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 H-m(i) (T (R)) are concentrated in degree - n, where m is an irrelevant maximal ideal of R.
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