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 $\mathcal{D}(R,V)$-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 $\mathbb{Z}_2 [[x_0, \dotsc, x_5]]$ that is annihilated by $2$, thereby answering a question of Hochster in the negative. The same example also provides a counterexample to a conjecture of Lyubeznik and Yildirim.