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Base Change for Cochains on Finite-Type Spaces

Author: Wei Yang · Added: September 18, 2026

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Article date: September 18, 2026.

Abstract

We prove that cochains on a CW complex with finitely many cells in each dimension commute with extension of scalars between bounded-above $E_\infty$-rings. Neither connectivity nor finite relative Tor amplitude is assumed. The proof uses uniform presentation bounds over the Steenrod algebra to establish nilpotence for sequences of spectra with finite $p$-primary homotopy in intervals of fixed width. Perfect approximations then give a comparison over sphere Postnikov sections, from which the theorem follows by bounded-above dévissage. Two rational examples show that neither upper bound can be omitted: the comparison need not be surjective or injective on $\pi_0$, even when the other coefficient ring is $H\mathbb{Q}$. For fixed $A$ and $X$, we also prove that comparison with every commutative $A$-algebra is equivalent to perfectness of the $A$-module of chains on $X$. Without any upper bounds, the comparison remains valid after taking the inverse limit of the finite-skeleton comparisons.