H0 of the cotangent complex and the polynomial case

The cokernel presents derivations. Write $P_0=A[B]$ with symbols $[b]$ and $P_1=A[P_0]$; the two face maps $d_0,d_1\text{\hspace{0.17em}⁣}:P_1\to P_0$ send the outer variable $[p]$ to $p$ and to the variable $[ϵ(p)]$, where $ϵ\text{\hspace{0.17em}⁣}:P_0\to B$ is the augmentation. The cokernel of the two face maps on differentials is therefore the free $B$-module on the symbols $d[b]$ modulo the relations $d(p)-d[ϵ(p)]$ as $p$ ranges over $P_0$. Taking $p=a\in A$, $p=[b]+[c]$ and $p=[b][c]$ forces $d[a]=0$, additivity and the Leibniz rule; conversely these derivation relations give $d(p)=d[ϵ(p)]$ for every polynomial $p$ by induction on sums and products. Hence the cokernel represents $A$-derivations $B\to -$ and is ${\mathrm{\Omega }}_{B\mathrm{/}A}$ by [F2], and by [F1] it is $H^0(L_{B\mathrm{/}A})$; this proves clause (1).

The polynomial case. If $B$ is a polynomial $A$-algebra, the constant simplicial $A$-algebra $B$ with the identity augmentation is a polynomial resolution and its augmentation is a trivial Kan fibration; by [F3] it may be used to compute $L_{B\mathrm{/}A}$. Its associated differential complex is the constant simplicial module ${\mathrm{\Omega }}_{B\mathrm{/}A}$, whose alternating differential is the identity in positive even chain degrees and zero in odd degrees; pairing consecutive positive degrees contracts them, leaving ${\mathrm{\Omega }}_{B\mathrm{/}A}$ in degree zero. Hence $L_{B\mathrm{/}A}$ is quasi-isomorphic to ${\mathrm{\Omega }}_{B\mathrm{/}A}$ placed in degree $0$, proving clause (2).

Choice accounting. The only construction depending on AC is the comparison of resolutions in [F3], used in step 2.1; the computation of step 1.1 uses only the universal property of Kähler differentials.