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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08SG. Beware of the difference between the letter 'O' and the digit '0'.