Abstract We present a more general homological characterization of the direct summand theorem (DST). Specifically, we state two new conjectures: the socle-parameter conjecture (SPC) in its weak and strong forms. We give a proof for the weak form by showing that it is equivalent to the DST. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller than or equal to two. Finally, we present a new proof of the DST in the equicharacteristic case, based on the techniques thus developed.

中文翻译：

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对直接被加数定理的同调推广

摘要 我们提出了直接被加数定理 (DST) 的更一般的同调特征。具体来说，我们陈述了两个新的猜想：弱和强形式的社会参数猜想（SPC）。我们通过证明它等价于 DST 来证明弱形式。此外，我们在参数的多重性小于或等于 2 的情况下证明了 SPC 的强形式。最后，基于由此开发的技术，我们在等特征情况下提出了 DST 的新证明。