  We present a more general (parametric-) homological characterization of the
Direct Summand Theorem. Specifically, we state two new conjectures: the
Socle-Parameter conjecture (SPC) in its weak and strong forms. We give a proof
for the week form by showing that it is equivalent to the Direct Summand
Conjecture (DSC), now known to be true after the work of Y. Andr\'{e}, based on
Scholze's theory of perfectoids. Furthermore, we prove the SPC in its strong
form for the case when the multiplicity of the parameters is smaller or equal
than two. Finally, we present a new proof of the DSC in the equicharacteristic
case, based on the techniques thus developed.

Commutative Algebra