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The Stacks project

Lemma 92.4.1. 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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