Tangent space to Milnor $K$-groups of rings

TL;DR

This paper establishes an isomorphism between the tangent space of Milnor K-groups of certain rings and their Kähler differentials, under specific stability conditions.

Contribution

It proves a new isomorphism linking tangent spaces of Milnor K-groups to Kähler differentials for rings with particular invertibility properties.

Findings

Tangent space to Milnor K-groups is isomorphic to Kähler differentials.

The result applies to rings containing 1/2 and with weak 5-fold stability.

Provides a new perspective on the structure of Milnor K-groups.

Abstract

We prove that the tangent space to the $(n+1)$-th Milnor $K$-group of a ring $R$ is isomorphic to group of $n$-th absolute K\"ahler differentials of $R$ when the ring $R$ contains $\frac{1}{2}$ and has sufficiently many invertible elements. More precisely, the latter condition is that $R$ is weakly $5$-fold stable in the sense of Morrow.