The Stacks project

Lemma 35.3.5. Suppose that $R \to A$ has a section. Then for any $R$-module $M$ the extended cochain complex ( is exact.

Proof. By Simplicial, Lemma 14.28.5 the map $R \to (A/R)_\bullet $ is a homotopy equivalence of cosimplicial $R$-algebras (here $R$ denotes the constant cosimplicial $R$-algebra). Hence $M \to (A/R)_\bullet \otimes _ R M$ is a homotopy equivalence in the category of cosimplicial $R$-modules, because $\otimes _ R M$ is a functor from the category of $R$-algebras to the category of $R$-modules, see Simplicial, Lemma 14.28.4. This implies that the induced map of associated complexes is a homotopy equivalence, see Simplicial, Lemma 14.28.6. Since the complex associated to the constant cosimplicial $R$-module $M$ is the complex

\[ \xymatrix{ M \ar[r]^0 & M \ar[r]^1 & M \ar[r]^0 & M \ar[r]^1 & M \ldots } \]

we win (since the extended version simply puts an extra $M$ at the beginning). $\square$

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