On the equality of de Rham depth and formal grade in characteristic zero

Abstract

Let $Y \subset \mathbb{P}^n_k$ be a non-singular proper closed subset of projective $n$-space over a field $k$ of characteristic zero and let $I \subset R=k[x_0, \ldots, x_n]$ be the homogeneous defining ideal of $Y$. We show that in this case, the de Rham depth of $Y$ is the same as the so-called formal grade of $I$ in $R$.