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We generalize the functorial quasi-isomorphism in \cite{Davis2011} fromoverconvergent Witt de-Rham cohomology to rigid cohomology on smooth varietiesover a finite field $k$, dropping the quasi-projectiveness condition. We do soby constructing an \et ale hypercover for any smooth scheme $X$, refined ateach level to be a disjoint union of open standard smooth subschemes of $X$. Wethen find, for large $N$, an $N$-truncated closed embedding into a simplicialsmooth 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 thecomparison.

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