Tag: rigidity

I had a number of questions regarding Gabber’s rigidity, I’m not sure whether I am understanding it correctly, so please let me know if I’m making any mistakes.

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 $K_*(A/I, mathbb{Z}/lmathbb{Z})simeq K_*(hat{A}, mathbb{Z}/lmathbb{Z})$. 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. I don’t think these two coincide in general. (Correct me if I am wrong)

My original question is about whether there is a global version of this rigidity in the setting of formal completions or not? The following is my attempt on this question. (I am not sure whether I am making any mistakes or not)

Now we have Noetherian schemes $X$ and a closed subscheme $Z$ and we are looking at the formal completion $X_Z$. Well as 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 $mathbb{Z}/lmathbb{Z}$. But this comparison possibly fails even in the affine case according to the observation above. So I think the natural generalization would be instead of considering $X_Z$ as a formal scheme we can consider it as a scheme just as in the affine case. This means if on an affine chart $X_Z$ is the formal scheme $hat{A}$, we consider it just as a scheme i.e. $text{Spec}(hat{A})$ (Please let me know if this construction is flawed because I’ve never since a formal scheme to be considered as a scheme but I do not see any issues on this case). Since completion with respect to an ideal preserves being Noetherian and having a finite Krull dimension and because of Zariski descent of connective algebraic $K$-theory this implies that $K_*(Z,mathbb{Z}/lmathbb{Z})simeq K_*(X_Z,mathbb{Z}/lmathbb{Z})$ where $X_Z$ is regarded as a scheme rather than a formal scheme.

For the question when the $K$-theory of formal scheme $X_Z$ should match with its $K$-theory as a scheme, I think if the pair $(X,Z)$ satisfy the property that every formal vector bundle on $X_Z$ can be extended to a neighborhood of $Z$ they should be compatible.