Homology pro stability for Tor-unital pro rings

Abstract

Let $\{A_m\}$ be a pro system of associative commutative, not necessarily unital, rings. Assume that the pro systems $\{\mathrm{Tor}^{\mathbb{Z}\ltimes A_m}_i(\mathbb{Z},\mathbb{Z})\}_m$ vanish for all $i>0$. Then we prove that the sequence \[ \{H_l(\mathrm{GL}_n(A_m))\}_m \to \{H_l(\mathrm{GL}_{n+1}(A_m))\}_m \to \{H_l(\mathrm{GL}_{n+2}(A_m)\}_m \to \cdots \] stabilizes up to pro isomorphisms for $n$ large enough than $l$ and the stable range of $A_m$'s.