Independence of the cotangent complex from the chosen simplicial resolution

A common factorization category and contractibility. Choose a bounded small polynomial-factorization category as in [F1] containing the standard resolution together with the specified $P_{\bullet }$ and $Q_{\bullet }$. For an object $V=(A[E]\to B)$ and an admissible resolution $P_{\bullet }$, the required simplicial set is ${\mathrm{Hom}}_{{\mathcal{C}}_{B\mathrm{/}A}^{\kappa }}(P_{\bullet },V)={\mathrm{Hom}}_{A\text{-}\mathrm{A}\mathrm{l}\mathrm{g}\mathrm{/}B}(A[E],P_{\bullet })$, since ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }$ is the opposite of the presentation category. It is the product over $e\in E$ of the augmentation fibres over the prescribed images of the variables, and these fibres together with all their set-indexed products are nonempty and contractible by [F2]. Hence the hypothesis of the contractible-evaluation lemma [F3] is satisfied for the standard resolution and for both $P_{\bullet }$ and $Q_{\bullet }$.

Canonical identification of the three complexes. Apply [F3] to the cotangent diagram $F$ of [F1]. Evaluation on the standard resolution is the definition of $L_{B\mathrm{/}A}$, while evaluation on $P_{\bullet }$ and on $Q_{\bullet }$ is ${\mathrm{\Omega }}_{P_{\bullet }\mathrm{/}A}{\otimes }_{P_{\bullet }}B$ and ${\mathrm{\Omega }}_{Q_{\bullet }\mathrm{/}A}{\otimes }_{Q_{\bullet }}B$; the canonical augmentation roofs from these three evaluations into the same diagram-derived-colimit therefore give canonical isomorphisms of all three complexes in $D(B)$. The standard resolution is genuinely admissible by [F2], and enlarging the bound is harmless by the category-change half of [F3] using the shared standard resolution. This proves the first assertion, including that the comparison lives in $D(B)$ and need not be a direct chain map.

Base change of the comparison. For a commutative ordinary ring square, the functor $u\text{\hspace{0.17em}⁣}:P\mapsto P{\otimes }_A{A}^{\mathrm{\prime }}$ between bounded factorization categories sends a nested variable $[p]$ to the nested variable of its image, giving the canonical map of standard simplicial algebras and hence the canonical cotangent comparison. By the coefficient-change half of [F3], extending coefficients turns $L{\mathrm{colim}}_{B}F$ into $L{\mathrm{colim}}_{B^{\mathrm{\prime }}}(F{\otimes }_B^{\mathbf{L}}{B}^{\mathrm{\prime }})$; every value of $F$ is a free $B$-module because polynomial differentials are free, so this coefficient tensor is ordinary, and by [F4] it is identified with $u^{\ast }{F}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}$.

The isomorphism criterion. Assume the canonical derived tensor map $B{\otimes }_A^{\mathbf{L}}{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$ is a quasi-isomorphism. The associated $A$-module complex of the standard resolution is bounded above and free by [F2], so its tensor with $A^{\mathrm{\prime }}$ computes this derived tensor product by [F5]; therefore the tensored augmentation $P_{\bullet }{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$ is a quasi-isomorphism. It is termwise surjective: in degree zero the augmentation maps onto $B{\otimes }_A{A}^{\mathrm{\prime }}$, whose canonical map to $B^{\mathrm{\prime }}$ is an isomorphism on $H^0$, and the degeneracies supply the higher termwise surjections. The normalized additive lifting criterion of [F6] then makes it a trivial Kan fibration, so both the source standard resolution and its image under $u$ satisfy the contractibility hypothesis of step 1.2. The category-change half of step 2.1 identifies $L{\mathrm{colim}}_{\mathrm{o}\mathrm{l}\mathrm{d}}{u}^{\ast }{F}_{\mathrm{n}\mathrm{e}\mathrm{w}}$ with $L{\mathrm{colim}}_{\mathrm{n}\mathrm{e}\mathrm{w}}{F}_{\mathrm{n}\mathrm{e}\mathrm{w}}$, and composing with the coefficient identification shows that the canonical cotangent map $L_{B\mathrm{/}A}{\otimes }_B^{\mathbf{L}}{B}^{\mathrm{\prime }}\to L_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}$ is an isomorphism in $D(B^{\mathrm{\prime }})$.

Tor criterion and the same-target case. For the ordinary pushout $B^{\mathrm{\prime }}=B{\otimes }_A{A}^{\mathrm{\prime }}$, the canonical derived tensor map is a quasi-isomorphism precisely when ${\mathrm{Tor}}_i^A(B,A^{\mathrm{\prime }})=0$ for all $i>0$, by the Tor identification of [F5] under AC (which implies the supplier's Dependent Choice); a flat $A\to A^{\mathrm{\prime }}$ gives this vanishing. If $B^{\mathrm{\prime }}=B{\otimes }_A^{\mathbf{L}}{A}^{\mathrm{\prime }}$ is concentrated in degree zero, the ordinary statement applies to that discrete algebra. Finally, for composable maps $A\to A^{\mathrm{\prime }}\to B$ with $B{\otimes }_A^{\mathbf{L}}{A}^{\mathrm{\prime }}\to B$ a quasi-isomorphism, take $B^{\mathrm{\prime }}=B$ and the identity on $B$ in the square of step 2.1; the criterion of step 3.1 gives $L_{B\mathrm{/}A}\cong L_{B\mathrm{/}{A}^{\mathrm{\prime }}}$ in $D(B)$, which applies in particular to a localization $A^{\mathrm{\prime }}=S^{-1}A$ through which $A\to B$ factors.