Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic rings. In addition, we show that $K$-theory with finite coefficients satisfies continuity for complete noetherian rings which are $F$-finite modulo $p$. Our main new ingredient is a basic finiteness property of $\mathrm{TC}$ with finite coefficients.

中文翻译：

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$K$-Henselian 对的理论和拓扑循环同调

给定交换环的亨塞对 $(R, I)$，我们证明了相对 $K$ 理论和具有有限系数的相对拓扑循环同源性是通过分圆轨迹 $K \to \mathrm{TC}$ 来识别的。这产生了经典 Gabber-Gillet-Thomason-Suslin 刚性定理（对于 mod $n$ 系数，$n$ 在 $R$ 中可逆）和麦卡锡关于相对 $K$ 理论的定理（当 $I$ 是幂零）。我们推导出圆环迹是 $p$-adic $K$-theory 和大类 $p$-adic 环的拓扑循环同调之间在很大程度上的等价。此外，我们证明了具有有限系数的 $K$-理论满足 $F$-有限模 $p$ 的完整诺特环的连续性。我们的主要新成分是具有有限系数的 $\mathrm{TC}$ 的基本有限性属性。