The fixed-base simplicial cotangent module represents derived derivations

Cofibrancy in the slice and its kernel. Let $P\to B$ be a free-cell cofibrant replacement of $B$ in simplicial $A$-algebras augmented to $B$, and put $D=B{\otimes }_{A}P$ with its multiplication augmentation to $B$ and section from $B$. Extension and restriction along $A\to B$ are left and right Quillen because restriction creates fibrations and weak equivalences, so $D$ is cofibrant in the augmented $B$-algebra category. By [F1] the strict augmented and nonunital equivalences transport $D$ to the cofibrant nonunital $B$-algebra $I=\ker (D\to B)$, and $Q$ is left Quillen because $Z$ preserves underlying fibrations and weak equivalences; hence $Q(I)$ is a cofibrant simplicial $B$-module.

The chain of enriched adjunctions. For a $B$-module $M$, whose underlying additive object is fibrant in the model of [F2], the strict adjunctions of [F1] give natural isomorphisms ${\mathrm{M}\mathrm{a}\mathrm{p}}_{A\text{-}\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{/}B}(P,B\oplus M)\cong {\mathrm{M}\mathrm{a}\mathrm{p}}_{{\mathrm{A}\mathrm{u}\mathrm{g}\mathrm{A}\mathrm{l}\mathrm{g}}_B}(B{\otimes }_{A}P,B\oplus M)\cong {\mathrm{M}\mathrm{a}\mathrm{p}}_{{\mathrm{N}\mathrm{U}\mathrm{A}\mathrm{l}\mathrm{g}}_B}(I,Z(M))\cong {\mathrm{M}\mathrm{a}\mathrm{p}}_{{\mathrm{M}\mathrm{o}\mathrm{d}}_B}(Q(I),M)\mathrm{.}$ All sources and targets are cofibrant and fibrant as required, so these are derived mapping spaces by [F2]; this proves that $L_{B\mathrm{/}A}^{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}=Q(I)$ represents relative derived derivations.

The Kähler description. There is an elementwise natural isomorphism between $Q(I)$ and $B{\otimes }_P{\mathrm{\Omega }}_{P\mathrm{/}A}$: an augmented $B$-linear derivation of $D$ into $M$ is the same as an $A$-derivation of $P$ into $M$ through $P\to B$, and both are represented by the displayed modules; explicitly, one maps $p$ to the class of $1\otimes p$ minus its augmentation and verifies the Leibniz relation modulo the products $I^2$. The universal derivation corresponds to the identity of the representing module.

Independence of the replacement. For two cell cofibrant replacements $P_1,P_2\to B$, lift $P_1\to P_2$ over $B$ against the trivial fibration $P_2\to B$; the lift is a weak equivalence by two-out-of-three. The mapping corner $\mathrm{M}\mathrm{a}\mathrm{p}(P_1,P_2)\to \mathrm{M}\mathrm{a}\mathrm{p}(P_1,B)$ has boundary lifting by [F2], so the fibre over the prescribed augmentation is contractible and all such lifts give the same homotopy class. Both $P_i$ are fibrant in the augmented slice because $P_i\to B$ is a trivial fibration, so the weak comparison is a simplicial homotopy equivalence by the replacement argument of [F2]. The composite enriched left adjoint (extension, kernel equivalence and indecomposables) preserves simplicial homotopies, hence sends that comparison to a simplicial homotopy equivalence of modules and therefore to a normalized quasi-isomorphism. Thus the representing modules are canonically compared in the model homotopy category. Hence $L_{B\mathrm{/}A}^{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}$ is independent of the choice of $P$.

Ordinary specialization and scope. For a discrete map $A\to B$, choose $P\to B$ by the actual $I$-cell factorization of the initial map in ordinary $A$-algebras: in each degree the generating map is a polynomial-ring inclusion on a subset of the simplex variables, a pushout adjoins complementary variables, and a sequential union of polynomial extensions is again a polynomial ring on the union of the variable sets. Hence each $P_k$ is polynomial over ordinary $A$ and its augmentation is a trivial Kan fibration; the ordinary comparison packet [F3] then identifies the fixed-base object with the ordinary cotangent complex of The cotangent complex of a ring map computed on the standard resolution. No invariance under weak replacement of $A$ and no global derived-scheme gluing is asserted here.