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External qcqs classical perfect generator

PDF p. 3: Nonconnective algebraic K theory: enhanced prerequisite interface

Statement

Assume AC and let X be a quasi-compact quasi-separated scheme. In the ordinary perfect subcategory of D(OX) there is a perfect object P such that every perfect object is obtained from P by finitely iterating integer shifts, finite sums, cones and direct summands. This is the classical generator statement of Stacks Remark 36.17.2; its proof combines the perfect generator of DQCoh with compact=perfect and the compact-generation proposition.

Source contract

Given: qcqs X and AC; perfection is the supplied sheaf-derived local condition Perfect complexes on schemes and their support.

Goal: Import one substantial scheme theorem needed only to enumerate perfect objects.

[F1] Stacks Remark 36.17.2, Proposition 36.17.1, Theorem 36.15.3 and Proposition 13.37.6 give the stated classical generator.

External input

Import [F1]. The proof is not reconstructed. Its use below establishes essential smallness through finite constructions and small Hom sets; it supplies no coherent product, enhanced equivalence, or scheme localization theorem.

Depends on

Used by · 1

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External dependencies

The Stacks Project, Derived Categories of Schemes, Compact and perfect objects · Remark 36.17.2; Proposition 36.17.1; Theorem 36.15.3 (09IS), together with Proposition 13.37.6 (09SR)

For a qcqs scheme X there is a perfect complex P that is a classical generator of D_perf(O_X): every perfect complex lies in its closure under finite sums, shifts, cones and direct summands.

Hypotheses: X quasi-compact and quasi-separated. This is a statement about the ordinary derived category of O_X-module sheaves with quasi-coherent cohomology.

Reading limits: Read Remark 36.17.2 and full statements/proofs of Proposition 36.17.1, Theorem 36.15.3 and Proposition 13.37.6. Theorem 36.15.3 itself says a small detail is omitted; its affine-gluing, compact-generation and Koszul prerequisites are imported, not reconstructed. This is used only for object smallness, never to infer a coherent enhancement or monoidal equivalence.

Sources