TheoremStatement: AI-adaptedProof: Literature-sourced

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

External actual graded algebra and multiplicative tower spectral sequence

PDF p. 4: Motivic graded spectral sequence; PDF p. 9: Eventual filtration vanishing

Statement

Assume the Axiom of Choice and fix the compatible size conventions of the local filtration realization. On qcqs schemes over $Q$ or $F_{p}$ for a prime $p$ , the EM filtration used by the completed local filtration definition is the actual pullback filtered $E_{\infty}$ -algebra of EM Definition4.9 in characteristic zero and Definition4.23 in characteristic $p$ . Its natural graded equivalence is $γ_{r} : {gr}^{r} F_{m o t} K \tilde{\to} Z (r)_{m o t} [2 r] .$ For $r \geq 0$ these are the very complexes defining the completed motivic cohomology. For $r < 0$ both sides are zero under the explicit source convention. Associated graded is strong symmetric monoidal for Day convolution, so this is the specified graded algebra structure, with products induced from the actual filtered multiplication. Its associated tower spectral sequence has multiplicative page pairings, the graded product on its initial page, differential derivation rules and compatibility with the filtered abutment product where bounded convergence applies. No convergence hypothesis is needed to define the products on finite pages.

Facts & Assumptions

[A1]

The Axiom of Choice and the fixed compatible size conventions are inherited from the completed local filtration realization. This assumption discharges the local realization premise when that definition is invoked; it is not an extra geometric condition on $X$ . The Axiom of Choice Motivic filtration and image groups

Given: The Axiom of Choice, the fixed compatible size conventions, a qcqs prime-field scheme, and the same actual multiplicative filtered object as in the completed local supplier.

Goal: Supply the missing multiplicative and graded source data, rather than infer them from additive exact-couple groups.

[F1]

EM Definitions4.9/4.23 and Theorems4.10/4.24 identify motivic complexes as shifted actual graded pieces, including vanishing in negative weights. Its §2.2 specifies the Day convolution algebra structures.

[F2]

EM Theorems4.12/4.27 identify the associated tower spectral sequence as multiplicative (and bounded in finite valuative dimension). The actual pairings are those of its filtered algebra, not arbitrary products placed on matching groups. Dugger’s tower-pairing construction describes their finite-page/graded and abutment compatibility; all derivation checks use finite sections, as explicitly explained in§6.3, and do not need a global bound. Its realizability hypotheses are not weakened to a bare homotopy-category pairing here.

External input

Import [F1]–[F2] for this actual source tower, with the precise statements and reading limits recorded in the references. The source product on motivic cohomology is by this graded algebra. The local additive exact-couple supplier is used below only to calculate and identify its underlying pages and indices; it does not establish [F2]. No general model-category rectification theorem is invented or reconstructed by this item.

Proof

1.1A1F1given

Under [A1], use the local filtration realization and apply [F1] to its underlying qcqs prime-field scheme. The defining graded equivalences and negative-weight zero convention are those of this actual filtered algebra, so the strong symmetric monoidal associated-graded functor transports its multiplication to the specified motivic products.

2.1A1F2step 1.1∎

Apply [F2] to the same actual multiplicative tower. Its finite-page pairings and derivation rules do not use convergence; under the stated boundedness hypothesis its filtered abutment compatibility applies as well. These are imported source data with the stated realization qualifications, rather than products inferred from additive groups or an unproved rectification claim. This gives all the stated comparisons and scope.

Source qualifications

In Elmanto–Morrow v1, the classical paragraph on printed p.13 gives the homological lower-bound order $\dim X - j$ . Its compatible order is $j - \dim X$ : the printed order at $X = A^{1}$ , $j = 0$ would incorrectly force $K_{0} (X) = 0$ . The boundedness proofs on pp.23 and29 similarly print the lower bounds $d + 1 - j$ and $d + 2 - j$ for the motivic stages. At stage zero over the respective prime field ( $d = 0$ ), these would kill $K_{0} = Z$ . These are retained source errors; they do not refute the separately stated boundedness theorem. Only boundedness in the §2.2 sense, with some finite $B (X)$ and connectivity growing as $j - B (X)$ , is imported. No corrected explicit numerical constant is asserted here. The characteristic-zero defining square is (4.3), although the p.23 proof refers to (4.6); (4.6) is the characteristic- $p$ square.

