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External local perfection and summand closure
PDF p. 16: Perfect complexes with support
Statement
Assume the Axiom of Choice. Perfection on a ringed space is invariant under isomorphism in the derived category and is local. When all stalk rings are local, it means local equivalence to a bounded complex of finite locally free modules. Perfect objects satisfy two-out-of-three in distinguished triangles and are closed under direct summands. Scheme stalk rings are local.
Source contract
Given: A ringed space, with local stalk rings for the finite locally free characterization.
Goal: Record local invariance and closure under triangles and direct summands.
[F1] The cited Stacks definition and lemmas supply these assertions.
External input
Import [F1] with its stated reading limits. The local strictification and direct-summand arguments are not claimed as local reconstructions.
Depends on
Dependency tree · 3 levels
Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.
External dependencies
Perfection is a local derived property; for local stalk rings it is characterized by local bounded finite locally free representatives. Perfect objects satisfy two-out-of-three in distinguished triangles and are closed under direct summands.
Hypotheses: Ringed spaces for local invariance, triangle and summand closure; local stalk rings for the finite locally free characterization.
Reading limits: Inspected the definition and displayed triangle and direct-summand statements. The referenced strictification, pseudo-coherence and Tor-amplitude arguments are imported and not reconstructed.
Sources
- The Stacks Project, Cohomology of Sheaves, Perfect complexes · Definition 20.49.1 and Lemmas 20.49.2, 20.49.3, 20.49.7 and 20.49.9