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

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

By Cohomology on Sites, Lemma 21.39.10 we have

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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