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A bifunctor can be derived in either variable when the relevant resolution data are supplied

Statement

Let $A, C, D$ be abelian categories, and let $B : A^{o p} \times C \to D$ be additive in each variable. Let $P$ be supplied projective resolution data on a class $D_{A}$ of objects of $A$ , and let $I$ be supplied injective resolution data on a class $D_{C}$ of objects of $C$ . Then:

for each fixed $A \in D_{A}$ , the covariant functor $B (A, -)$ has right derived objects at every $C \in D_{C}$ , $R_{I}^{n} (B (A, -)) (C);$
for each fixed $C \in D_{C}$ , the contravariant functor $B (-, C)$ is derived at every $A \in D_{A}$ on $A^{o p}$ using the projective datum on $A$ .

These are the two candidate one-variable derived constructions. No equality between them is asserted here.

Facts & Assumptions

Given: Abelian categories $A, C, D$ , the displayed bifunctor $B$ , and the supplied data $P$ on $D_{A}$ and $I$ on $D_{C}$ .

[L1]

Right derived objects are defined for covariant functors from supplied injective resolution data (Right derived objects relative to supplied injective resolution data).

[L2]

Left derived objects are defined for covariant functors from supplied projective resolution data (Left derived objects relative to supplied projective resolution data).

[L3]

Contravariant functors are derived on the opposite category (Contravariant derived functors are derived on the opposite category).

Proof

technique · direct

1.1

Fix $A \in D_{A}$ . Then $B (A, -) : C \to D$ is a covariant additive functor between abelian categories, so [L1] gives the right derived objects $R_{I}^{n} (B (A, -)) (C)$ for each $C \in D_{C}$ .

L1givenconstruct

1.2

Fix $C \in D_{C}$ . Then $B (-, C)$ is contravariant and additive in the $A$ -variable. By [L3], it is derived at each $A \in D_{A}$ on $A^{o p}$ using $P$ , equivalently the corresponding injective datum $P^{o p}$ on $A^{o p}$ .

L2L3construct

2.1

Steps 1.1 and 1.2 give the two candidate one-variable derived constructions. Since no comparison map between them has yet been supplied, no balance conclusion follows here.

step 1.1step 1.2∎

Depends on

Used by

A balanced derived bifunctor Definition
FALSE: derived functors in two variables are automatically balanced False statement

Dependency tree · two levels

16 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)