André-Quillen homology of Rees algebras and extended Rees algebras

TL;DR

The paper establishes a precise correspondence between the geometric property that Proj$\mathcal{R}(I)$ (and Proj$\widehat{\mathcal{R}}(I)$) is a complete intersection and sophisticated homological data encoded by André-Quillen homology and Koszul homology. By relating $D_3(\mathcal{R}(I)|A,\mathcal{R}(I))_n$ and the freeness of $H_1(J)_P$, the authors give equivalent conditions for CI-ness in terms of high-degree vanishing and local freeness, with parallel results for extended Rees algebras. They also provide rank formulas for $H_1(J)$ and $H_1(\widehat{J})$ in Cohen-Macaulay domains, bounds on minimal generators of these modules, and polynomial-growth statements for the associated homology lengths. Collectively, these results give a robust toolkit to diagnose complete intersections in Rees-type algebras via homological invariants and to study the impact of base-change and filtration structures on these invariants. The work advances understanding of how André-Quillen theory governs the defining equations of Rees objects and their geometric properties, with implications for resolutions of singularities and invariants in commutative algebra.

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 $ψ\colon S = A[X_1, \ldots, X_r] \rightarrow \mathcal{R}(I)$ which maps $X_i \mapsto a_it$ for all $i$. Let $J = \ker ψ$ 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 André-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.

Theorems & Definitions (52)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Proposition 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9