The Stacks project

Remark 92.3.3. Let $A \to B$ be a ring map. Let $\mathcal{A}$ be the category of arrows $\psi : C \to B$ of $A$-algebras and let $\mathcal{S}$ be the category of maps $E \to B$ where $E$ is a set. There are adjoint functors $V : \mathcal{A} \to \mathcal{S}$ (the forgetful functor) and $U : \mathcal{S} \to \mathcal{A}$ which sends $E \to B$ to $A[E] \to B$. Let $X_\bullet $ be the simplicial object of $\text{Fun}(\mathcal{A}, \mathcal{A})$ constructed in Simplicial, Section 14.34. The diagram

\[ \xymatrix{ \mathcal{A} \ar[d] \ar[r] & \mathcal{S} \ar@<1ex>[l] \ar[d] \\ \textit{Alg}_ A \ar[r] & \textit{Sets} \ar@<1ex>[l] } \]

commutes. It follows that $X_\bullet (\text{id}_ B : B \to B)$ is equal to the standard resolution of $B$ over $A$.

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