DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Right total derived functor on the bounded below derived category

Definition

Let $F : A \to B$ be additive and supply bounded-below injective replacements $j_{X} : X \to I_{X}$ under the hypotheses of Injective complexes model the bounded below derived category. Put $T = Q_{B} K (F)$ . The right total derived functor is $R F : D^{+} (A) \to D^{+} (B)$ with $R F (X) = Q_{B} F (I_{X})$ and maps induced by the injective model equivalence. It has coaugmentation $η : T \to R F Q_{A}$ induced by $F (j_{X})$ .

Its universal property is initial: for every $G : D^{+} (A) \to D^{+} (B)$ and natural $γ : T \to G Q_{A}$ there is a unique natural $ν : R F \to G$ such that $(ν Q_{A}) η = γ$ . Here $F$ is additive in the sense of Additive functor.

Depends on

Used by

fs-an-unbounded-total-derived-functor-exists-from-enough-injectives-alone.md False statement
Existence of the bounded below right total derived functor Theorem

Dependency tree · two levels

8 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

10.5.1–10.5.8, pp. 391–393; restrict to supplied replacement data (standard reference, not scraped)