Note

A Counterexample to Tate Blueshift for E-infinity Rings

Author: Wei Yang · Added: August 24, 2026

Article date: August 11, 2026.

Abstract

At the prime $2$ , we construct a connective $\mathbb{E}_\infty$ -ring $C$ such that

K(1) \wedge C \simeq 0 \qquad\text{and}\qquad H\mathbb{Q} \wedge C^{tC_2} \not\simeq 0.

Thus $K(1)$ -acyclicity of a connective $\mathbb{E}_\infty$ -ring does not imply rational acyclicity of its $C_2$ -Tate construction. The example is the versal $ku_{(2)}$ -algebra obtained by adjoining a degree- $4$ class $b$ and a nullhomotopy of $2b-u^2$ , where $u$ is the Bott class.

The proof uses an affine weight filtration whose layers are extended powers of the mod- $2$ Moore spectrum. McClure’s finite-coefficient Bockstein calculation, followed by a Milnor inverse-limit argument, bounds the denominators detected by completed periodic $K$ -theory. A Laurent-series model for the Tate constructions of the finite Postnikov sections then gives, for every $m\geq0$ ,

\operatorname{ord}\!\left(1\in \pi_0\bigl((\tau_{\leq 4m}C)^{tC_2}\bigr)\right) \geq 2^{s_2(m)+1}.

Here $s_2(m)$ is the binary digit sum of $m$ . These orders are unbounded, which implies that $C^{tC_2}$ is not rationally acyclic.