The Stacks project

Proof. We use the description of $E$ as the cone on $L_{B/R} \otimes _ B^\mathbf {L} (A \otimes _ R B) \to L_{A \otimes _ R B/A}$. By Lemma 92.13.3 the canonical truncations $\tau _{\geq -2}L_{B/R}$ and $\tau _{\geq -2}L_{A \otimes _ R B/A}$ are computed by the Lichtenbaum-Schlessinger complex (92.13.0.1). These isomorphisms are compatible with functoriality (Remark 92.13.4). Thus in this proof we work with the Lichtenbaum-Schlessinger complexes.

Choose a polynomial algebra $P$ over $R$ and a surjection $P \to B$. Choose generators $f_ t \in P$, $t \in T$ of the kernel of this surjection. Let $Rel \subset F = \bigoplus _{t \in T} P$ be the kernel of the map $F \to P$ which maps the basis vector corresponding to $t$ to $f_ t$. Set $P_ A = A \otimes _ R P$ and $F_ A = A \otimes _ R F = P_ A \otimes _ P F$. Let $Rel_ A$ be the kernel of the map $F_ A \to P_ A$. Using the exact sequence

and standard short exact sequences for Tor we obtain an exact sequence

Note that $P_ A \to A \otimes _ R B$ is a surjection whose kernel is generated by the elements $1 \otimes f_ t$ in $P_ A$. Denote $TrivRel_ A \subset Rel_ A$ the $P_ A$-submodule generated by the elements $(\ldots , 1 \otimes f_{t'}, 0, \ldots , 0, - 1 \otimes f_ t \otimes 1, 0, \ldots )$. Since $TrivRel \otimes _ R A \to TrivRel_ A$ is surjective, we find a canonical exact sequence

The map of Lichtenbaum-Schlessinger complexes is given by the diagram

Note that vertical maps $-1$ and $-0$ induce an isomorphism after applying the functor $A \otimes _ R - = P_ A \otimes _ P -$ to the source and the vertical map $-2$ gives exactly the map whose cokernel is the desired Tor module as we saw above. $\square$