Annihilators of $D$-modules in mixed characteristic

Abstract

Let $R$ be a polynomial or formal power series ring with coefficients in a DVR $V$ of mixed characteristic with a uniformizer $\pi$. We prove that the $R$-module annihilator of any nonzero $\D(R,V)$-module is either zero or is generated by a power of $\pi$. In contrast to the equicharacteristic case, nonzero annihilators can occur; we give an example of a top local cohomology module of the ring ${\mathbb{Z}}_2[[x_0,\dots ,x_5]]$ that is annihilated by $2$, thereby answering a question of Hochster in the negative.

Some changes according to the referee's comments

References in corpus (1)

• Local Cohomology and Matlis duality

Cited by in corpus (2)

• The Second Vanishing Theorem for Local Cohomology Modules
• Local cohomology -- an invitation