The Stacks project

Example 14.34.4. Let $R$ be a ring. As an example of the above we can take $i : \text{Mod}_ R \to \textit{Sets}$ to be the forgetful functor and $F : \textit{Sets} \to \text{Mod}_ R$ to be the functor that associates to a set $E$ the free $R$-module $R[E]$ on $E$. For an $R$-module $M$ the simplicial $R$-module $X(M)$ will have the following shape

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

which comes with an augmentation towards $M$. 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 $M$ to be the only nonzero cohomology group of the chain complex associated to the simplicial module $X(M)$.


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