6.2 Left and right derived functors

Left and right derived functors measure how non-exact functors behave. They're built using projective or injective resolutions and taking homology. These tools help us understand functor behavior in complex algebraic structures.

Derived functors come in two flavors: left and right. Left derived functors use projective resolutions, while right derived functors use injective resolutions. They're key to studying important concepts like Ext and Tor in homological algebra.

Derived Functors

Constructing Derived Functors

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• Left derived functor $\mathbf{L}F$ constructed by applying functor $F$ to a projective resolution of an object and taking homology
  • Measures the extent to which $F$ fails to be exact when applied to the left
  • Independent of the choice of projective resolution up to natural isomorphism
• Right derived functor $\mathbf{R}F$ constructed by applying functor $F$ to an injective resolution of an object and taking homology
  • Measures the extent to which $F$ fails to be exact when applied to the right
  • Independent of the choice of injective resolution up to natural isomorphism

Types of Functors

• Covariant functor $F: \mathcal{C} \to \mathcal{D}$ preserves the direction of morphisms
  • For morphism $f: A \to B$ in $\mathcal{C}$, $F(f): F(A) \to F(B)$ in $\mathcal{D}$
  • Composes morphisms in the same order as the original category: $F(g \circ f) = F(g) \circ F(f)$
• Contravariant functor $G: \mathcal{C} \to \mathcal{D}$ reverses the direction of morphisms
  • For morphism $f: A \to B$ in $\mathcal{C}$, $G(f): G(B) \to G(A)$ in $\mathcal{D}$
  • Composes morphisms in the opposite order of the original category: $G(g \circ f) = G(f) \circ G(g)$
  • Examples include $\mathrm{Hom}(-, A): \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ and $\mathrm{Hom}(A, -): \mathcal{C} \to \mathbf{Set}$

Specific Derived Functors

Ext and Tor Functors

• Ext functor $\mathrm{Ext}^n_R(A, B)$ is the $n$-th right derived functor of $\mathrm{Hom}_R(A, -)$
  • Measures the failure of $\mathrm{Hom}_R(A, -)$ to be exact
  • Can be computed using a projective resolution of $A$ or an injective resolution of $B$
  • $\mathrm{Ext}^0_R(A, B) \cong \mathrm{Hom}_R(A, B)$ and $\mathrm{Ext}^1_R(A, B)$ classifies extensions of $B$ by $A$
• Tor functor $\mathrm{Tor}_n^R(A, B)$ is the $n$-th left derived functor of $- \otimes_R B$
  • Measures the failure of $- \otimes_R B$ to be exact
  • Can be computed using a projective resolution of either $A$ or $B$
  • $\mathrm{Tor}_0^R(A, B) \cong A \otimes_R B$ and $\mathrm{Tor}_1^R(A, B)$ measures the abelian group of relations between $A$ and $B$

Cohomology and Satellites

• (Co)homology functors are derived functors in various settings
  • Singular cohomology is the right derived functor of the global sections functor on sheaves
  • Group cohomology is the right derived functor of the invariants functor on $G$-modules
  • Lie algebra cohomology is the right derived functor of the invariants functor on Lie modules
• Satellite functors generalize the construction of derived functors
  • Include local cohomology functors and local homology functors
  • Defined using the language of triangulated categories and localization