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.



You must log in to answer this question.