The Stacks project

Remark 92.5.5 (Resolutions). Let $A \to B$ be any ring map. Let us call an augmented simplicial $A$-algebra $\epsilon : P_\bullet \to B$ a resolution of $B$ over $A$ if each $P_ n$ is a polynomial algebra and $\epsilon $ is a trivial Kan fibration of simplicial sets. If $P_\bullet \to B$ is an augmentation of a simplicial $A$-algebra with each $P_ n$ a polynomial algebra surjecting onto $B$, then the following are equivalent

  1. $\epsilon : P_\bullet \to B$ is a resolution of $B$ over $A$,

  2. $\epsilon : P_\bullet \to B$ is a quasi-isomorphism on associated complexes,

  3. $\epsilon : P_\bullet \to B$ induces a homotopy equivalence of simplicial sets.

To see this use Simplicial, Lemmas 14.30.8, 14.31.9, and 14.31.8. A resolution $P_\bullet $ of $B$ over $A$ gives a cosimplicial object $U_\bullet $ of $\mathcal{C}_{B/A}$ as in Cohomology on Sites, Lemma 21.39.7 and it follows that

\[ L\pi _!\mathcal{F} = \mathcal{F}(P_\bullet ) \]

functorially in $\mathcal{F}$, see Lemma 92.4.1. The (formal part of the) proof of Proposition 92.5.2 shows that resolutions exist. We also have seen in the first proof of Lemma 92.4.2 that the standard resolution of $B$ over $A$ is a resolution (so that this terminology doesn't lead to a conflict). However, the argument in the proof of Proposition 92.5.2 shows the existence of resolutions without appealing to the simplicial computations in Simplicial, Section 14.34. Moreover, for any choice of resolution we have a canonical isomorphism

\[ L_{B/A} = \Omega _{P_\bullet /A} \otimes _{P_\bullet , \epsilon } B \]

in $D(B)$ by Lemma 92.4.3. The freedom to choose an arbitrary resolution can be quite useful.

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