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The product on graded motivic groups is the source cup product

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

Statement

Assume the Axiom of Choice. Under the actual graded comparison, the filtered multiplication induces the source cup product on the page2 groups. Explicitly it gives $E_{2}^{a, b} \otimes E_{2}^{c, e} ⟶ E_{2}^{a + c, b + e}$ corresponding to the motivic product of degree/weight $(a - b, - b)$ and $(c - e, - e)$ . The commutativity sign is $(- 1)^{(a + b) (c + e)}$ , equivalently the usual cohomological cup-product sign $(- 1)^{(a - b) (c - e)}$ .

Facts & Assumptions

[A1]

The Axiom of Choice and the fixed size conventions of the cited constructions are inherited through External actual graded algebra and multiplicative tower spectral sequence. They discharge that supplier’s local-realization premise at its first application below. The Axiom of Choice

Given: The Axiom of Choice and the fixed size conventions of the cited constructions, together with the actual filtered algebra and its source-induced graded algebra.

Goal: Identify products by actual multiplication maps, not by a coincidence of additive groups.

[F1]

Motivic filtration and image groups gives coherent filtered products. External actual graded algebra and multiplicative tower spectral sequence makes associated graded strong symmetric monoidal and supplies its actual source comparison.

[F2]

Actual cofiber triangles and source grading of the motivic tower specifies that comparison; Cohomological indexing of the actual graded groups gives its page2 degree labels.

Proof

1.1A1F1F2

Use the supplied realization under [A1]. Associated graded in [F1] sends filtered multiplication to actual maps $Q_{r} \land Q_{t} \to Q_{r + t}$ . Transport these maps along the source $γ_{r}, γ_{t}, γ_{r + t}$ ; the resulting maps on the shifted motivic complexes are their specified source products by [F1], not independently chosen pairings. After taking homotopy groups the degrees add: $r = - b, t = - e$ and homotopy degrees $- a - b, - c - e$ give target weight $- b - e$ and degree $- a - b - c - e$ , namely the page2 group at $(a + c, b + e)$ .

2.1F1F2step 1.1algebra∎

Symmetry of sphere smash in homotopy degrees $- a - b$ and $- c - e$ contributes $(- 1)^{(a + b) (c + e)}$ . The shifts $2 r$ and $2 t$ are even, and $a + b$ has the same parity as $a - b$ , likewise for $c, e$ . Hence the transported sign is exactly the displayed cohomological sign. Associativity and unit maps are those of the actual graded algebra in [F1], so these are the source cup products with all indicated laws.

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