Lemma 10.134.4 (Jacobi-Zariski sequence). Let A \to B \to C be ring maps. Choose a presentation \alpha : A[x_ s, s \in S] \to B with kernel I. Choose a presentation \beta : B[y_ t, t \in T] \to C with kernel J. Let \gamma : A[x_ s, y_ t] \to C be the induced presentation of C with kernel K. Then we get a canonical commutative diagram
\xymatrix{ 0 \ar[r] & \Omega _{A[x_ s]/A} \otimes C \ar[r] & \Omega _{A[x_ s, y_ t]/A} \otimes C \ar[r] & \Omega _{B[y_ t]/B} \otimes C \ar[r] & 0 \\ & I/I^2 \otimes C \ar[r] \ar[u] & K/K^2 \ar[r] \ar[u] & J/J^2 \ar[r] \ar[u] & 0 }
with exact rows. We get the following exact sequence of homology groups
H_1(\mathop{N\! L}\nolimits _{B/A} \otimes _ B C) \to H_1(L_{C/A}) \to H_1(L_{C/B}) \to C \otimes _ B \Omega _{B/A} \to \Omega _{C/A} \to \Omega _{C/B} \to 0
of C-modules extending the sequence of Lemma 10.131.7. If \text{Tor}_1^ B(\Omega _{B/A}, C) = 0 and \text{Tor}_2^ B(\Omega _{B/A}, C) = 0, then H_1(\mathop{N\! L}\nolimits _{B/A} \otimes _ B C) = H_1(L_{B/A}) \otimes _ B C.
Proof.
The precise definition of the maps is omitted. The exactness of the top row follows as the \text{d}x_ s, \text{d}y_ t form a basis for the middle module. The map \gamma factors
A[x_ s, y_ t] \to B[y_ t] \to C
with surjective first arrow and second arrow equal to \beta . Thus we see that K \to J is surjective. Moreover, the kernel of the first displayed arrow is IA[x_ s, y_ t]. Hence I/I^2 \otimes C surjects onto the kernel of K/K^2 \to J/J^2. Finally, we can use Lemma 10.134.2 to identify the terms as homology groups of the naive cotangent complexes.
The final assertion is a statement in homological algebra. Recall that \mathop{N\! L}\nolimits _{B/A} = (N^{-1} \to N^0) is a two term complex of B-modules with N^0 free and cohomology modules H^0 = \Omega _{B/A} and H^{-1} = H_1(L_{B/A}). Write M \subset N^0 for the image of the differential. If \text{Tor}_1^ B(H^0, C) = 0, then we have an exact sequence
0 \to M \otimes _ B C \to N^0 \otimes _ B C \to N^0 \otimes _ B C \to 0
Since N^0 is free, we also see that \text{Tor}_2^ B(H^0, C) = \text{Tor}_1^ B(M, C). Hence if \text{Tor}_2^ B(H^0, C) = 0 then we also have an exact sequence
0 \to H^{-1} \otimes _ B C \to N^{-1} \otimes _ B C \to M \otimes _ B C \to 0
Putting everything together we see that if \text{Tor}_1^ B(H^0, C) = 0 and \text{Tor}_2^ B(H^0, C) = 0, then H^{-1} \otimes _ B C is the kernel of N^{-1} \otimes _ B C \to N^0 \otimes _ B C as desired.
\square
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