  In this paper, we describe a novel way of identifying Adams spectral sequence
$E_2$-terms in terms of homological algebra of quiver representations. Our
method applies much more broadly than the standard techniques based on
descent-flatness, bearing on a varied array of ring spectra. In the particular
case of $p$-local integral homology, we are able to give a decomposition of the
$E_2$-term, describing it completely in terms of the classical Adams spectral
sequence. In the appendix, which can be read independently from the main body
of the text, we develop functoriality of deformations of $\infty$-categories of
the second author and Patchkoria.
