K-theory of admissible Zariski-Riemann spaces

Abstract

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 ) → ...