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Truncation and closure of the native derived smooth Zariski site

PDF p. 11: Lemma5.3, second Chern additivity

Statement

Assume the Axiom of Choice and compatible universes. Retain the native derived geometry and the usual projective line of the supplied framework. Classical truncation $t_{0} Y$ of a derived scheme $Y$ has the same underlying Zariski topological space, and its open subschemes correspond to the open derived subschemes of $Y$ . A derived scheme is qcqs exactly when its classical truncation is qcqs. If $S$ is qcqs and $Y$ is a smooth qcqs derived $S$ -scheme, each quasi-compact open derived subscheme of $Y$ is again smooth and qcqs over $S$ . The projective line $P_{S}^{1}$ and the derived product $Y \times_{S} P_{S}^{1}$ are smooth qcqs $S$ -schemes. The assertions hold in animated derived geometry and in connective spectral geometry with its flat, fiber-smooth convention; no equivalence between these categories is asserted.

Facts & Assumptions

[A1]

Use The Axiom of Choice in the imported categorical foundations and the site, compatible universes and usual projective line of Source-specific non-A¹ motivic spectrum framework and hypotheses. Apply the assertions below in that framework's native geometry. In the spectral instance, smooth means fiber smooth; the projective line is the flat chart-defined one, not the differentially smooth replacement.

[F1]

Source record 0 supplies the spectral truncation, open-restriction, finite-limit, qcqs-product and smoothness interfaces. Its truncation replaces the structure sheaf by $π_{0}$ without replacing the topological space; its open immersions are etale.

[F2]

Source records 1 and 2 supply the animated truncation and affine-open comparison. Source record 3 supplies animated smoothness, locality, composition and base change, together with the finite-limit property of classical truncation. On an affine chart, a principal open is obtained by inverting an element of $π_{0}$ ; its homotopy groups are localized at that element.

[F3]

Quasi-compact and quasi-separated schemes supplies the ordinary finite-cover and affine-intersection meaning of qcqs. Relative projective space from standard charts supplies the ordinary projective-line charts and their localization overlaps. Source record 4 supplies qcqs closure under ordinary fiber products. Source record 0 supplies the native spectral projective-space charts and base change; source record 3 permits ordinary smooth charts, whose ordinary smoothness is supplied in [F4], to be used in animated geometry. Under [A1], Every affine scheme is quasi-compact makes every ordinary affine scheme quasi-compact, and Affine schemes are quasi-separated makes it quasi-separated. These apply in particular to $Spec Z$ and to all the ordinary affine charts used below. Use also the affine-cover diagonal test imported in [F2] and applied in step 1.1: quasi-compactness of the intersections of members of an affine cover tests quasi-separatedness. The ordinary case is included in that criterion. Thus the compactness assertions needed before invoking the ordinary fiber-product theorem are supplied, rather than inferred from the definition of qcqs.

[F4]

Under [A1], Polynomial rings are flat and smooth makes $Spec A [T] \to Spec A$ smooth for every ordinary commutative ring $A$ ; use $A = Z$ for both ordinary projective-line charts. Choice is used here through that supplier's Jacobian-criterion proof of ordinary smoothness. Open immersions are etale makes every ordinary principal open immersion etale, and Étale morphism of schemes defines an etale morphism to be smooth of relative dimension zero. Thus the ordinary localization maps on $π_{0}$ needed in [F2] are smooth as well. These are ordinary smoothness premises; the animated inclusion, scalar-extension criterion, locality and base-change rules remain those of source record 3.

External input

The exact primitive geometric interfaces are imported from source records 0-4 under their individual hypotheses. Source record 0 treats spectral geometry directly; records 1-3 treat animated geometry directly. Applying the appropriate instance uses no change-of-model equivalence. The ordinary scheme interfaces in [F3] are used only after classical truncation or to specify the ordinary projective-line charts. The global open comparison and the site-closure consequences are composed locally below.

Proof

Given: The native geometric convention, Choice, universes, and $S, Y$ in the statement.

Goal: Establish the truncation comparison and the two closure properties needed by smooth-site descent and projective-line evaluation.

