Let R be a polynomial ring in n indeterminates with coefficients in the field K of characteristic p > 0 and 𝒟 be the ring of differential operators over R. In this paper, we introduce the notion of generalized Eulerian 𝒟-modules for characteristic p > 0 and establish their properties. We show that if 𝒯 is any graded Lyubeznik functor on the category of modules over R, then 𝒯 (R) is a generalized Eulerian 𝒟-module. As a consequence, we prove that all socle elements of module H𝔪i(𝒯 (R)) are concentrated in degree − n, where 𝔪 is an irrelevant maximal ideal of R.

中文翻译：

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char p > 0 中的广义欧拉 𝒟 模

让R是一个多项式环n与领域中的系数不确定ķ有特色的p > 0和𝒟是微分算子的环R. 在本文中，我们介绍了广义欧拉的概念𝒟- 特征模块p > 0并确定它们的属性。我们证明如果𝒯是模块类别上的任何分级 Lyubeznik 函子R,然后𝒯 (R)是广义的欧拉𝒟-模块。因此，我们证明了 module 的所有 socle 元素H𝔪一世(𝒯 (R))集中在程度 - n,在哪里𝔪是一个无关的最大理想R.