Let $X$ be a smooth hypersurface in $\mathbb{P}^n$ and let $\mathfrak{P}$ be the completion of $\mathbb{P}^n$ along $X$.

Let $\mathcal{R}$ denote the category of reflexive sheaves on $\mathbb{P}^n$ which are bundles outside a finite set of points in $X$ (the finite set is not fixed, it could/will be different for different sheaves). The motivation for this is Grothendieck's effective Lefschetz condition.

Let $\mathcal{V}$ denote the category of (formal) locally free sheaves on $\mathfrak{P}$ and consider the functor

$$F: \mathcal{R} \to \mathcal{V}.$$ Then by Grothendieck's Lefschetz conditions, this functor is an equivalence of categories. This therefore implies that

$$K_0(\mathbb{P}^n) \cong K_0(\mathfrak{P}).$$

On the other hand, the subgroup $CH^{n}(\mathbb{P}^n) \subset K_0(\mathbb{P}^n)$ maps to zero in $K_0(\mathfrak{P})$. This can be seen in the following way:

take any point $x \in \mathbb{P}^n \setminus X$ and consider the resolution of the skyscraper sheaf $k_x$ $$0 \to F_{\bullet} \to k_x.$$ On restricting this resolution to $\mathfrak{P}$ (restrict to any thickening $X_m$ otherwise), we get an exact sequence $$ 0 \to F_{\bullet}\otimes O_{\mathfrak{P}} \to 0.$$

Therefore we see that $\sum(-1)^iF_i \mapsto 0$ under the map $K_0(\mathbb{P}^n) \to K_0(\mathfrak{P})$.

Clearly I am making some (stupid) mistake. I would be glad if someone points out what it is. In particular which of the arguments is wrong.

Thanks!