I'm reading chapter 3 of Central Simple Algebras and Galois Cohomology by Gille and Szamuely on group cohomology, and I'm really hung up on the notation.

For context, $G$ is any group, and $A$ is a left $G$-module. We form the standard resolution of the trivial $G$-module $\mathbb{Z}$, $$ \ldots \to \mathbb{Z}[G^2] \to \mathbb{Z}[G] \to \mathbb{Z} \to 0 $$ with $\delta^i:\mathbb{Z}[G^{i+1}] \to \mathbb{Z}[G^i]$ given by a typical formula (unimportant for my question). Then we apply the contravariant functor $\operatorname{Hom}_G(-,A)$ to this and drop the $\mathbb{Z}$ term, and write $C^i(G,A) = \operatorname{Hom}_G(\mathbb{Z}[G^{i+1}],A)$ to get the chain complex $$ 0 \to C^0(G,A) \to C^1(G,A) \to C^2(G,A) \to \ldots $$ The group $C^i(G,A)$ is called the group of $i$-cochains. They say,

We may identify $i$-cochains with functions $[\sigma_1, \ldots, \sigma_i] \to a_{\sigma_1, \ldots, \sigma_i}$ and compute the maps $\delta_{i-1}^*:C^{i-1}(G,A) \to C^i(G,A)$ by the formula $$a_{\sigma_1, \ldots, \sigma_{i-1}} \mapsto \sigma_1 a_{\sigma_2, \ldots, \sigma_i} + \sum_{j=1}^i (-1)^j a_{\sigma_1, \ldots, \sigma_j \sigma_{j+1}, \ldots, \sigma_i} + (-1)^{i+1} a_{\sigma_1, \ldots \sigma_{i-1}} $$

First, I suspect the indices are wrong in this formula, and hope that someone can confirm this. I think that the lower index on $\delta^*_{i-1}$ should be $i$ instead of $i-1$, because $\delta^i:\mathbb{Z}[G^{i+1}] \to \mathbb{Z}[G^i]$ and thus the induced map is $$\delta^i_*:\operatorname{Hom}_G(\mathbb{Z}[G^{i}],A) = C^{i-1}(G,A) \to \operatorname{Hom}_G(\mathbb{Z}[G^{i+1}],A) = C^i(G,A)$$

Second, I think the index on the left hand side of their formula is wrong, and should be $a_{\sigma_1, \ldots, \sigma_i}$ instead of $a_{\sigma_1, \ldots, \sigma_{i-1}}$, because otherwise it comes from the wrong domain, and besides the right hand side has $\sigma_i$, which wouldn't make sense otherwise. Can someone please confirm that I'm not crazy, and that the book has wrong indices in both these cases?


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