K-theory and topological cyclic homology of henselian pairs
| Autor: | Clausen, Dustin, Mathew, Akhil, Morrow, Matthew |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 59 pages, revised and final version |
| Databáze: | arXiv |
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