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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Dependency tree · 2 levels

Sources

10.5.1–10.5.8, pp. 391–393; restrict to supplied replacement data