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Local perfection implies quasi-coherent cohomology
PDF p. 16: Perfect complexes with support
Statement
Assume the Axiom of Choice. On any scheme, local equivalence to a bounded complex of finite locally free sheaves is independent of the chosen derived representative and local on the scheme. Every object satisfying it has quasi-coherent cohomology. The finite bounds may depend on the open set.
Facts & Assumptions
Given: A scheme and a locally perfect object of .
Goal: Prove well-definedness and quasi-coherence of its cohomology.
External local perfection and summand closure gives invariance and locality of perfection.
Bounded finite-projective complexes and affine chain models identifies finite locally free complexes on affine opens with finite-projective complexes, and External affine derived and perfect comparisons gives exactness of associated sheaf.
Restriction, stalks and cohomological support gives commutation of restriction with cohomology.
Proof
The local condition is invariant under derived equivalence and can be checked on an open cover by [F1]. Restricting a local representative to a smaller open set preserves its bounded finite locally free form.
Refine the local presentations to affine opens. By [F2], on each such open the representative is the associated sheaf of a bounded finite-projective complex. Exactness of associated sheaf identifies its cohomology with associated sheaves of module cohomology, which are quasi-coherent. By [F3] these are the restrictions of . Quasi-coherence is local on affine covers, so every is quasi-coherent. The construction imposes no common global bound.
Depends on
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