Eulerian graded $\mathscr{D}$-modules

Let $R=K[x_1,\dots,x_n]$ with $K$ a field of arbitrary characteristic and $\mathscr{D}$ be the ring of differential operators over $R$. Inspired by Euler formula for homogeneous polynomials, we introduce a class of graded $\mathscr{D}$-modules, called Eulerian graded $\mathscr{D}$-modules. It is proved that a vast class of $\mathscr{D}$-modules, including all local cohomology modules $H^{i_1}_{J_1}\ldots H^{i_s}_{J_s}(R)$ where $J_1,\dots,J_s$ are homogeneous ideals of $R$, are Eulerian. As an application of our theory of Eulerian graded $\mathscr{D}$-modules, we prove that all socle elements of each local cohomology module $H^{i_0}_{\mathfrak{m}}H^{i_1}_{J_1}\cdots H^{i_s}_{J_s}(R)$ must be in degree $-n$ in all characteristic. This answers a question raised in [12]. It is also proved that graded $F$-modules are Eulerian and hence the main result in [12] is recovered. An application of our theory of Eulerian graded $\mathscr{D}$-modules to the graded injective hull of $R/P$, where $P$ is a homogeneous prime ideal of $R$, is discussed as well.

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Published 25 July 2014