ( n ≥ 0 ) , where H n H^n H n cohomology
.
Key facts (standard in homological algebra):
R 0 F ≅ F R^0F \cong F R 0 F ≅ F R n F ( A ) R^nF(A) R n F ( A ) A short exact sequence in A \mathcal A A long exact sequence
in the R n F R^nF R n F Left derived functors (from projective resolutions) Assume F F F A \mathcal A A projective resolutions
exist).
For A ∈ A A\in\mathcal A A ∈ A P ∙ → A P_\bullet\to A P ∙ → A F F F F ( P ∙ ) F(P_\bullet) F ( P ∙ ) B \mathcal B B left derived functors are
L n F ( A ) : = H n ( F ( P ∙ ) ) ( n ≥ 0 ) ,
L_nF(A)\;:=\;H_n\bigl(F(P_\bullet)\bigr)\qquad (n\ge 0),
L n F ( A ) := H n ( F ( P ∙ ) ) ( n ≥ 0 ) , where H n H_n H n homology
.
Again:
L 0 F ≅ F L_0F \cong F L 0 F ≅ F L n F ( A ) L_nF(A) L n F ( A ) Short exact sequences yield long exact sequences in the L n F L_nF L n F Fundamental examples: Ext and Tor In A = R -Mod \mathcal A = R\text{-Mod} A = R -Mod
The functor Hom R ( M , − ) \operatorname{Hom}_R(M,-) Hom R ( M , − ) left exact
, and its right derived functors are
R n Hom R ( M , − ) ≅ H A H A H U G O S H O R T C O D E 288 s 8 H B H B .
R^n\operatorname{Hom}_R(M,-)\;\cong\; Ext_R^n(M,-)
.
R n Hom R ( M , − ) ≅ H A H A H U GOS H ORTCO D E 288 s 8 H B H B . The functor − ⊗ R N -\otimes_R N − ⊗ R N right exact
, and its left derived functors are
L n ( − ⊗ R N ) ≅ H A H A H U G O S H O R T C O D E 288 s 10 H B H B .
L_n(-\otimes_R N)\;\cong\; Tor_n^R(-,N)
.
L n ( − ⊗ R N ) ≅ H A H A H U GOS H ORTCO D E 288 s 10 H B H B . Ext and Tor as derived functors
. Examples (explicit computations) Example 1: Computing T o r 1 Z ( Z / n , Z / m ) \mathrm{Tor}_1^{\mathbb Z}(\mathbb Z/n,\mathbb Z/m) Tor 1 Z ( Z / n , Z / m ) Let F ( − ) = − ⊗ Z Z / m Z F(-)= -\otimes_{\mathbb Z}\mathbb Z/m\mathbb Z F ( − ) = − ⊗ Z Z / m Z
0 → Z → × n Z → Z / n Z → 0.
0\to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathbb Z/n\mathbb Z \to 0.
0 → Z × n Z → Z / n Z → 0. Applying F F F
0 → Z / m → × n Z / m → 0 ,
0\to \mathbb Z/m \xrightarrow{\times n} \mathbb Z/m \to 0,
0 → Z / m × n Z / m → 0 , so
L 1 F ( Z / n ) ≅ H 1 ≅ ker ( × n : Z / m → Z / m ) ≅ Z / gcd ( m , n ) Z .
L_1F(\mathbb Z/n)\cong H_1\cong \ker(\times n:\mathbb Z/m\to\mathbb Z/m)\cong \mathbb Z/\gcd(m,n)\mathbb Z.
L 1 F ( Z / n ) ≅ H 1 ≅ ker ( × n : Z / m → Z / m ) ≅ Z / g cd( m , n ) Z . Thus H A H A H U G O S H O R T C O D E 288 s 12 H B H B 1 Z ( Z / n , Z / m ) ≅ Z / gcd ( m , n ) Z Tor_1
_1^{\mathbb Z}(\mathbb Z/n,\mathbb Z/m)\cong \mathbb Z/\gcd(m,n)\mathbb Z H A H A H U GOS H ORTCO D E 288 s 12 H B H B 1 Z ( Z / n , Z / m ) ≅ Z / g cd( m , n ) Z
Example 2: Computing E x t Z 1 ( Z / n , Z ) \mathrm{Ext}^1_{\mathbb Z}(\mathbb Z/n,\mathbb Z) Ext Z 1 ( Z / n , Z ) Let F ( − ) = Hom Z ( Z / n , − ) F(-)=\operatorname{Hom}_{\mathbb Z}(\mathbb Z/n,-) F ( − ) = Hom Z ( Z / n , − )
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. Applying F F F
0 → Hom ( Z / n , Q ) → Hom ( Z / n , Q / Z ) → 0 ,
0\to \operatorname{Hom}(\mathbb Z/n,\mathbb Q)\to \operatorname{Hom}(\mathbb Z/n,\mathbb Q/\mathbb Z)\to 0,
0 → Hom ( Z / n , Q ) → Hom ( Z / n , Q / Z ) → 0 , with Hom ( Z / n , Q ) = 0 \operatorname{Hom}(\mathbb Z/n,\mathbb Q)=0 Hom ( Z / n , Q ) = 0 Hom ( Z / n , Q / Z ) ≅ Z / n \operatorname{Hom}(\mathbb Z/n,\mathbb Q/\mathbb Z)\cong \mathbb Z/n Hom ( Z / n , Q / Z ) ≅ Z / n
R 1 F ( Z ) ≅ H 1 ≅ Z / n Z ,
R^1F(\mathbb Z)\cong H^1 \cong \mathbb Z/n\mathbb Z,
R 1 F ( Z ) ≅ H 1 ≅ Z / n Z , i.e. H A H A H U G O S H O R T C O D E 288 s 13 H B H B Z ( Z / n , Z ) ≅ Z / n Z Ext^1
_{\mathbb Z}(\mathbb Z/n,\mathbb Z)\cong \mathbb Z/n\mathbb Z H A H A H U GOS H ORTCO D E 288 s 13 H B H B Z ( Z / n , Z ) ≅ Z / n Z
Example 3: Exact functors have no higher derived functors If F F F exact functor
(e.g. in module categories, localization is exact on suitable classes of modules), then applying F F F
R n F = 0 ( n > 0 ) and L n F = 0 ( n > 0 ) ,
R^nF=0 \ (n>0)\quad\text{and}\quad L_nF=0 \ (n>0),
R n F = 0 ( n > 0 ) and L n F = 0 ( n > 0 ) , whenever the relevant derived functors are defined (right/left).