[Paper Review] $\eta$-periodic motivic stable homotopy theory over Dedekind domains

Tom Bachmann|arXiv (Cornell University)|Jun 3, 2020

Algebraic structures and combinatorial models26 references4 citations

TL;DR

This paper establishes $η$-periodic motivic stable homotopy theory over mixed-characteristic Dedekind schemes where 2 is invertible, extending key motivic spectra like $ω$-K-theory and Witt cohomology to such bases. By leveraging framed transfers and $η$-localization, it proves base change invariance and lifts the fundamental fiber sequence from [BH20] to arbitrary Dedekind schemes, enabling computation of $η$-periodized algebraic cobordism groups.

ABSTRACT

We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of $\\eta$-periodic motivic stable homotopy theory established in [arxiv:2005.06778] from fields to arbitrary base schemes, and use this to determine (among other things) the $\\eta$-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1/2.

Motivation & Objective

• To extend $η$-periodic motivic spectra—such as $ω$-K-theory, Witt cohomology, and higher Chow–Witt groups—from fields to mixed-characteristic Dedekind schemes.
• To establish base change invariance for these spectra over Dedekind domains where 2 is invertible, enabling integrality arguments in arithmetic geometry.
• To lift the fundamental $η$-periodic fiber sequence from [BH20] to arbitrary base schemes, extending results on algebraic cobordism and $SL$-cobordism.
• To provide explicit, computationally useful descriptions of spectra like $ω$-K-theory via sheaf-theoretic and framed transfer techniques.

Proposed method

• Uses the equivalence between motivic spectra and framed spectra ($\mathcal{SH}^{S^1\mathrm{fr}}(D) \simeq \mathcal{SH}^{\mathrm{fr}}(D)$) to simplify $η$-localization and avoid $\mathbb{P}^1$-stabilization.
• Applies framed transfers and the Hopf map $\eta$ in $\mathcal{SH}^{S^1\mathrm{fr}}(D)$ to define $\eta$-periodic $S^1$-spectra and relate them to known spectra.
• Employs the Gersten conjecture for Witt rings over discrete valuation rings (via [Gei04]) to prove strict $\mathbb{A}^1$-invariance of the Witt sheaf $\underline{W}$.
• Uses base change detection via pullback to residue fields (from [BH21, Prop. B.3]) to verify equivalences and connectivity in $\mathcal{SH}(D)$.
• Leverages stability under base change of $\mathrm{H}\mathbb{Z}$ (Spitzweck) and Jacobson's results on Witt rings to construct and compare spectra.
• Applies realizations and localization at 2 and $\mathbb{Z}[1/2]$ to reduce problems to known cases over $\mathbb{R}$ and $\mathbb{Q}$.

Experimental results

Research questions

• RQ1 Can the $η$-periodic motivic stable homotopy theory fiber sequence from [BH20] be extended from fields to arbitrary Dedekind schemes with $1/2$?
• RQ2 Are the spectra $\mathrm{kw}$, $\mathrm{HW}$, $\mathrm{H}_{W}\mathbb{Z}$, and $\underline{K}^W$ stable under base change over Dedekind domains where $2$ is invertible?
• RQ3 What is the structure of the $η$-periodized algebraic $\mathrm{SL}$-cobordism and symplectic cobordism groups over mixed-characteristic Dedekind schemes?
• RQ4 Can the $η$-periodic spectra be described explicitly via framed transfers and sheaf-theoretic data over such bases?

Key findings

• The fundamental $η$-periodic fiber sequence $\mathbbm{1}[\eta^{-1}]_{(2)} \to \mathrm{kw}_{(2)} \to \Sigma^4 \mathrm{kw}_{(2)}$ holds in $\mathcal{SH}(D)$ for any Dedekind scheme $D$ with $1/2$, extending [BH20] to general bases.
• The spectrum $\underline{K}^W$ has homotopy sheaves $\underline{\pi}_0(\underline{K}^W)_* = \underline{I}^*$ and $\underline{\pi}_i(\underline{K}^W)_* = 0$ for $i \neq 0$, confirming strict $\mathbb{A}^1$-invariance of $\underline{W}$.
• All spectra in the list—including $\mathrm{kw}$, $\mathrm{HW}$, $\mathrm{H}_{W}\mathbb{Z}$, $\underline{K}^W$, and $\mathrm{H}\tilde{\mathbb{Z}}$—are cellular in $\mathcal{SH}(S)$ for any Dedekind scheme $S$ with $1/2$, as shown via base change and cellularity of $\mathrm{MSp}$.
• The $\mathrm{kw}^{*}_{(2)}\mathrm{kw}$-algebra has a non-commutative structure with $\varphi\beta = 9\beta\varphi + 8$, generalizing the known case over $\mathbb{Q}$.
• The smash product $\mathrm{HW} \wedge \mathrm{HW}_{(2)}$ is identified as $\bigvee_{n \geq 0} \Sigma^{4n} \mathrm{HW}/8n$, with generators satisfying $x_m x_n = \binom{m+n}{n} x_{m+n}$.
• The $\eta$-periodized $\mathrm{MSL}$-spectrum satisfies $\mathrm{MSp}/(y_1,y_3,\dots) \simeq \mathrm{MSL}$ and $\mathrm{MSL}/(y_4,y_6,\dots) \simeq \mathrm{kw}$, with the latter an equivalence after pullback to fields.