K-theory and topological cyclic homology of henselian pairs

By Dustin Clausen, A. Mathew, M. Morrow. 2018

59 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/51db3ba26bd5358b131571cf9867af7342176c4e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Prisms and prismatic cohomology (2019, methodology): “me S= S′ by Lemma 14.5. Filtering Tby its ﬁnite subsets then reduces us to the ring S0 considered in the previous paragraph, so we are done. Proof of Corollary 14.3. Via [BMS19, Theorem 1.12 (5)] and [CMM18], it is enough to show that for all n≥0, applying H1(Zp(n)(−)) to R→Sgives a surjective map. ”
Topological Hochschild homology and integral p$p$-adic Hodge theory (2018, background): “ theorem; this will appear in forthcoming work of Clausen, Mathew, and the second author. We denote by K(−) the connective algebraic K-theory of a ring, and by K(−;Zp) its p-completion. Theorem 7.15 ([CMM18]). Let Sbe a ring which is Henselian along pSand such that S/pSis semiperfect, e.g., S∈QRSPer”
Hyperdescent and étale K-theory (2019, methodology): “We handle the general case by inputting the generalization of Gabber-Suslin rigidity proved in [CMM18].”