The Stacks project

Lemma 90.4.2. Let $A \to B$ be a ring map. Let $\epsilon : P_\bullet \to B$ be the standard resolution of $B$ over $A$. Let $\pi $ be as in (90.4.0.1). Then

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

in $D(\textit{Ab})$, resp. $D(B)$ functorially in $\mathcal{F}$ in $\textit{Ab}(\mathcal{C})$, resp. $\textit{Mod}(\underline{B})$.

First proof. We will apply Lemma 90.4.1. Since the terms $P_ n$ are polynomial algebras we see the first assumption of that lemma is satisfied. The second assumption is proved as follows. By Simplicial, Lemma 14.34.3 the map $\epsilon $ is a homotopy equivalence of underlying simplicial sets. By Simplicial, Lemma 14.31.9 this implies $\epsilon $ induces a quasi-isomorphism of associated complexes of abelian groups. By Simplicial, Lemma 14.31.8 this implies that $\epsilon $ is a trivial Kan fibration of underlying simplicial sets. $\square$

Second proof. We will use the criterion of Cohomology on Sites, Lemma 21.38.7. Let $U = (Q, \beta )$ be an object of $\mathcal{C}$. We have to show that

\[ S_\bullet = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}((Q, \beta ), (P_\bullet , \epsilon )) \]

is homotopy equivalent to a singleton. Write $Q = A[E]$ for some set $E$ (this is possible by our choice of the category $\mathcal{C}$). Using the notation of Remark 90.3.3 we see that

\[ S_\bullet = \mathop{\mathrm{Mor}}\nolimits _\mathcal {S}((E \to B), i(P_\bullet \to B)) \]

By Simplicial, Lemma 14.34.3 the map $i(P_\bullet \to B) \to i(B \to B)$ is a homotopy equivalence in $\mathcal{S}$. Hence $S_\bullet $ is homotopy equivalent to

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {S}((E \to B), (B \to B)) = \{ *\} \]

as desired. $\square$


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