Andr\'e-Quillen homology of Rees algebras and extended Rees algebras

TL;DR

This paper characterizes when the Rees algebra of an ideal over an excellent local complete intersection ring is a complete intersection using Andre9-Quillen homology and Koszul homology, and extends results to the associated extended Rees algebra.

Contribution

It provides a new homological criterion involving Andre9-Quillen and Koszul homology for the complete intersection property of Rees algebras, including the extended case.

Findings

Equivalence between the complete intersection property and vanishing of certain Andre9-Quillen homology groups.

Identification of conditions on Koszul homology modules for the Rees algebra.

Explicit computation of the rank of the first Koszul homology module in Cohen-Macaulay cases.

Abstract

Let $(A,\mathfrak{m})$ be an excellent local complete intersection ring and let $I = (a_1, \ldots, a_r)$ be an ideal of positive height. Let $\mathcal{R}(I) = A[It]$ be the Rees algebra of $I$. Consider the map $\psi \colon S = A[X_1, \ldots, X_r] \rightarrow \mathcal{R}(I)$ which maps $X_i \mapsto a_it$ for all $i$. Let $J = \ker \psi$ and let $H_*(J)$ be the Koszul homology of $J$. We prove that the following assertions are equivalent: (i) $\text{Proj} \ \mathcal{R}(I)$ is a complete intersection. (ii) (a) $D_3(\mathcal{R}(I)|A, \mathcal{R}(I))_n = 0$ for $n \gg 0$ and, (ii) (b) For $P \in \text{Proj} \ \mathcal{R}(I)$ we have $H_1(J)_P$ is a free $\mathcal{R}(I)_P$-module. Here $D_3(\mathcal{R}(I)|A, \mathcal{R}(I))$ is the third Andr\'e-Quillen homology of $\mathcal{R}(I)$ with respect to $A \rightarrow \mathcal{R}(I)$. We prove an analogous result for the extended Rees algebra $\widehat{\mathcal{R}} = A[It, t^{-1}]$. When $A$ is a Cohen-Macaulay domain (not necessarily a complete intersection) we compute that rank of $H_1(J)$ and hence compute its free locus.