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

TL;DR

This paper proves that for smooth projective varieties over a field of characteristic zero, the de Rham depth equals the formal grade of their defining ideal, linking geometric and algebraic invariants.

Contribution

It establishes the equality of de Rham depth and formal grade for non-singular projective varieties in characteristic zero, clarifying their relationship.

Findings

De Rham depth equals formal grade for smooth projective varieties.

Provides a bridge between geometric and algebraic invariants.

Enhances understanding of the structure of defining ideals in algebraic geometry.

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 $R$.