Note

On E-infinity- and DAlg-Descendability

Author: Wei Yang · Added: September 3, 2026

Article date: August 30, 2026.

Abstract

For a map $R\to S$ of $\mathbb{E}_\infty$ -rings, its $n$ th finite Amitsur stage is

\operatorname{Tot}^n\bigl(S^{\otimes_R(\bullet+1)}\bigr).

Antieau and Stefanich asked five questions about multiplicative retractions from these finite stages. We answer all five.

For every prime $p$ and every $k\geq 2$ , the restriction map

C^*(BC_{p^k};\mathbb{F}_p)\longrightarrow C^*(BC_p;\mathbb{F}_p)

is descendable but not $\mathbb{E}_\infty$ -descendable, and hence is not $\mathrm{DAlg}$ -descendable; for $k=1$ the map is the identity. Thus ordinary and $\mathbb{E}_\infty$ -descendability differ even for cochains on finite cyclic $p$ -groups.

Complexification $KO\to KU$ is $\mathbb{E}_\infty$ -descendable: the unit

KO\longrightarrow \operatorname{Tot}^5\bigl(KU^{\otimes_{KO}(\bullet+1)}\bigr)

admits a retraction in $\mathrm{CAlg}_{KO}$ . The units at stages $0$ , $1$ , and $2$ do not split in $\mathrm{CAlg}_{KO}$ , and the unit at stage $3$ does not split even as an $\mathbb{E}_1$ - $KO$ -algebra map. Thus the least $\mathbb{E}_\infty$ -splitting stage is either $4$ or $5$ .

Let $A$ be a Noetherian $\mathbb{F}_p$ -algebra of finite Krull dimension, put $d_A=\max\{1,\dim A\}$ , and let $A\to B$ be a faithfully flat map of discrete $\mathbb{F}_p$ -algebras. Without any finite-type hypothesis on $B$ , the unit admits a retraction in derived commutative $A$ -algebras

\operatorname{Tot}^{d_A+1}\bigl(B^{\otimes_A(\bullet+1)}\bigr) \longrightarrow A;

specializing to $\mathbb{F}_p[x]\to\mathbb{F}_p[x^{1/p^\infty}]$ gives a retraction at stage $2$ . The proof uses the first two natural layers of a free derived symmetric algebra, a two-term induced resolution over the Frobenius skew-polynomial ring $A[F]$ , and integral $\mathrm{DAlg}$ flattening.