The Stacks project

Remark 91.12.2. In the situation of Lemma 91.12.1 one can show that $H_ k(\mathcal{C}, \text{Sym}^ k(\mathcal{F})) = \wedge ^ k_ B(H_1(\mathcal{C}, \mathcal{F}))$. Namely, it can be deduced from the proof that $H_ k(\mathcal{C}, \text{Sym}^ k(\mathcal{F}))$ is the $S_ k$-coinvariants of

\[ H^{-k}(L\pi _!(\mathcal{F}) \otimes _ B^\mathbf {L} L\pi _!(\mathcal{F}) \otimes _ B^\mathbf {L} \ldots \otimes _ B^\mathbf {L} L\pi _!(\mathcal{F})) = H_1(\mathcal{C}, \mathcal{F})^{\otimes k} \]

Thus our claim is that this action is given by the usual action of $S_ k$ on the tensor product multiplied by the sign character. To prove this one has to work through the sign conventions in the definition of the total complex associated to a multi-complex. We omit the verification.

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