## \$l\$-adic rigidity for Milnor \$K\$-theory

Given a local henselian ring $$A$$ with the maximal ideal $$m$$, does the quotient map $$Amapsto A/m$$ induce isomorphisms on $$l$$-adic Milnor $$K$$-theories? ($$K_n^M(R)otimes mathbb{Z}_l$$, where $$l$$ is an invertible prime.)

## ag.algebraic geometry – Global version of Gabber’s rigidity theorem

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.

## dg.differential geometry – A strong form of Mostow rigidity without geometrization?

Gabai proved that homotopy hyperbolic 3-manifolds are virtually hyperbolic, in the paper of that name:

Gabai, David, Homotopy hyperbolic 3-manifolds are virtually
hyperbolic.
J. Amer. Math. Soc. 7 (1994), no. 1, 193–198.

I suspect this is the best you can do without geometrisation.

## probability theory – Criteria for the rigidity of the sequence of distribution functions

Leave $$φ_n$$ Be a sequence of characteristic functions. We know that converge at each point of a not empty
open interval around 0
yet function $$φ$$ which is also continued at 0.

Can you say that the sequence
of the distribution functions corresponding to $$φ_n$$They are tight

I am aware of the test of a different version where convergence occurs in the entire real line. But what to do in this case?

## differential equations – Non-autonomous ODE uses NDSolve, error: Step Size is effectively zero; The singularity or rigidity of the system is suspected.

I have seen this error `NDSolve :: ndsz` many times when I use `NDSolve` To obtain the solution of a non-autonomous ODE. I try but everything has failed. Thanks, Here is the code, very simple.

``````    s = NDSolve[{x'
Y
X[0] == 1, and[0] == 1}, {x
``````