Finite-field computation of weight-zero Kato homology

Let $F$ be a finite field, let $X$ be the variety under consideration, let $p=\operatorname{ch}(F)$, and let $r\geq 1$. The groups $KH_j^{(0)}(X,\mathbb{Z}/p^r)$ are the weight-zero Kato homology groups.

Finite-field computation. There is an isomorphism

$$KH_j^{(0)}(X,\mathbb{Z}/p^r)\xrightarrow{\simeq}\begin{cases}\mathbb{Z}/p^r,&j=0,\\0,&j>0.\end{cases}$$

Thus the weight-zero Kato homology is concentrated in degree zero and equals $\mathbb{Z}/p^r$ there.