The Simplicial Flabby Resolution of a Sheaf

Question

I study sheaf cohomology by Demailly's book and I have a trouble. Is the inductive formula at the end of page 198 $$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]})^{[0]}$$ right? I think that $(\mathcal{A}^{[0]})^{[0]} = \mathcal{A}^{[0]}$, is it true?

May be

$$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]}/\mathcal{A})^{[0]}$$ or $$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]}/\mathcal{A}^{[q-2]})^{[0]}?$$ Is this resolution have any other name?

Thanks a lot!

Answer 1

The explicit expression is indeed $$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]}/\mathcal{A}^{[q-2]})^{[0]}.$$

And the flabby sheaves $\mathcal A^{[q]}$ are also called the Godement sheaves of $\mathcal A$, while the long exact sequence

$$0\longrightarrow \mathcal A\longrightarrow \mathcal A^{[0]}\longrightarrow\cdots\longrightarrow \mathcal A^{[q]}\longrightarrow \mathcal A^{[q+1]}\longrightarrow \cdots$$ is also called the Godement canonical flabby resolution.