I was hoping someone could help me with the understanding of a particular truncated object. Here are some background:

For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an object in the category of complexes $\mathbf{C}(\mathcal{A})$ in $\mathcal{A}$ by setting $A$ as the degree zero object and the other terms in the complex as 0. Two complexes $A^\bullet$ and $B^\bullet$ in $\mathbf{C}(\mathcal{A})$ are *quasi-isomorphic* if the hypercohomology groups $H^i(A^\bullet)$ and $H^i(B^\bullet)$ are isomorphic for all $i$. By formally inverting all quasi-isomorphisms, we get the so-called *derived category* $\mathcal{D}(\mathcal{A})$ of $\mathcal{A}$.

In my setting, let $p : X \rightarrow Z$ be a faithfully flat morphism of finite type, where $Z$ is an integral regular Noetherian scheme and $X$ is a smooth geometrically integral scheme, both defined over a number field $k$. Thus we have a functor $p_*$ from the category of étale sheaves on $X$ to the category of étale sheaves on $Z$. Let $\mathcal{D}(Z)$ denote the category of bounded complexes of étale sheaves on $Z$ (same for $X$), this means that terms of sufficiently small or large degree simply vanish. Then we have derived functor $\mathbf{R}p_*: \mathcal{D}(X) \rightarrow \mathcal{D}(Z)$.

One obtains a complex $\mathbf{R}p_*\mathbb{G}_{m,X}$ in $\mathcal{D}(Z)$. As above, we write this complex as $$\ldots \rightarrow 0 \rightarrow \mathbf{R}p_*\mathbb{G}_{m,X} \rightarrow 0 \rightarrow \ldots$$ where the sheaf is of degree zero and it is the only nonzero term in the complex.

Now we apply the *truncation functor* $\tau_{\leq 1}$, which sends a complex $A^\bullet$ to
$$\tau_{\leq 1}A^\bullet = \ldots \rightarrow A^{-2} \rightarrow A^{-1} \rightarrow A^0 \rightarrow \mathrm{ker}(d^1) \rightarrow 0 \rightarrow \ldots$$
and the *shift functor* $(-)[1]$ which sends $A^\bullet$ to the complex $A^\bullet[1]$ where the terms of degree $i$ in $A^\bullet$ are of degree $i-1$ in $A^\bullet[1]$.

**Question.** The truncated object $(\tau_{\leq 1}\mathbf{R}p_*\mathbb{G}_{m,X})[1]$ has trivial cohomology outside the degrees -1 and 0. But I don't see why it has nontrivial cohomology at 0 because this truncated object is simply a complex where the nonzero term is of degree -1 and everywhere else is 0. Is it possible that I understood the complex wrongly?

**Context.** This object was defined in the paper *Descent theory for open varieties* by Harari and Skorobogatov and it was denoted as $KD(X)$. This idea of using Galois hypercohomology was subsequently adopted by authors who attempt to generalise results about descent obstructions to the case for varieties $X$ where $\bar{k}[X]^* \neq \bar{k}^*$.

a priorithat this is concentrated in degree $0$. $\endgroup$derived functorbetween the two derived categories, this is different from $R^ip_*$, which is the $i$-th derived functor of the functor $p_*$ between étale sheaves over $X$ and étale sheaves over $Z$. $\endgroup$2more comments