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Negative coefficients of the actual EM projective-line model

Statement

Assume the Axiom of Choice and the fixed site and universe bounds. Let $S$ be a qcqs scheme over a field, and let $E_{S}$ be the actual non- $A^{1}$ EM projective-line spectrum in the supplied model. For every smooth qcqs $S$ -scheme $Y$ , every integer $a$ and every integer $b < 0$ , its actual coefficient group is zero: $E_{S}^{a, b} (Y) = 0.$ Here the bidegree convention is $E_{S}^{a, b} (Y) = π_{2 b - a} {Map}_{{S p}_{P^{1}} (S p)} (Σ_{P^{1}}^{\infty} Y_{+}, Σ_{P^{1}}^{b} E_{S})$ , using the spectrum-valued coefficient stabilization. The pointed projective line uses its infinity section. This asserts vanishing for the represented model itself, rather than merely assigning the symbol for a negative motivic complex the value zero.

Facts & Assumptions

Given: $S, Y, a, b$ , Choice and size bounds as stated.

Goal: Compute the negative twists of the actual coefficient model at level zero.

[F1]

An actual non-A¹ P¹-spectrum for the supplied EM complexes constructs an actual coefficient-valued $Ω$ - $P^{1}$ spectrum whose nonnegative levels are $A_{n} = Z (n)_{m o t} [2 n]$ . Its multiplication and restrictions are those of the actual EM complexes.

[F2]

The actual projective-line formula of External actual EM graded algebra, Picard class and complex PBF at weight zero says that pullback $A_{0} (Y) \to A_{0} (P_{Y}^{1})$ is an equivalence. Its composite with restriction to infinity is identity. This is an assertion about the actual nonnegative level $A_{0}$ , not about an unspecified negative level.

[F3]

External symmetric P¹ model, orientation criterion and fundamental stabilization gives the shift/evaluation rule for $Ω$ - $P^{1}$ spectra and their spectrum-valued coefficient stabilization. Evaluation of $Σ_{P^{1}}^{- m} E_{S}$ at level zero is $Ω_{P^{1}}^{m} A_{0}$ for $m \geq 1$ . On coefficient sheaves, $(Ω_{P^{1}} A) (Y) = fib (A (P_{Y}^{1}) \to A (Y))$ , with the arrow induced by infinity. The Axiom of Choice states the inherited Choice assumption.

Proof

1.1F1F2F3given

By [F2], restriction $A_{0} (P_{Y}^{1}) \to A_{0} (Y)$ is inverse to the displayed pullback equivalence. Its fiber is therefore contractible. This is natural for every $Y$ in the smooth qcqs site, so $Ω_{P^{1}} A_{0} = 0$ as an actual sheaf of spectra.

2.1F1F3step 1.1

Write $b = - m$ with $m \geq 1$ . By the shift/evaluation rule in [F3], the zeroth coefficient sheaf of $Σ_{P^{1}}^{b} E_{S}$ is $Ω_{P^{1}}^{m} A_{0}$ . Iterating the reduced-loop functor on the zero sheaf in step 1.1 gives zero for this sheaf. This uses the actual negative twists of the constructed spectrum.

3.1F3step 2.1∎

The suspension/evaluation adjunction identifies the mapping spectrum from $Σ_{P^{1}}^{\infty} Y_{+}$ to that twist with the evaluation of its zeroth coefficient sheaf on $Y$ , which is zero by step 2.1. All of its homotopy groups vanish, in particular degree $2 b - a$ for every $a$ . This proves the assertion without any dimension or Noetherian hypothesis on $S$ .

Depends on

Used by · 4

Dependency tree · 3 levels

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Sources

Elmanto and Morrow, Motivic cohomology of equicharacteristic schemes, arXiv 2309.08463v1 · Theorem 5.24(1), actual map (5.15), weight zero and projective-line case
Annala and Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · Notations 2.2.7–2.2.9 and coefficient-stabilization contract in the existing supplier