Depends on

Used by · 7

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External dependencies

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023): §2.2 pp.11–12; Definition4.9 and Theorem4.10(1)–(2) pp.21–22; Definition4.23 pp.27–28 and Theorem4.24 p.28; Theorem4.12 pp.22–23 and Theorem4.27 p.29; §3.1 p.13.

The actual motivic filtered algebra is the pullback in filtered E_infinity algebras of the displayed trace diagram. Z(r)_mot is defined as gr^r Fil_mot K[−2r]; the resulting graded comparison is natural and inherits multiplication from the same graded algebra. Negative graded terms vanish. The source spectral sequence is the multiplicative tower spectral sequence of this same actual filtration; finite valuative dimension supplies its bounded convergence.

Hypotheses: Any qcqs Q-scheme or F_p-scheme; all weights r for the graded construction, with Z(r)=0 for r<0 explicitly in4.10(1)/4.24(1). Finite valuative dimension only for boundedness/convergence.

Reading limits: This adjudication read the primary arXiv v1 PDF text via the web PDF extraction: §2.2, Definitions4.9/4.23, Theorems4.10(1)–(2)/4.24(1)–(2) and their graded-piece arguments, Theorems4.12/4.27 statements and boundedness paragraphs, and the §3.1 classical-connectivity paragraph. The characteristic-zero square label and the characteristic-p square label were checked in that extraction. The external construction and qualitative boundedness are imported; trace methods, descent foundations and optimal numerical bounds are not reconstructed. No page-image inspection is claimed this round. The printed homological estimates are excluded from the application.

Daniel Dugger, Multiplicative structures on homotopy spectral sequences I · §5 Proposition5.1 and Proposition5.3 p.11; §6 Theorem6.1 p.13, Theorem6.2 and Proposition6.4 p.14; proofs pp.16–17

Actual pairings of towers of spectra and their cofiber augmentations induce pairings of their spectral sequences, agreeing with the specified graded pairings on the first page and satisfying the derivation formula; where convergence identifies associated graded groups, the abutment pairing is the filtered one. A merely commuting pairing in the homotopy category needs local realizability; no such weakening is assumed for the actual source filtered algebra.

Hypotheses: The point-set statements use rigid/cofibrant towers or locally realizable homotopy pairings as specified in§6. Used here to describe and compare the source-induced pairing construction; the actual motivic multiplication and its spectral sequence are imported from EM, not obtained by asserting a general unproved rectification theorem.

Reading limits: This adjudication read arXiv math/0305173v1 (12 May 2003), the primary PDF extraction of §5 Propositions5.1/5.3 and their adjacent derivation/abutment discussion; §6 Theorems6.1/6.2, the finite-section local-realizability definition and Propositions6.4/6.5; and the proofs of6.1/6.2/6.4 on pp.16–17. AppendixB replacements and AppendixC foundations are imported rather than reconstructed. The application retains the actual EM filtered algebra; it asserts no realization theorem for arbitrary homotopy-category pairings.

Sources

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023): §2.2 pp.11–12; Definition4.9 and Theorem4.10(1)–(2) pp.21–22; Definition4.23 pp.27–28 and Theorem4.24 p.28; Theorem4.12 pp.22–23 and Theorem4.27 p.29; §3.1 p.13.
Daniel Dugger, Multiplicative structures on homotopy spectral sequences I · §5 Proposition5.1 and Proposition5.3 p.11; §6 Theorem6.1 p.13, Theorem6.2 and Proposition6.4 p.14; proofs pp.16–17