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The zero-th left derived functor of a right exact functor recovers the functor

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum on a class $D$ , and let $F : A \to B$ be an additive right exact functor between abelian categories. Then for every $A \in D$ there is a canonical isomorphism $L_{0}^{P} F (A) \tilde{\to} F (A),$ natural in $A$ .

Facts & Assumptions

Given: An object $A \in D$ .

[L1]

The chosen projective resolution of $A$ is an exact augmented complex $\dots \to P_{1} (A) \to P_{0} (A) \to A \to 0$ (Projective resolutions in an abelian category).

[L2]

The zeroth homology of the deleted complex is the cokernel of the boundary map into degree $0$ (Homology object of a chain complex).

[L3]

Right exactness means that $F$ preserves the cokernel appearing at the end of the displayed augmented resolution (Left exact and right exact functors).

[L4]

The assignments $A \mapsto L_{0}^{P} F (A)$ are already functorial (Left derived functors relative to supplied data are additive functors).

Proof

technique · direct

1.1L1L3givenalgebra

By [L1], the morphism $P_{1} (A) \to P_{0} (A) \to A \to 0$ is exact. Applying $F$ and using [L3] gives an exact sequence $F (P_{1} (A)) \to F (P_{0} (A)) \to F (A) \to 0.$ Hence $F (A)$ is the cokernel of $F (P_{1} (A)) \to F (P_{0} (A))$ .

2.1L2step 1.1

By [L2], that same cokernel is exactly $H_{0} (F (P (A)_{d e l})) = L_{0}^{P} F (A)$ . Therefore there is a canonical isomorphism $L_{0}^{P} F (A) \tilde{\to} F (A)$ .

3.1L4step 2.1∎

The construction in steps 1.1 and 2.1 is functorial in $A$ , and [L4] already supplies the functoriality of $L_{0}^{P} F$ . Thus the isomorphism is natural in $A$ .

Depends on

Dependency tree · 2 levels

Sources

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed.