On the Cohen-Macaulay property of the Rees algebra of the module of differentials

TL;DR

This paper investigates conditions under which the Rees algebra of the module of Kähler differentials over a complete intersection algebra is Cohen-Macaulay, focusing on the linear type property in characteristic zero.

Contribution

It establishes that for homogeneous complete intersections over a field of characteristic zero, the Rees algebra of the module of differentials is Cohen-Macaulay and of linear type under specific local embedding dimension conditions.

Findings

Rees algebra of differentials is Cohen-Macaulay under given conditions

Module of differentials is of linear type when Rees algebra is Cohen-Macaulay

Local embedding dimension constraints are crucial for the property

Abstract

Let $R$ be an algebra essentially of finite type over a field $k$ and let $\Omega_k(R)$ be its module of K\"ahler differentials over $k$. If $R$ is a homogeneous complete intersection and $\mathrm{char}(k)=0$, we prove that $\Omega_k(R)$ is of linear type whenever its Rees algebra is Cohen-Macaulay and locally at every homogeneous prime $\mathfrak{p}$ the embedding dimension of $R_{\mathfrak{p}}$ is at most twice its dimension.