The Stacks project

Lemma 91.4.1. With notation as above let $P_\bullet $ be a simplicial $A$-algebra endowed with an augmentation $\epsilon : P_\bullet \to B$. Assume each $P_ n$ is a polynomial algebra over $A$ and $\epsilon $ is a trivial Kan fibration on underlying simplicial sets. 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. $\textit{Mod}(\underline{B})$.

Proof. We will use the criterion of Cohomology on Sites, Lemma 21.39.7 to prove this. Given an object $U = (Q, \beta )$ 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 $\mathcal{C}$). We see that

\[ S_\bullet = \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}((E, \beta |_ E), (P_\bullet , \epsilon )) \]

Let $*$ be the constant simplicial set on a singleton. For $b \in B$ let $F_{b, \bullet }$ be the simplicial set defined by the cartesian diagram

\[ \xymatrix{ F_{b, \bullet } \ar[r] \ar[d] & P_\bullet \ar[d]_\epsilon \\ {*} \ar[r]^ b & B } \]

With this notation $S_\bullet = \prod _{e \in E} F_{\beta (e), \bullet }$. Since we assumed $\epsilon $ is a trivial Kan fibration we see that $F_{b, \bullet } \to *$ is a trivial Kan fibration (Simplicial, Lemma 14.30.3). Thus $S_\bullet \to *$ is a trivial Kan fibration (Simplicial, Lemma 14.30.6). Therefore $S_\bullet $ is homotopy equivalent to $*$ (Simplicial, Lemma 14.30.8). $\square$

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