1.1A1F1F2F3F4given

In spectral geometry [F1] gives the same topological space for $Y$ and $t_{0} Y$ and identifies open restrictions; its schematic qcqs comparison gives the claimed qcqs equivalence. In animated geometry [F2] identifies opens on every affine chart with opens of its ordinary truncation. The local identifications respect restriction to an overlap, since they come from base change to the truncation and the universal property of localization. They therefore glue along an affine Zariski cover to identify the posets of all opens. A chosen derived affine cover truncates to an ordinary affine cover. Quasi-compactness agrees because all open covers and their finite subcovers correspond. To test quasi-separatedness, pull the diagonal back to the affine target charts formed by pairs of these affine charts; the pullbacks are exactly the open intersections. Quasi-compactness of a morphism is checked on an affine-open cover of its target, so this diagonal test is precisely quasi-compactness of these intersections. Their quasi-compactness agrees under the open comparison already proved. The same diagonal test for the ordinary affine cover gives the qcqs equivalence of [F3]. This argument uses the chosen affine cover and does not infer affineness of a lift from an equivalence of posets. An open subset of a quasi-separated scheme is quasi-separated: its affine opens are affine opens of the whole scheme and have the same intersections. Consequently a quasi-compact open of a qcqs $Y$ is qcqs. Its inclusion is smooth: in the spectral instance it is etale by [F1]; in the animated instance it is locally a principal localization. The induced map on $π_{0}$ is an ordinary principal open immersion, hence smooth by [F4]; the homotopy groups localize by the Lemma 4.2 input of [F2]. These two facts satisfy the animated smoothness criterion of [F2]. Smoothness is local and preserved by composition in both instances. Its composite to $S$ is therefore smooth.

2.1A1F1F2F3F4step 1.1∎

The usual projective line over an affine base has two affine charts with affine localization overlap. Each of these ordinary affine schemes is quasi-compact by the affine compactness input in [F3]. The two charts therefore make the ordinary projective line quasi-compact: an open cover restricts to a cover of each chart, and the union of their two finite subcovers is finite and covers the whole scheme. Their intersection is the indicated affine localization, hence is quasi-compact by [F3]. The affine-cover diagonal test of [F2], used in step 1.1, now proves quasi-separatedness. Thus its ordinary truncation is qcqs. The ordinary base $Spec Z$ is itself qcqs by the two affine-scheme inputs of [F3]. Over a general qcqs $S$ , its ordinary truncation is $P_{t_{0} S}^{1}$ : this follows chartwise from the native construction and truncation, and the chart identifications glue. The ordinary scheme $P_{t_{0} S}^{1} = t_{0} S \times_{Spec Z} P_{Z}^{1}$ is qcqs by the ordinary fiber-product assertion of [F3]: $t_{0} S$ is qcqs by step 1.1 and the hypothesis on $S$ , while $Spec Z$ and $P_{Z}^{1}$ are qcqs by the preceding affine and two-chart checks. Thus both factors and their base meet the hypotheses of that input before it is applied. Step 1.1 now makes $P_{S}^{1}$ qcqs. Its smoothness is the projective-space assertion of [F1] in spectral geometry; in animated geometry the two ordinary charts $Spec Z [T] \to Spec Z$ are first smooth by [F4]. The ordinary-scheme inclusion preserves smooth maps by [F2], so their animated realizations are smooth. Base change along $S \to Spec Z$ gives the two polynomial charts over $S$ , and smoothness is local on the source by [F2]. These are the charts of the usual projective line by [F3], so $P_{S}^{1} \to S$ is smooth. The product $Y \times_{S} P_{S}^{1}$ exists in the native category. In the spectral instance it is schematic and qcqs by [F1]. In the animated instance its truncation is $t_{0} Y \times_{t_{0} S} P_{t_{0} S}^{1}$ by [F2], which is qcqs by [F3], and step 1.1 applies again. Finally its projection to $Y$ is a base change of the smooth projective line, and composition with $Y \to S$ is smooth. All hypotheses of the ordinary fiber-product input are met: the two factors and the base are qcqs. This proves both site-closure claims without a Noetherian, dimension or separatedness bound.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Lurie, Spectral Algebraic Geometry, February 2018 author manuscript · Definition 1.1.2.8 and Remark 1.1.2.10; Proposition 1.1.7.10 (n=0); Definition 1.6.7.2, Proposition 1.6.7.3 and Corollary 1.6.7.5; Proposition 2.3.4.2 and Corollary 2.3.5.4; Construction 5.4.1.3, Remark 5.4.1.6, Proposition 5.4.1.7 and Example 5.4.1.8; Example 11.2.3.2, Definition 11.2.5.5, Example 11.2.5.7 and Proposition 11.2.5.9

For connective spectral schemes the underlying ordinary scheme and zero truncation have the same topological space. Open substacks are restrictions to topological opens and are etale; schematic objects are closed under finite limits. Qcqs schematic objects correspond to the infinity-quasi-compact objects, which are closed under fiber products. The usual projective space has the standard affine charts with affine intersections, commutes with base change and is fiber smooth. Fiber smoothness is local on the source and stable under composition and base change; etale maps are fiber smooth.

