Towards an Homological Generalization of the Direct Summand Theorem

TL;DR

This paper introduces a homological framework for the Direct Summand Theorem, proposing new conjectures, proving their relations, and providing a novel proof of the theorem in specific cases using advanced algebraic techniques.

Contribution

It formulates the Socle-Parameter conjecture, establishes its equivalence to the DSC, and offers a new proof of the DSC in equicharacteristic cases, advancing homological understanding.

Findings

Weak SPC is equivalent to the DSC.

Strong SPC holds when parameter multiplicity ≤ 2.

New proof of DSC in equicharacteristic case.

Abstract

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.