Homology pro stability for Tor-unital pro rings

Let ${\lbrace A_m \rbrace}_m$ be a pro system of associative commutative, not necessarily unital, rings. Assume that the pro systems of Tor-groups ${\lbrace \operatorname{Tor}^{\mathbb{Z} \ltimes A_m}_{i} (\mathbb{Z}, \mathbb{Z}) \rbrace }_m$ vanish for all $i \gt 0$. Then we prove that the pro systems ${\lbrace H_l (\operatorname{GL}_n (A_m)) \rbrace }_m$ stabilize up to pro isomorphisms for $n$ large enough relative to $l$ and the stable range of $A_m$’s.

Keywords

homology stability, $K$-theory excision, Tor-unitality

2010 Mathematics Subject Classification

13D15, 16E20, 19B14

Full Text (PDF format)

Received 16 May 2018

Received revised 16 July 2019

Accepted 22 July 2019

Published 8 January 2020