A Comparison of Overconvergent Witt de-Rham Cohomology and Rigid Cohomology on Smooth Schemes

We generalize the functorial quasi-isomorphism in \cite{Davis2011} from overconvergent Witt de-Rham cohomology to rigid cohomology on smooth varieties over a finite field $k$, dropping the quasi-projectiveness condition. We do so by constructing an \et ale hypercover for any smooth scheme $X$, refined at each level to be a disjoint union of open standard smooth subschemes of $X$. We then find, for large $N$, an $N$-truncated closed embedding into a simplicial smooth scheme over $W(k)$, which allows us to use the results of \textit{loc. cit} at the simplicial level, and use cohomological descent to prove the comparison.