Let $X$ be an affinoid variety over a discretely valued non-archimedean field $k$ with valuation ring $\mathcal{R}$. Fix a uniformizer $\omega$. On the section 3.2 of the paper https://arxiv.org/abs/1501.02215 it is stated that, for every flat $O(X)^{\circ}$-module $Q$, the functor $$A\mapsto \widehat{A\otimes_{O(X)^{\circ}}Q}\otimes_{\mathcal{R}}k$$ where the completion is with respect to the $\omega$-adic topology, is left exact. Showing this would involve showing that the kernel of the compled tensor is torsion, however, I do not understand why this is the case.
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This proposition has been removed from the paper in the published paper, available in the author's webpage.
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2$\begingroup$ Just to confirm as an author of the paper that this argument is at best missing details and plausibly simply wrong. However this does not affect the main results as it was replaced in the final published version by an a different argument. $\endgroup$ Commented Apr 4, 2022 at 11:50