# Global version of Gabber's rigidity theorem

I had a question regarding Gabber's rigidity.

Let $$A$$ be a ring (let's assume Noetherian) and $$I$$ be an ideal, since the pair $$(\hat{A},I)$$ is a henselian pair ($$\hat{A}$$ is the completion along $$I$$), Gabber's rigidity in algebraic $$K$$-theory for henselian pairs implies that $$K_*(A/I, \mathbb{Z}/l\mathbb{Z})\simeq K_*(\hat{A}, \mathbb{Z}/l\mathbb{Z})$$. Here algebraic $$K$$-theory of the completion is taken in the sense that $$\hat{A}$$ is just a Noetherian ring and we can construct $$BGL(\hat{A})^+$$ just as we can do for any ring i.e. the $$K$$-theory of finitely generated projective modules on $$\hat{A}$$. There is another way to define $$K$$-theory and that is to take the $$K$$-theory of formal vector bundles on $$\hat{A}$$, which means a system of vector bundles on all finite thickenings of $$I$$ with compatibility conditions under pullbacks. These two probably are the same. (There is an equivalence between the formal vector bundles and $$\hat{A}$$ bundles given by taking limit and restriction to infinitesimal neighborhoods.)

Q) My question is about whether there is a global version of this rigidity in the setting of formal completions or not?

Let's assume we have Noetherian schemes $$X$$ and a closed subscheme $$Z$$ and we are looking at the formal completion $$X_Z$$. Since we are dealing with formal schemes it makes sense to consider formal vector bundles and compare its $$K$$-theory with the $$K$$-theory of $$Z$$ with coefficient in $$\mathbb{Z}/l\mathbb{Z}$$. One question is whether there is a Zariski descent for formal schemes with finite coefficients? if so that would imply $$K_*(X_Z,\mathbb{Z}/l)\simeq K_*(Z, \mathbb{Z}/l)$$. Or there could be possibly a different approach. This could also be wrong in the global case, which I'd like to know some examples.