Hypotheses: Connective spectral schemes, or their associated schematic spectral Deligne-Mumford stacks. For the projective-space assertion use the flat, chart-defined space of Construction 5.4.1.3 and fiber smoothness, not the differentially smooth replacement of Definition 19.2.6.3. No Noetherian hypothesis is imposed.

Reading limits: Historical reading record retained from the incoming generated item: Freshly read the named definitions and statements in the web tool's indexed extraction of the primary author PDF, including the chart intersections, n=0 truncation formula, qcqs comparison and fiber-product closure. Also read the displayed proof of Proposition 1.6.7.3 and the statements and proof of Proposition 11.2.5.9. Direct full-PDF retrieval exceeded the web tool size limit; the indexed primary text was accessible. Foundational proofs are imported. This record supplies the native spectral instance only; it asserts no equivalence with animated geometry. Current consumer maintenance read this generated body and its supplier contracts, including the two added published affine-scheme interfaces; no fresh primary-source or page-image reading, or independent acceptance, is claimed.

Adeel Khan, Descent in algebraic K-theory, Lecture 1: Derived algebraic geometry · Definition 1.3, paragraph 1.8, paragraphs 2.7-2.11 (printed pp. 1-2, 5-6)

Animated rings are product-preserving space-valued functors on polynomial rings. Derived affine fiber products are given by derived tensor products. Derived schemes admit affine Zariski covers. Restriction to ordinary rings gives the underlying ordinary scheme, with Spec(A) truncated to Spec(pi_0 A); this truncation is right adjoint to the ordinary-scheme inclusion.

Hypotheses: Animated derived schemes as in SAG Chapter 25, without replacing them by all connective spectral schemes. All constructions are in fixed compatible universes.

Reading limits: Historical reading record retained from the incoming generated item: Freshly opened the primary lecture PDF and read Definition 1.3, paragraph 1.8 and all of paragraphs 2.7-2.11. Only these derived-ring, open-cover and truncation interfaces are imported; module descent and perfect-complex results were not used. Current consumer maintenance read this generated body and its supplier contracts, including the two added published affine-scheme interfaces; no fresh primary-source or page-image reading, or independent acceptance, is claimed.

Adeel Khan, Descent in algebraic K-theory, Lecture 3: Compact generation of quasi-coherent sheaves · Definition 3.2 and Exercise 3.3, printed pp. 3-4; paragraph 4.1 and Lemma 4.2 with its proof, printed p. 5; Proposition 4.9, printed p. 6; Theorem 4.12 with its proof, printed p. 7

For an affine animated derived scheme, passage to its classical truncation identifies the poset of open derived subschemes with the poset of ordinary open subschemes. Every open is covered by principal localizations. Quasi-compactness is the finite-subcover property; quasi-separatedness is quasi-compactness of the diagonal and is tested on intersections of affine opens. For an animated ring R and a point f of its underlying space, paragraph 4.1 constructs R[f^{-1}] and identifies pi_0(R[f^{-1}]) with pi_0(R)[f^{-1}]. The proof of Lemma 4.2 gives pi_*(R[f^{-1}]) = pi_*(R)[f^{-1}], hence pi_n(R[f^{-1}]) = (pi_n R)[f^{-1}] for every n >= 0, and proves Spec(R[f^{-1}]) -> Spec(R) is an open immersion.

Hypotheses: Theorem 4.12 is stated for an affine derived scheme. The global open comparison used here is obtained by gluing along an affine Zariski cover, not attributed to a global statement of that theorem. The affine-intersection characterization is explicitly presented as an exercise by the source. For the localization calculation R is an animated (simplicial commutative) ring and f is a point representing a class in pi_0(R); the graded localization is at that class. No Noetherian or discreteness assumption is used.

