Bloch-Wigner theorem over rings with many units

The purpose of this article is to provide a version of Bloch-Wigner theorem over the class of rings with many units.

💡 Research Summary

The paper “Bloch‑Wigner theorem over rings with many units” extends the classical Bloch‑Wigner exact sequence, originally formulated for algebraically closed fields of characteristic zero, to a broad class of commutative rings that possess “many units”. A ring R is said to have many units if for every integer n ≥ 2 and any finite collection of surjective linear forms f_i : Rⁿ → R, there exists a vector v ∈ Rⁿ such that each f_i(v) is a unit of R. This condition, introduced by van der Kallen in the context of K₂‑theory, is satisfied by infinite fields, semilocal rings with infinite residue fields, and any finite‑dimensional algebra over an infinite field, among others.

The authors first construct the pre‑Bloch group p(R) as the quotient of the free abelian group generated by symbols