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Coherent descent for shifted derived abelian sections

PDF p. 9: Actual low-weight motivic projections and their normalization

Statement

Assume the Axiom of Choice and fixed compatible Grothendieck universe bounds, including enlarged bounds for spaces and sheaf categories; universe existence is a separate assumption. Let $C$ be a small site presented by covering families, with the iterated fibre products of their members in $C$ . Let $A$ be an abelian sheaf on this site, and fix one nonnegative injective resolution $0 \to A \to I^{0} \overset{d^{0}}{\to} I^{1} \overset{d^{1}}{\to} I^{2} \to \dots$ in abelian sheaves on $C$ .

Use cochain shifts $C [1]^{q} = C^{q + 1}$ with differential $- d_{C}^{q + 1}$ , and the specified Eilenberg–Mac Lane realization $H C$ , with $π_{n} (H C) = H^{- n} (C)$ . Set $D_{I} (X) = Ω^{\infty} H (I^{∙} (X) [1])$ , where $Ω^{\infty}$ is the zeroth space. Restrictions of this single complex of sheaves give a simultaneous pointed space-valued presheaf. It is a sheaf of spaces and computes $Ω^{\infty} H (R Γ (X, A) [1])$ using derived evaluation on this same site. Its loop sheaf is canonically the discrete abelian sheaf $A$ .

Here a sheaf of spaces means homotopy descent: for every specified cover ${U_{i} \to X}$ , its value at $X$ is equivalent to the homotopy limit of the cosimplicial diagram whose degree $r$ is $\prod_{i_{0}, \dots, i_{r}} D_{I} (U_{i_{0}} \times_{X} \dots \times_{X} U_{i_{r}})$ , with maps induced by the projections and diagonals. Limits are coherent limits of spaces. The construction and comparison are pointed and additive, natural in $X$ and in maps of the chosen sheaf resolutions. The loop identification uses $A ↪ I^{0}$ without negating $A$ .

Statement references

The Axiom of Choice

Small sites and the plus construction

Imported derived sections on a small site

Dold-Kan equivalence for simplicial modules with explicit inverse

Eilenberg–Mac Lane spectrum of a chain complex

Complex shifts and spectral homotopy indices

Stable spectra and exact triangles

Facts & Assumptions

Given: The fixed site, compatible universe bounds, abelian sheaf $A$ and single resolution $I^{∙}$ in the Statement.

Goal: Supply coherent restriction maps and descent for the space-valued realization, including its loop normalization.

[A1]

Choice is assumed as in The Axiom of Choice for the resolution and homotopical inputs. Keep the fixed compatible bounds throughout; no universe existence follows from Choice.

[F1]

Small sites and the plus construction supplies small sites, sheaves, restriction functors and the relative universe bounds. A covering-family presentation means that isomorphisms cover, composites of covering families cover, and pullbacks of covering families exist and cover; take the topology generated by these families.

[F2]

Imported derived sections on a small site supplies the abelian category of sheaves, existence of the stated resolution, and the fact that $I^{∙} (X)$ computes right-derived evaluation. The injective resolution is an exact coaugmented sheaf complex; its kernels are computed sectionwise.

[F3]

Eilenberg–Mac Lane spectrum of a chain complex fixes the underlying Omega spectrum used for $H$ . Complex shifts and spectral homotopy indices fixes cochain-to-chain reindexing, the shift differential and the natural homotopy-group formula. Stable spectra and exact triangles supplies the coherent spectrum associated to this Omega tower, its zeroth space and coherent limits of spaces.

[F4]

Dold-Kan equivalence for simplicial modules with explicit inverse supplies normalization and its inverse on nonnegative chain complexes, here for the constant ring $Z$ . Write $DK$ for that inverse. Apply it sectionwise to a presheaf of complexes; functoriality gives all restrictions and their compositions.

[F5]

Source reference 0 supplies covering Čech descent for $E_{I} = DK (T_{I})$ , where $T_{I}$ is the chain complex of sheaves with $T_{I, 1} = I^{0}$ , $T_{I, 0} = Z^{1} I = \ker (d^{1} : I^{1} \to I^{2})$ , boundary $d^{0}$ , and zero other terms. In the exact sheaf resolution, $Z^{1} I$ is the sheaf image of $d^{0}$ . Its sections need not be the image of $d^{0} (X)$ .

[F6]

Source reference 1 supplies the natural zeroth-space model $Ω^{\infty} H B ≃ W (B) ≅ DK (τ_{\geq 0} B)$ for a chain complex $B$ . Here $H B$ denotes the chain-indexed version of [F3]. The smart truncation $τ_{\geq 0} B$ has $B_{n}$ for $n > 0$ , $\ker (B_{0} \to B_{- 1})$ for $n = 0$ , zero for $n < 0$ , and the induced differential.

External input

Reference 0 imports the descent fact [F5], using the $n = 1$ case of its injective-resolution construction. Reference 1 imports the zeroth-space construction [F6] for the same underlying Omega-spectrum realization as [F3]. Their proofs and foundations remain imported within the recorded reading limits. The signed comparison and its application to the chosen derived evaluation below are local.

