ABSTRACT. Consider a complete discrete valuation ring O with quotient field F and finite residue field. Then the inclusion map O ↪ F induces a mapK M * O →K M * F on improved Milnor K-theory. We show that this map is an isomorphism in degrees bigger or equal to 3. This implies the Gersten conjecture for improved Milnor K-theory for O. This result is new in the p-adic case.

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly non-vanishing continuous K-group and the highest possibly non-vanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of a Zariski-Riemann space with respect to different topologies; namely, the rh-topology which is related to K-theory as well as the Zariski topology whereon the cohomology groups in question rely.

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly non-vanishing continuous K-group and the highest possibly non-vanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of an admissible Zariski-Riemann space with respect to different topologies; namely, the rh-topology which is related to K-theory as well as the Zariski topology whereon the cohomology groups in question rely.

We study relative algebraic K-theory of admissible Zariski-Riemann spaces and prove that it is equivalent to G-theory and satisfies homotopy invariance. Moreover, we provide an example of a non-noetherian abelian category whose negative K-theory vanishes. CONTENTS 1. Introduction 1 2. Admissible Zariski-Riemann spaces 2 3. Modules on admissible Zariski-Riemann spaces 3 4. K-theory of admissible Zariski-Riemann spaces 8 Appendix A. Limits of locally ringed spaces 12 Appendix B. Localisations of exact categories 14 References 15 CHRISTIAN DAHLHAUSENThe notion of Zariski-Riemann spaces goes back to Zariski [Zar44] who called them "Riemann manifolds" and was further studied by Temkin [Tem11]. Recently, Kerz-Strunk-Tamme [KST18] used them to prove that homotopy algebraic K-theory [Wei89] is the cdh-sheafification of algebraic K-theory, and Elmanto-Hoyois-Iwasa-Kelly applied them on reults on Milnor excision for motivic spectra [EHIK20a,EHIK20b].Combining part (i) of the theorem with a result of Kerz about the vanishing of negative relative K-theory, we provide an example of a non-noetherian abelian category whose negative K-theory vanishes (Example 4.19). This gives evidence to a conjecture by Schlichting (which was shown to be false at the generality it was stated), see Remark 4.18.Notation. A scheme is said to be divisorial iff it admits an ample family of line bundles [TT90, 2.1.1]; such schemes are quasi-compact and quasi-separated.Acknowledgements. The author thanks Andrew Kresch for providing a fruitful research environment at the University of Zurich where (most of) this paper was written as well as Georg Tamme, Matthew Morrow, and Moritz Kerz for helpful conversations. ADMISSIBLE ZARISKI-RIEMANN SPACESNotation. In this section let X be a reduced quasi-compact and quasi-separated scheme and let U be a quasi-compact open subscheme of X . Definition 2.1. A U-modification of X is a projective morphism X ′ → X of schemes which is an isomorphism over U. Denote by Mdf(X ,U) the category of U-modifications of X with morphisms over X . We define the U-admissible Zariski-Riemann space of X to be the limitin the category of locally ringed spaces; it exists due to Proposition A.7.Example 2.2. Let V be a valuation ring with fraction field K . Then the canonical projection ⟨Spec(V )⟩ Spec(K) → Spec(V ) is an isomorphism as every Spec(K )-modification of Spec(V ) is split according to the valuative criterion for properness.Lemma 2.3. The full subcategory Mdf red (X ,U) spanned by reduced schemes is cofinal in Mdf(X ,U).Proof. As U is reduced by assumption, the map X ′ red ↪ X ′ is a U-modification for every X ′ ∈ Mdf(X ,U).Lemma 2.4. The underlying topological space of ⟨X ⟩ U is coherent and sober (Definition A.4) and for any XProof. This is a special case of Proposition A.7 or [FK18, ch. 0, 2.2.10].The notion of a U-modification is convenient as it is stable under base change. In practice, one can restrict to the more concrete notion of a U-admissible blow-up. Definition 2.5. A U-admissible blow-up is a blow-up Bl Z (X ) → ...

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