Lemma 92.11.3. Let A \to B be a ring map. Then \tau _{\geq -1}L_{B/A} is canonically quasi-isomorphic to the naive cotangent complex.
Proof. Consider P = A[B] \to B with kernel I. The naive cotangent complex \mathop{N\! L}\nolimits _{B/A} of B over A is the complex I/I^2 \to \Omega _{P/A} \otimes _ P B, see Algebra, Definition 10.134.1. Observe that in (92.11.1.2) we have already constructed a canonical map
c : \mathop{N\! L}\nolimits _{B/A} \longrightarrow \tau _{\geq -1}L_{B/A}
Consider the distinguished triangle (92.7.0.1)
L_{P/A} \otimes _ P^\mathbf {L} B \to L_{B/A} \to L_{B/P} \to (L_{P/A} \otimes _ P^\mathbf {L} B)[1]
associated to the ring maps A \to A[B] \to B. We know that L_{P/A} = \Omega _{P/A}[0] = \mathop{N\! L}\nolimits _{P/A} in D(P) (Lemma 92.4.7 and Algebra, Lemma 10.134.3) and that \tau _{\geq -1}L_{B/P} = I/I^2[1] = \mathop{N\! L}\nolimits _{B/P} in D(B) (Lemma 92.11.2 and Algebra, Lemma 10.134.6). To show c is a quasi-isomorphism it suffices by Algebra, Lemma 10.134.4 and the long exact cohomology sequence associated to the distinguished triangle to show that the maps L_{P/A} \to L_{B/A} \to L_{B/P} are compatible on cohomology groups with the corresponding maps \mathop{N\! L}\nolimits _{P/A} \to \mathop{N\! L}\nolimits _{B/A} \to \mathop{N\! L}\nolimits _{B/P} of the naive cotangent complex. We omit the verification. \square
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