Reading limits: Historical reading record retained from the incoming generated item: Freshly read Definition 3.2, Exercise 3.3, Proposition 4.9 and the complete extracted proof of Theorem 4.12. The local proof below explains the global gluing and the qcqs application. No claim of solving the source's unrelated exercises or reconstructing its K-theory results is made. Current consumer maintenance read this generated body and its supplier contracts, including the two added published affine-scheme interfaces; no fresh primary-source or page-image reading, or independent acceptance, is claimed. Current assigned-item repair: freshly read the primary PDF text extraction of Definition 3.2, Exercise 3.3, all of paragraph 4.1 and Lemma 4.2 with its proof, Proposition 4.9, and Theorem 4.12 with its proof. In particular the displayed graded-homotopy localization calculation is in the proof of Lemma 4.2 on printed p. 5. PDF-page screenshots returned cache misses, so this is extracted-text reading only. The source presents Exercise 3.3 as an exercise; its characterization is imported as stated, and no solution or independent foundational audit is claimed.

Can Yaylali, Notes on derived algebraic geometry, arXiv:2208.01506v1 · Definition 2.53, p. 23; Definition 3.3 and Remark 3.4, p. 27; Definition 3.13 and Lemma 3.14, pp. 29-30; Remark and Definition 3.35, pp. 32-33

An animated-ring map is smooth precisely when its map on pi_0 is classically smooth and the higher homotopy groups are obtained by scalar extension. Smooth morphisms are local on source and target and stable under base change and composition. Classical truncation preserves finite limits and smoothness; the ordinary-scheme inclusion preserves smooth maps and pullbacks along flat maps.

Hypotheses: Animated rings and derived schemes of the source, with the fixed size conventions. These assertions are used only in that setting; they do not identify animated rings with connective E-infinity rings.

Reading limits: Historical reading record retained from the incoming generated item: Freshly read the complete displayed Definition 2.53 and its scalar-extension condition, Remark 3.4, Definition 3.13/Lemma 3.14 and the four truncation/inclusion properties in Remark and Definition 3.35. The substantial cotangent-complex criteria and the proofs in Toën-Vezzosi cited there remain imported, rather than reconstructed. Current consumer maintenance read this generated body and its supplier contracts, including the two added published affine-scheme interfaces; no fresh primary-source or page-image reading, or independent acceptance, is claimed.

Stacks Project, qcqs fiber products of algebraic spaces · Lemma 67.8.10(3), statement and proof (Section 67.8)

A fiber product of quasi-compact quasi-separated algebraic spaces is quasi-compact and quasi-separated. This includes ordinary schemes.

Hypotheses: All three objects in the fiber-product diagram are qcqs. Apply over Spec(Z); the diagram of ordinary schemes is also a diagram of algebraic spaces, with the same scheme fiber product.

Reading limits: Historical reading record retained from the incoming generated item: Freshly read the complete statement and proof of Lemma 67.8.10 in the primary live section. Its cited transitivity, base-change and diagonal results are imported. Only its ordinary-scheme special case is used below. Current consumer maintenance read this generated body and its supplier contracts, including the two added published affine-scheme interfaces; no fresh primary-source or page-image reading, or independent acceptance, is claimed.

Sources

Lurie, Spectral Algebraic Geometry, February 2018 author manuscript · Definition 1.1.2.8 and Remark 1.1.2.10; Proposition 1.1.7.10 (n=0); Definition 1.6.7.2, Proposition 1.6.7.3 and Corollary 1.6.7.5; Proposition 2.3.4.2 and Corollary 2.3.5.4; Construction 5.4.1.3, Remark 5.4.1.6, Proposition 5.4.1.7 and Example 5.4.1.8; Example 11.2.3.2, Definition 11.2.5.5, Example 11.2.5.7 and Proposition 11.2.5.9
Adeel Khan, Descent in algebraic K-theory, Lecture 1: Derived algebraic geometry · Definition 1.3, paragraph 1.8, paragraphs 2.7-2.11 (printed pp. 1-2, 5-6)
Adeel Khan, Descent in algebraic K-theory, Lecture 3: Compact generation of quasi-coherent sheaves · Definition 3.2 and Exercise 3.3, printed pp. 3-4; paragraph 4.1 and Lemma 4.2 with its proof, printed p. 5; Proposition 4.9, printed p. 6; Theorem 4.12 with its proof, printed p. 7
Can Yaylali, Notes on derived algebraic geometry, arXiv:2208.01506v1 · Definition 2.53, p. 23; Definition 3.3 and Remark 3.4, p. 27; Definition 3.13 and Lemma 3.14, pp. 29-30; Remark and Definition 3.35, pp. 32-33
Stacks Project, qcqs fiber products of algebraic spaces · Lemma 67.8.10(3), statement and proof (Section 67.8)