The Stacks project

Lemma 91.12.1. 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. $\square$


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