Lemma 92.12.1. Notation and assumptions as in Cohomology on Sites, Example 21.39.1. Assume $\mathcal{C}$ has a cosimplicial object as in Cohomology on Sites, Lemma 21.39.7. Let $\mathcal{F}$ be a flat $\underline{B}$-module such that $H_0(\mathcal{C}, \mathcal{F}) = 0$. Then $H_ l(\mathcal{C}, \text{Sym}_{\underline{B}}^ k(\mathcal{F})) = 0$ for $l < k$.
Proof. We drop the subscript ${}_{\underline{B}}$ from tensor products, wedge powers, and symmetric powers. We will prove the lemma by induction on $k$. The cases $k = 0, 1$ follow from the assumptions. If $k > 1$ consider the exact complex
\[ \ldots \to \wedge ^2\mathcal{F} \otimes \text{Sym}^{k - 2}\mathcal{F} \to \mathcal{F} \otimes \text{Sym}^{k - 1}\mathcal{F} \to \text{Sym}^ k\mathcal{F} \to 0 \]
with differentials as in the Koszul complex. If we think of this as a resolution of $\text{Sym}^ k\mathcal{F}$, then this gives a first quadrant spectral sequence
\[ E_1^{p, q} = H_ p(\mathcal{C}, \wedge ^{q + 1}\mathcal{F} \otimes \text{Sym}^{k - q - 1}\mathcal{F}) \Rightarrow H_{p + q}(\mathcal{C}, \text{Sym}^ k(\mathcal{F})) \]
By Cohomology on Sites, Lemma 21.39.10 we have
\[ L\pi _!(\wedge ^{q + 1}\mathcal{F} \otimes \text{Sym}^{k - q - 1}\mathcal{F}) = L\pi _!(\wedge ^{q + 1}\mathcal{F}) \otimes _ B^\mathbf {L} L\pi _!(\text{Sym}^{k - q - 1}\mathcal{F})) \]
It follows (from the construction of derived tensor products) that the induction hypothesis combined with the vanishing of $H_0(\mathcal{C}, \wedge ^{q + 1}(\mathcal{F})) = 0$ will prove what we want. This is true because $\wedge ^{q + 1}(\mathcal{F})$ is a quotient of $\mathcal{F}^{\otimes q + 1}$ and $H_0(\mathcal{C}, \mathcal{F}^{\otimes q + 1})$ is a quotient of $H_0(\mathcal{C}, \mathcal{F})^{\otimes q + 1}$ which is zero. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)