Example 14.34.5. Let $A$ be a ring. Let $\textit{Alg}_ A$ be the category of commutative $A$-algebras. As an example of the above we can take $i : \textit{Alg}_ A \to \textit{Sets}$ to be the forgetful functor and $F : \textit{Sets} \to \textit{Alg}_ A$ to be the functor that associates to a set $E$ the polynomial algebra $A[E]$ on $E$ over $A$. (We apologize for the overlap in notation between this example and Example 14.34.4.) For an $A$-algebra $B$ the simplicial $A$-algebra $X(B)$ will have the following shape

$X(B) = \left( \ldots \xymatrix{ A[A[A[B]]] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & A[A[B]] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & A[B] \ar@<0ex>[l] } \right)$

which comes with an augmentation towards $B$. We will also show this augmentation is a homotopy equivalence of sets. By Lemmas 14.30.8, 14.31.9, and 14.31.8 this is equivalent to asking $B$ to be the only nonzero cohomology group of the chain complex of $A$-modules associated to $X(B)$ viewed as a simplicial $A$-module.

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