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External extension and morphism input for perfect complexes

PDF p. 7: Zariski locality; PDF p. 15: Localization diagram

Statement

Let $X$ be qcqs and $j : U ↪ X$ a quasi-compact open. In the ordinary sheaf-derived perfect categories:

(a) For $E, F \in D_{p e r f} (O_{X})$ and $b : j^{*} E \to j^{*} F$ , there exist a perfect $Q$ and maps $c : Q \to E$ , $a : Q \to F$ with $j^{*} c$ invertible and $b j^{*} c = j^{*} a$ .

(b) If $a, b : E \to F$ have the same restriction, there are a perfect $Q$ and $c : Q \to E$ with $j^{*} c$ invertible and $a c = b c$ .

(c) A perfect object on $U$ extends to $X$ whenever its Euler class lies in the image of restriction on the classical perfect Euler groups. In particular $F \oplus F [1]$ extends for every perfect $F$ on $U$ .

The Euler group here is the classical perfect-complex Grothendieck group, with $[B] = [A] + [C]$ for triangles and $[F [1]] = - [F]$ . This is a source-side extension criterion, not an identification with $π_{0}$ of the K spectrum defined in this run. Verdier quotient by a full thick triangulated subcategory is localization at maps with cone in it.

Source contract

Given: qcqs $X$ , quasi-compact open $U$ , and ordinary derived categories of sheaf complexes with locally bounded finite locally free representatives.

Goal: Import exactly the established extension/roof inputs needed before applying enhanced localization.

[F1] TT Propositions 5.2.2(a), 5.2.3(a), 5.2.4(a) supply (a)–(c), under the hypotheses of 5.1 with its auxiliary $Z = X$ ; 5.6.3 removes the temporary ample-family/noetherian hypotheses in their proof.

[F2] TT 1.5.6–1.5.7 give the Euler relations, so $[F \oplus F [1]] = 0$ is in the image of any restriction map. Stacks Lemma 13.6.6 and Definition 13.6.7 give the quotient convention.

External input

Import [F1] and [F2] with the exact references and reading limits in the metadata. Only their perfect-complex and ordinary morphism assertions are used. No TT nonconnective spectrum, model comparison, or claim of a reconstructed extension theorem is included.

Depends on

Used by · 2

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

R. W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories · §5, hypotheses of Theorem 5.1 p. 337; Propositions 5.2.2–5.2.4 pp. 338–339; general-qcqs completion Lemma 5.6.2 and 5.6.3 pp. 348–350; Euler relations 1.5.6–1.5.7 p. 261. PDF pages 91–93, 102–104 and 15.

For qcqs X and quasi-compact open j:U→X, restriction of ordinary perfect derived categories has the following properties: every morphism between restricted objects admits a roof on X whose denominator becomes invertible on U; two morphisms equal on U become equal after such a denominator; a perfect F on U extends exactly when its Euler class is in the image of restriction. In particular F⊕F[1] extends because its Euler class is zero.

Hypotheses: X quasi-compact and quasi-separated, U a quasi-compact open. These are the unrestrained-support clauses 5.2.2(a), 5.2.3(a), 5.2.4(a); set the additional Z of 5.1 equal to X. No ample-family, noetherian, regularity or resolution-property assumption survives 5.6.3.

Reading limits: Read full statements on printed pp. 337–339, the equalizer proof 5.2.6 p. 339, general-qcqs Lemma 5.6.2 statement and start of proof p. 348, and closeout 5.6.3 p. 350. Read Euler relations 1.5.6–1.5.7 p. 261. The lengthy extension/approximation proofs in 5.3–5.6 remain imported. Only ordinary perfect complexes and actual derived restriction are imported here; neither TT K^B nor a comparison of K-theory models is used.

The Stacks Project, Derived Categories · Lemma 13.6.6 and Definition 13.6.7 (05RI)

For a full thick triangulated subcategory A of B, the Verdier quotient B/A is localization at morphisms with cone in A; this class admits the usual calculus of fractions.

Hypotheses: A full triangulated subcategory; thickness ensures saturation. Apply to the kernel of an exact restriction functor.

Reading limits: Read the full lemma statement and its displayed proof at 05RA, and the exact quotient definition at 05RI. General categorical fraction construction is imported; the application proof explicitly checks fullness and faithfulness.

Sources

R. W. Thomason and Thomas Trobaugh, Higher Algebraic K-Theory of Schemes and of Derived Categories · §5, hypotheses of Theorem 5.1 p. 337; Propositions 5.2.2–5.2.4 pp. 338–339; general-qcqs completion Lemma 5.6.2 and 5.6.3 pp. 348–350; Euler relations 1.5.6–1.5.7 p. 261. PDF pages 91–93, 102–104 and 15.
The Stacks Project, Derived Categories · Lemma 13.6.6 and Definition 13.6.7 (05RI)