André-Quillen homology of Rees algebras and extended Rees algebras

Abstract

Let $(A,\mathfrak{m})$ be an excellent local complete intersection ring and let $I=(a_1,\dots ,a_r)$ be an ideal of positive height. Let $\mathcal{R}(I)=A[It]$ be the Rees algebra of $I$. Consider the map $\psi :S=A[X_1,\dots ,X_r]\to \mathcal{R}(I)$ which maps $X_i\mapsto a_{i}t$ for all $i$. Let $J=\ker \psi$ and let $H_{\ast }(J)$ 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 $A\to \mathcal{R}(I)$. We prove an analogous result for the extended Rees algebra $\hat{\mathcal{R}}=A[It,t^{-1}]$. 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.

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