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The EM weight-one Picard normalization is positive identity

PDF p. 9: Actual low-weight motivic projections and their normalization; PDF p. 10: Lemma 5.2, smooth-case first-Chern normalization

Statement

Assume the Axiom of Choice and the fixed compatible universe and site-size bounds of the supplied constructions.

On Sm_k for any field k, write S=Ω∞Fil¹K and identify Ω∞gr¹K with $B G_{m}$ by the canonical EM c₁ comparison. Let κ: $B G_{m}$ →S lift L↦[L]−1. Then the based map q₁κ: $B G_{m}$ → $B G_{m}$ is identity as a map of Nisnevich stacks, hence on both the Picard classes and their automorphism units. This identifies the sign of the actual projection and not merely its target group.

Facts & Assumptions

[A1]

Assume the Axiom of Choice as formulated in The Axiom of Choice and the stated universe and site-size bounds. They are inherited when applying the determinant/one-type input in [F2] at step 1.2. The stack comparison is on a universe-bounded smooth big Nisnevich site; derived sections use the compatible small sites. The Laurent-unit calculation and the integer sign test add no further choice principle.

Given: The normalized maps and the field hypothesis in the Statement. All maps and homotopies below are in pointed Nisnevich sheaves of spaces on ${S m}_{k}$ . Assume Choice and the universe and site-size bounds in [A1].

Goal: Identify the chosen lift with the normalized lift, then prove the identity on the whole line-bundle stack.

[F1]

External EM low-weight maps with fixed positive normalization supplies the particular pointed lift $a$ of $L \mapsto 1 - [L]$ , the canonical identification $Ω^{\infty} {gr}^{1} K ≃ B G_{m}$ , and $q_{1} (- a) (O (- 1)) = c_{1} (O (- 1))$ . It also fixes $Pic (P_{k}^{1}) = Z$ with generator $c_{1} (O (1))$ . This fact does not assert that an arbitrary lift is $- a$ .

[F2]

External determinant and pointed one-type inputs supplies the fully faithful classification of pointed connected one-types by sheaves of groups. In particular pointed maps $B G_{m} \to B G_{m}$ correspond to sheaf-group endomorphisms of $G_{m}$ .

[F3]

Weight zero is the locally constant rank function, including its weight-zero comparison and stage computation, identifies $q_{0}$ with rank and uses the actual fiber sequence ${Fil}^{1} K \to K \to {gr}^{0} K$ . On smooth schemes the comparison of [F1] identifies the zeroth space of the last term with the locally constant integer sheaf: its positive homotopy groups are negative cohomology groups of the degree-zero constant sheaf and vanish.

Proof

1.1F3given

Put $T = Ω^{\infty} {gr}^{0} K$ and $f : B G_{m} \to Ω^{\infty} K$ , $L \mapsto [L] - 1$ . By [F3], $T$ is the discrete locally constant integer sheaf and $q_{0} f$ is the zero map, since every line bundle has rank one. Since $Ω^{\infty}$ preserves fibers, $S$ is the fiber over zero of $Ω^{\infty} K \to T$ . Applying the pointed mapping-space functor from $B G_{m}$ identifies the space of lifts of this fixed $f$ with the space of paths from $q_{0} f$ to zero in ${Map}_{*} (B G_{m}, T)$ . The latter mapping space is discrete because $T$ is zero-truncated. The path space in question is therefore contractible.

1.2F2givenA1

By [F2], $q_{1} κ$ corresponds to an endomorphism $h : G_{m} \to G_{m}$ . The sheaf $G_{m}$ is represented on ${S m}_{k}$ by $Spec k [t, t^{- 1}]$ , so Yoneda identifies its underlying natural map with a unit of $k [t, t^{- 1}]$ . Every such unit is $b t^{n}$ for $b \in k^{\times}$ and $n \in Z$ : the highest and lowest exponents of a product add over a field, and a product equal to one forces both factors to have a single exponent. Preservation of the identity gives $b = 1$ . Hence $h (u) = u^{n}$ , and the associated pushforward of torsors sends a line bundle $L$ to $L^{\otimes n}$ . This Laurent-unit calculation uses the field hypothesis.

2.1F1step 1.1

Inversion in the infinite loop space sends the particular lift $a$ in [F1] to a pointed lift $- a$ of $f$ . Both the specified $κ$ and $- a$ are thus lifts of the same fixed map. By step 1.1 they agree up to pointed homotopy over $Ω^{\infty} K$ . Consequently the normalized evaluation from [F1] applies to $q_{1} κ$ , giving $q_{1} κ (O (- 1)) = c_{1} (O (- 1))$ .

3.1F1F2step 2.1step 1.2∎

Apply step 1.2 to $O (- 1)$ on $P_{k}^{1}$ . Step 2.1 says its image is $O (- 1)$ , while step 1.2 gives $O (- n)$ . The specified infinite cyclic Picard group in [F1] therefore forces $n = 1$ . Full faithfulness in [F2] identifies the whole pointed stack map $q_{1} κ$ with the identity, including its action $u \mapsto u$ on automorphism units. The sign is inherited from the actual normalized projection $1 - [O (- 1)] \mapsto + c_{1} (O (1))$ , rather than chosen through a new target identification.

Depends on

Used by · 1

Actual first-Chern and determinant maps of the first EM layer

Dependency tree · 3 levels

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