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

Statement

Assume the Axiom of Dependent Choice.

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

Facts & Assumptions

Given: An object $A \in D$ .

[L1]

The chosen injective resolution of $A$ is an exact coaugmented complex $0 \to A \to I^{0} (A) \to I^{1} (A) \to \dots$ (Injective resolutions in an abelian category).

[L2]

The zeroth cohomology object is the quotient of the kernel of $d^{0} : F (I^{0} (A)) \to F (I^{1} (A))$ by the zero-th coboundary, which is $0$ (Cohomology object of a cochain complex).

[L3]

Left exactness means that $F$ preserves the kernel at the beginning of the displayed injective resolution (Left exact and right exact functors).

[L4]

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

Proof

technique · direct

1.1L1L3givenalgebra

By [L1], the morphism $0 \to A \to I^{0} (A) \to I^{1} (A)$ is exact. Applying $F$ and using [L3] gives an exact sequence $0 \to F (A) \to F (I^{0} (A)) \to F (I^{1} (A)) .$ Therefore $F (A) = \ker (F (I^{0} (A)) \to F (I^{1} (A)))$ .

2.1L2step 1.1

By [L2], the zeroth cohomology of $F (I (A)_{d e l})$ is that kernel, because the zero-th coboundary object is $0$ . Hence $R_{I}^{0} F (A) = H^{0} (F (I (A)_{d e l})) ≅ F (A)$ .

3.1L4step 2.1∎

Step 2.1 is natural in $A$ , and [L4] records the functoriality of $R_{I}^{0} F$ . Therefore the displayed isomorphism is natural.

Depends on

Used by

A balanced derived bifunctor Definition
Higher direct image of a sheaf Definition
The right derived functors of Hom from a fixed object Example
An exact base functor has the trivial universal delta functor Proposition
Degree-zero group (co)homology Proposition
The degree-zero injective construction of Ext is Hom Proposition
Degree-zero sheaf cohomology is global sections Theorem
Right derived functors form a cohomological delta functor Theorem
The acyclic-resolution theorem for right derived functors Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)