The Stacks project

Proof. Let $f_1, \ldots , f_ r \in I$ be a Koszul regular sequence generating $I$. Consider the ring map $\mathbf{Z}[x_1, \ldots , x_ r] \to A$ sending $x_ i$ to $f_ i$. Since $x_1, \ldots , x_ r$ is a regular sequence in $\mathbf{Z}[x_1, \ldots , x_ r]$ we see that the Koszul complex on $x_1, \ldots , x_ r$ is a free resolution of $\mathbf{Z} = \mathbf{Z}[x_1, \ldots , x_ r]/(x_1, \ldots , x_ r)$ over $\mathbf{Z}[x_1, \ldots , x_ r]$ (see More on Algebra, Lemma 15.30.2). Thus the assumption that $f_1, \ldots , f_ r$ is Koszul regular exactly means that $B = A \otimes _{\mathbf{Z}[x_1, \ldots , x_ r]}^\mathbf {L} \mathbf{Z}$. Hence $L_{B/A} = L_{\mathbf{Z}/\mathbf{Z}[x_1, \ldots , x_ r]} \otimes _\mathbf {Z}^\mathbf {L} B$ by Lemmas 92.6.2 and 92.14.1. $\square$