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.

中文翻译：

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Tor-unital pro 环的同源性 pro 稳定性

令 $\{A_m\}$ 成为结合交换环的亲系统，不一定是单位环。假设专业系统 $\{\mathrm{Tor}^{\mathbb{Z}\ltimes A_m}_i(\mathbb{Z},\mathbb{Z})\}_m$ 对于所有 $i>0$ 消失. 然后我们证明序列 \[ \{H_l(\mathrm{GL}_n(A_m))\}_m \to \{H_l(\mathrm{GL}_{n+1}(A_m))\}_m \到 \{H_l(\mathrm{GL}_{n+2}(A_m)\}_m \to \cdots \] 稳定到 $n$ 比 $l$ 大的同构和 $A_m 的稳定范围$的。