The standard simplicial resolution of a ring map

Let $A\to B$ be a homomorphism of commutative unital rings (Commutative ring, Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$), and for a set $S$ let $A[S]$ denote the polynomial $A$-algebra on the variable set $S$ (The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials). Write $U$ for the forgetful functor from $A$-algebras to sets, and write $F=A[-]$. The free polynomial universal property gives $F⊣U$, with unit $\eta \text{\hspace{0.17em}⁣}:{\mathrm{i}\mathrm{d}}_{\mathrm{S}\mathrm{e}\mathrm{t}}\to UF$ sending a set element to its variable, and counit $ϵ\text{\hspace{0.17em}⁣}:FU\to {\mathrm{i}\mathrm{d}}_{A\text{-}\mathrm{A}\mathrm{l}\mathrm{g}}$ sending a variable labelled by an algebra element to that element. Put $G=FU$ and $\delta =F\eta U:G\to G^2$.

The standard resolution of $B$ over $A$ is the augmented simplicial $A$-algebra $ϵ\text{\hspace{0.17em}⁣}:P_{\bullet }\to B$ (Simplicial objects, simplicial commutative rings and homotopy groups) with $P_0=A[B],P_1=A[A[B]],P_n=A[P_{n-1}](n\ge 1),$ equivalently $P_n=G^{n+1}(B)$. Its face maps are $d_i=G^iϵG^{n-i}:G^{n+1}B\to G^{n}B$ for $n\ge 1$, and its degeneracy maps are $s_i=G^i\delta G^{n-i}:G^{n+1}B\to G^{n+2}B$ for $0\le i\le n$; and the augmentation $P_0=A[B]\to B$ is induced by the structure map of the $A$-algebra $B$ on the free generators. The adjunction triangle identities imply the comonad identities $(ϵG)\delta =(Gϵ)\delta ={\mathrm{i}\mathrm{d}}_G$ and $(\delta G)\delta =(G\delta )\delta$. Substituting these identities in the face and degeneracy formulas gives the simplicial identities, so $P_{\bullet }$ is a simplicial $A$-algebra and $ϵ$ is a morphism of simplicial $A$-algebras to the constant simplicial algebra $B$.

Each $P_n=A[P_{n-1}]$ is a polynomial $A$-algebra, hence a free $A$-module on its monomials; in particular every $P_n$ is flat and the associated complex of $A$-modules with the alternating face differential is a complex of free $A$-modules. The augmentation admits an explicit homotopy contraction of underlying simplicial sets over $B$, making it a weak equivalence of simplicial rings once homotopy groups are read through the Moore complex; this is proved as The standard polynomial resolution has an augmentation contraction and is admissible ↗, which is the well-definedness statement for the present construction and which is where the simplex-level contraction is exhibited.

The well-definedness lemma uses only the displayed polynomial construction, its face and degeneracy formulas, and the polynomial universal property. It proves the augmentation properties just stated; those properties are not prerequisites of its proof.