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Covering-sieve descent for the actual two-open square

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and the fixed site and universe bounds of the enhanced motivic-complex interface. Let $k$ be algebraically closed, let $X$ be an affine finite-type $k$ -scheme of Krull dimension three, and let $T \subset X$ be closed and contained in $U = X_{s m o o t h}$ . Give $T$ its reduced induced structure and put $W = X ∖ T$ and $V = U ∖ T$ . With $C (Y) = Z (3)_{m o t} (Y)$ , restriction induces an equivalence in $D_{\infty} (Z)$ $C (X) ≃ C (U) \times_{C (V)} C (W) .$ Ordinary realization $E$ sends this to a homotopy pullback square of spectra, with its actual restriction maps.

External input

Reference 0 supplies pointwise presheaf colimits, Yoneda evaluation and the expression of a presheaf as a colimit over its category of elements. Here the presheaf category is $P = Fun (N (C_{k})^{o p}, S)$ , where $C_{k}$ is the size-bounded scheme category in [F2], $N$ is its nerve, and $S$ is the infinity-category of spaces in a compatible larger universe. Maps in $P$ are coherent natural transformations. Representable presheaves and the sieve below take discrete-space values. The deduction for the two-open cover is proved below.

Facts & Assumptions

Given: The stated geometry, Choice, and the compatible site and universe bounds.

Goal: Deduce the two-open pullback directly from covering-sieve descent.

[A1]

The Axiom of Choice and the size conventions in Elmanto–Morrow motivic complex input are assumed throughout. The nested universes have the closure convention of Grothendieck universe closure convention; their existence is an additional assumption.

[F1]

The actual smooth open pair and its choice scope supplies these actual qcqs opens, their scheme-theoretic intersection $V = U \times_{X} W$ , and the Nisnevich covering family ${U \to X, W \to X}$ . By Open immersions of schemes, an open immersion identifies its source isomorphically with an open subscheme, so a factorization through it is unique.

[F2]

Elmanto–Morrow motivic complex input supplies $C$ , its enhanced limits and its covering-sieve descent. Scheme sites and sheafification supplies sieves and the bounded site. For a scheme $Y$ write $h_{Y} (S) = {Hom}_{k} (S, Y)$ ; regard these sets as discrete spaces. For any test complex $K$ , put $Q_{K} (Y) = Map (K, C (Y))$ , a space-valued presheaf. Mapping out of $K$ preserves limits by their defining universal property.

[F3]

Reference 0 supplies pointwise colimits and the canonical identification $Map (R, Q) = \lim_{(S \to X) \in R^{o p}} Q (S)$ for a sieve $R \subset h_{X}$ , obtained by expressing $R$ as its colimit of representables and applying Yoneda evaluation.

[F4]

A coherent spectral realization of derived abelian complexes supplies the fixed ordinary realization functor and its preservation of limits. Take $E = H$ from that interface throughout. Stable spectra and exact triangles supplies the target spectra and their homotopy pullbacks.

[F5]

Article scheme conventions defines the algebraically closed base, affine finite-type and chain-dimension conventions. The smooth locus of a morphism defines the smooth locus; [F1] verifies its local-finite-presentation hypothesis and open structure here. Motivic complexes and cohomology names the weight-three complex supplied in [F2].

Proof

1.1A1F1F2F3F5givenconstruct

Let $R \subset h_{X}$ be the sieve generated by the two maps in [F1]. For every test scheme $S$ , $R (S)$ is the union of the subsets $h_{U} (S)$ and $h_{W} (S)$ of $h_{X} (S)$ : membership means factorization through one of the two opens. Open immersions give unique such factorizations, and their intersection is $h_{V} (S)$ by [F1]. Thus $R = h_{U} ⊔_{h_{V}} h_{W}$ in space-valued presheaves by [F3]. To check the objectwise homotopy pushout, map it into any space: an element present on both sides contributes two chosen points and a path between them. This is equivalent to one chosen point, with contractible extra data; an element present on only one side contributes one point. Thus the canonical map to the discrete union is an equivalence.

2.1F1F2F3F4step 1.1∎

Apply [F2] to the covering sieve $R$ and then map out of any $K$ . By [F3], descent identifies $Q_{K} (X)$ with $Map (R, Q_{K})$ . The pushout in step 1.1 identifies this mapping space with $Q_{K} (U) \times_{Q_{K} (V)} Q_{K} (W)$ , using Yoneda evaluation. This holds naturally for every $K$ , so the defining universal property of limits in [F2] gives the asserted equivalence in $D_{\infty} (Z)$ , with its restriction cone. Apply [F4] to that cone to obtain the spectral pullback square.

Depends on

Used by · 1

Excision and the exact fixed-support cycle fiber

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

Jacob Lurie, Higher Topos Theory (Princeton University Press, 2009) · Corollaries 5.1.2.3–5.1.2.4, printed p. 319 (PDF index 336); Section 5.1.5 opening and Lemmas 5.1.5.2–5.1.5.3, printed pp. 324–326 (PDF indices 341–343)

Space-valued presheaves on a small category C form the infinity-category Fun(N(C)^op, S). This category has small colimits computed at each object. Evaluation at an object is represented by its Yoneda presheaf. The identity on the presheaf category is the pointwise left Kan extension of the Yoneda embedding along itself, expressing a presheaf as the colimit of the representables mapping to it.

Hypotheses: The source is the nerve of a category small in the fixed site universe. Presheaves take values in spaces in a compatible larger universe; all diagrams and categories of elements used here are small there. Apply the density assertion to a set-valued sieve regarded as a presheaf of discrete spaces.

Reading limits: During adjudicate-b09c2795533e, read the primary PDF web extraction of both corollaries, both lemma statements and their displayed proofs, including the complete proof of Lemma 5.1.5.3. The model-category, left-fibration and Kan-extension antecedents are imported, not reconstructed. The generated-sieve union and the two-open derived application below are local arguments. No hyperdescent or excision theorem for arbitrary distinguished squares is imported.

Sources

Jacob Lurie, Higher Topos Theory (Princeton University Press, 2009) · Corollaries 5.1.2.3–5.1.2.4, printed p. 319 (PDF index 336); Section 5.1.5 opening and Lemmas 5.1.5.2–5.1.5.3, printed pp. 324–326 (PDF indices 341–343)