Proof

1.1A1F1F2F4F5construct

By [F1]–[F2], the resolution and its sheaf kernel $Z^{1} I$ are defined on one fixed site. Form the two-term chain complex $T_{I}$ in [F5] and evaluate it at each $X$ . Since $d^{1} d^{0} = 0$ , its boundary lands in $Z^{1} I (X)$ ; the next boundary is zero. All restrictions commute with this boundary. Apply [F4] to obtain the pointed additive simplicial presheaf $E_{I} (X) = DK (T_{I} (X))$ , with compatible restrictions. Its degree-zero term uses sections of the sheaf kernel, exactly as in [F5]. Under [A1], that imported fact gives the covering Čech descent in the Statement, so $E_{I}$ is a sheaf of spaces. The site is small in the fixed bounds, $A$ is already a sheaf, and $I^{∙}$ is its nonnegative injective sheaf resolution, which checks every hypothesis of this application.

2.1A1F2F3F4F6step 1.1∎

Reindex the shifted cochain complex $I^{∙} (X) [1]$ as the chain complex $B (X)_{n} = I^{1 - n} (X)$ with boundary $- d^{1 - n}$ , using [F3]. Its smart truncation in [F6] has $I^{0} (X)$ in degree one and $Z^{1} I (X)$ in degree zero, with boundary $- d^{0}$ . The map from $T_{I} (X)$ to this truncation which is the identity in degree one and minus the identity in degree zero is a chain isomorphism: both boundary composites equal $- d^{0}$ . It commutes with every restriction and is additive. Apply [F4] and [F6] to get a natural pointed additive equivalence $E_{I} (X) ≃ D_{I} (X)$ . Thus the coherent presheaf $D_{I}$ has the descent established in step 1.1, and [F2] identifies its evaluated complex with $R Γ (X, A) [1]$ . By [F3], $π_{n} D_{I} (X) = H^{1 - n} (I^{∙} (X))$ for $n \geq 0$ ; these vanish for $n \geq 2$ since $I^{∙}$ is nonnegative. Its loop space is therefore discrete with group $\ker d^{0} (X) = A (X)$ , using exactness at $I^{0}$ and sectionwise kernels in [F2]. The chain comparison is the identity on that degree-one kernel, so the loop identification is the identity on $A$ , as claimed. This also proves naturality in maps of the fixed resolutions, since kernels, truncation, the sign map and both realization functors commute with those maps.

Depends on

Used by · 1

Shifted Nisnevich units and the line-bundle groupoid

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

Daniel Dugger, Sharon Hollander and Daniel C. Isaksen, Hypercovers and simplicial presheaves (2004) · Definitions 4.2–4.3 and the Čech-family paragraph, printed p.16; Corollary 7.1 and proof, p.27; §7.5, pp.31–32 (PDF pages 8,19,23–24)

The Dold–Kan model obtained by truncating an injective sheaf resolution at weight n has hypercover descent; use n=1, hence covering Čech descent.

Hypotheses: A Grothendieck site, an abelian presheaf and an injective resolution of its sheafification; apply to the already sheaf-valued A on the fixed bounded site.

Reading limits: Read the named definitions, Čech paragraph, corollary with proof and all of §7.5 in primary PDF text. Import the descent theorem and its cited Illusie/model-category foundations; no reconstruction. Screenshot requests failed. The degree-zero boundary is the sheaf image in the exact resolution, equal to its cycle sheaf, not the image after evaluation.

Stefan Schwede and Brooke Shipley, Stable model categories are categories of modules · Appendix B.1, W construction and normalized-connective-cover paragraph, printed pp.38–39; level formula and (B.1.9), p.39; Theorem B.1.11 opening proof, p.40 (PDF pages 38–40)

For an unbounded chain complex B, the zeroth space of its specified H Omega spectrum is W(B); W is Dold–Kan after connective truncation, naturally in chain maps.

Hypotheses: The one-object ringoid Z and unbounded complexes of abelian groups; use the naive underlying Omega spectrum, as in the declared Eilenberg–Mac Lane supplier, not a different symmetric model.

Reading limits: Read the W definition, connective-cover paragraph, level formula, (B.1.9), Remark B.1.10 and opening proof of B.1.11 in the author-hosted PDF text. Import their realization facts; no reconstruction of the full Quillen-equivalence or Alexander–Whitney foundations.

Sources

Daniel Dugger, Sharon Hollander and Daniel C. Isaksen, Hypercovers and simplicial presheaves (2004) · Definitions 4.2–4.3 and the Čech-family paragraph, printed p.16; Corollary 7.1 and proof, p.27; §7.5, pp.31–32 (PDF pages 8,19,23–24)
Stefan Schwede and Brooke Shipley, Stable model categories are categories of modules · Appendix B.1, W construction and normalized-connective-cover paragraph, printed pp.38–39; level formula and (B.1.9), p.39; Theorem B.1.11 opening proof, p.40 (PDF pages 38–40)