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A unit-induced integral field coefficient zigzag

PDF p. 17: Corollary 6.9; integral Chern and support comparison

Statement

Fix a perfect field $k$ . Write $H_{S p i}$ for Spitzweck's integral field spectrum, $H_{A} = s^{0} K G L_{k}$ , and $U = s^{0} 1_{k}$ . There is a specified zigzag of unital motivic ring equivalences $H_{S p i} \overset{u_{S p i}}{\leftarrow} U \overset{u_{K}}{\to} H_{A} .$ Both arrows are induced by units and the zeroth-slice functor. Their integral equivalence, including in characteristic $p$ , is justified below using the canonical rank construction over $Spec Z$ and base change, rather than inferring it from a product-compatible field coefficient model. The shifted graded coefficient maps preserve products, units and the unit-adjoint reduced $P^{1}$ class, with the same Tate-coordinate convention on both sides.

Statement references

Exact primary unit, Picard and smooth-purity inputs for field compatibility

Facts & Assumptions

[F1]

Exact primary unit, Picard and smooth-purity inputs for field compatibility imports BEM Corollary9.7, which gives the integral E∞ unit equivalence $u_{K}$ and the base-change equivalence $g^{*} s^{0} 1_{Z} ≃ s^{0} 1_{k}$ , for $g : Spec k \to Spec Z$ . No prime is inverted.

[F2]

Bachmann, The very effective covers of KO and KGL over Dedekind schemes, Theorem1.1(1), Example1.3 and Lemma2.1, give over $Z$ the canonical map $Σ_{f r}^{\infty} V e c t ≃ f_{0} K G L$ and the equivalence induced by rank $(Σ_{f r}^{\infty} V e c t) / β ⟶ Σ_{f r}^{\infty} Z ≃ H_{S p i} .$ Here $f_{1} H_{S p i} = 0$ holds over the Dedekind base $Z$ as explicitly stated with Theorem1.1; the 1/2 assumption belongs only to the separate KO assertion. Bott periodicity identifies the displayed cofiber with $s^{0} K G L_{Z}$ .

[F3]

Hoyois, The localization theorem for framed motivic spaces, Theorem21 and proof, identify $Σ_{f r}^{\infty} Z$ with Spitzweck HZ by a natural E∞ equivalence, adjoint to the actual ring isomorphism on constant integers. It sends the rank-one constant to the ring unit, and is stable under arbitrary base change. Spitzweck Remark11.2 supplies the canonical unit-induced E∞ map $u_{S p i} : s^{0} 1 \to s^{0} H_{S p i} ≃ H_{S p i}$ over fields. The integral zero-slice/effectivity facts also hold over $Z$ by [F2]–[F3].

Proof

Given: The stated integral spectra and maps, the existing universes.

Goal: Establish the actual first arrow as an integral equivalence with its unit, independently of an unread characteristic range of the classical sphere-slice theorem.

1.1F2F3

Over $Z$ , use [F2] to obtain the equivalence $a : s^{0} K G L_{Z} \to H_{S p i, Z}$ . Its construction is rank on the framed vector-bundle model followed by the equivalence of [F3]. The vector-bundle tensor unit is $O$ , whose rank is the constant1. Consequently the composite of the sphere unit through $f_{0} K G L$ , its Bott cofiber, and $a$ is the actual unit of $H_{S p i}$ , by the adjunction construction in [F3]. This is a unit triangle of actual maps, not an inference that an arbitrary equivalence preserves units.

2.1F1F2F3step 1.1

Let $b : s^{0} 1_{Z} \to H_{S p i, Z}$ be the map induced by that unit. Because $H_{S p i, Z}$ is effective and has $f_{1} = 0$ , maps from $s^{0} 1$ to it are identified with maps from the effective unit: the $f_{1} 1$ term is orthogonal to a zero-slice target. Thus the triangle in step 1.1 identifies $b$ with $a \circ u_{K, Z}$ . Both factors are integral equivalences by [F1]–[F2], so $b$ is an equivalence. Its E∞ structure is the one supplied by multiplicative slices and the actual unit; no multiplicative structure on an arbitrary chosen Bott quotient is required for this inference.

3.1F1F2F3step 2.1

Pull this unit triangle back along $g$ . Spitzweck HZ is absolute, and BEM Corollary9.7 identifies $g^{*} s^{0} 1_{Z}$ with $s^{0} 1_{k}$ . Naturality of units and slice comparison identifies the resulting map with the canonical $u_{S p i, k}$ in [F3]: both factor the same unit into the field zero-slice target. Hence this particular map is an integral equivalence. Together with the integral $u_{K}$ of [F1] it gives the claimed zigzag. This argument never uses a characteristic-zero theorem to assert an integral characteristic- $p$ result.

4.1F1F3step 3.1∎

By BEM Remark 3.21 and Remark 3.23 in [F1], the normalized reduced $P^{1}$ class of a motivic ring $E$ is adjoint to $T \to E \otimes T$ induced by its unit. Since the two maps of the zigzag are the actual unital ring maps, they preserve this class. Spitzweck Proposition11.1/Lemma6.6 identify its original Picard normalization; the later Picard-space detection and EM comparison helpers therefore apply unchanged.

Depends on

Exact primary unit, Picard and smooth-purity inputs for field compatibility

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