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Actual cofiber triangles and source grading of the motivic tower

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

Statement

Assume the Axiom of Choice. For each qcqs scheme $X$ over the standing field and $r \geq 0$ , the actual natural maps of the supplied tower give a cofiber triangle $F_{m o t}^{r + 1} K (X) ⟶ F_{m o t}^{r} K (X) ⟶ Z (r)_{m o t} (X) [2 r] .$ The last arrow is the actual quotient map followed by the source equivalence $γ_{r}$ . Thus it is a map of functor diagrams, not an identification from equality of homotopy groups. Extending motivic complexes to negative weights by zero identifies the negative graded terms too.

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 completed tower, its constant nonpositive extension and the completed motivic complex functor.

Goal: Specify the actual quotient maps and comparison used by every subsequent calculation.

[F1]

Motivic filtration and image groups specifies the source tower by prime-field restriction and its constant nonpositive stages. Motivic complexes and cohomology likewise uses the prime-field EM complexes.

[F2]

External actual graded algebra and multiplicative tower spectral sequence supplies $γ_{r}$ for those very objects and the zero convention for negative weights.

Proof

1.1A1F1F2construct

Use the supplied realization under [A1]. Form $Q_{r} = cofib (F^{r + 1} \to F^{r})$ in the stable category and compose its canonical quotient map with $γ_{r}$ from [F2]. By [F1], both the stage diagram and complex diagram are the actual restrictions of the source diagrams. Their natural equivalence therefore gives precisely the displayed cofiber triangle and its naturality in $X$ .

2.1F1F2step 1.1∎

For $r < 0$ , the constant extension makes $F^{r + 1} \to F^{r}$ an equivalence, including $r = - 1$ . Hence $Q_{r} = 0$ . This agrees with the explicit source zero complex in [F2]. No negative-weight cohomology is assumed to have been previously defined: in subsequent full-plane formulas its symbol means this zero extension.

Depends on

Used by · 4

Dependency tree · 3 levels

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