Note
Adams-type Maps and Transfinite Factorizations of Ring Spectra
Article date: September 2026.
Abstract
We study three questions about Adams-type maps of ring spectra. We construct a descent-flat associative map which is not Adams-type, although its comodule-valued homology theory is adapted. We give an associative map which admits no continuous transfinite factorization into descent-flat maps, and show that the zero augmentation of the free commutative sphere algebra on a degree-zero generator admits no such factorization into Adams-type commutative maps. The latter obstruction uses primitive classes in bar homology, a Hopf structure on the relative bar filtration, and descent along coideal subalgebras. We also give an Adams-type map of ordinary commutative rings and a module with free base-change homology which cannot be expressed as a filtered colimit of perfect modules with projective base-change homology. Positive results hold for almost perfect modules over connective sources with discrete coherent targets, and for maps represented by strict commutative differential graded algebras.