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!



Grothendieck group of vector bundles
I will first assume that, by $K\_0$, you mean the quotient of the free abelian group generated by isomorphism classes of objects in your category, modulo the relation 
$$\sum\_i(-1)^i [F\_i] = 0 \quad (A)$$ 
for every exact sequence $0 \to F\_0 \to F\_1 \to \dots \to F\_{n-1} \to F\_n \to 0$. 


Well, since the functor does not preserve exactness of such long exact sequences, there is no reason that it should induce an isomorphism of $K\_0$ using this definition.


OTOH, if you use as definition of the K-group the quotient generated by relations for *short* exact sequences, then I do not see why the relation (A) should hold. Of course, in an abelian category, you can decompose a long exact sequence into short exact sequences, and deduce it from the relation . Then the above relation for long exact sequences follows from the relation coming from the short exact sequences. But in an exact category, I do not see how to deduce (A).


