We prove that algebraic K-theory satisfies `pro-descent' for abstract blow-up squares of noetherian schemes. As an application we derive Weibel's conjecture on the vanishing of negative K-groups.Comment: 48 pages, final versio

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We show that A 1 -localization decreases the stable connectivity by at most one over a Dedekind scheme with infinite residue fields. For the proof, we establish a version of Gabber's geometric presentation lemma over a henselian discrete valuation ring with infinite residue field.The authors are supported by the SFB/CRC 1085 Higher Invariants (Regensburg) funded by the DFG and the DFG-Forschergruppe 1920 Symmetrie, Geometrie und Arithmetik (Heidelberg-Darmstadt).1 Theorem B. Let o be a henselian discrete valuation ring with infinite residue field and let σ denote the closed point of S = Spec(o). Let X be a smooth S-scheme of finite type and let Z ֒→ X be a proper closed subscheme. Let z be a point in Z. If z lies in the special fibre Z σ , suppose that Z σ = X σ . Then, Nisnevich-locally around z, there exists a smooth o-scheme V of finite type and a cartesian squaresuch that p isétale, the restriction p| Z : Z ֒→ A 1 V is a closed subscheme and Z is finite over V . In particular, this square is a Nisnevich-distinguished square.The proof is based on [CTHK97, Thm. 3.1.1] combined with a Noether normalization over a Dedekind base (cf. [Kai15, Thm. 4.6]).Apart from this geometric input to the proof of Theorem A, we need a second key ingredient of a more homotopical kind: In Chapter 3, we examine a vanishing result for the non-sheafified homotopy classes of the A 1 -localization of a connected spectrum. This is a slight generalization of the argument in [Mor05, Lem. 4.3.1] to arbitrary noetherian base schemes of finite Krull-dimension. As a byproduct, we obtain that the S 1 -and the P 1 -homotopy t-structure over any base scheme is left complete, i.e., a presheaf of spectra is recovered as the homotopy limit over its Postnikov truncations (see Corollary 3.6 and 3.8).

Vorst's conjecture relates the regularity of a ring with the $\mathbb {A}^{1}$ -homotopy invariance of its $K$ -theory. We show a variant of this conjecture in positive characteristic.

We show a conditional exactness statement for the Nisnevich Gersten complex associated to an A 1 -invariant cohomology theory with Nisnevich descent for smooth schemes over a Dedekind ring with only infinite residue fields. As an application we derive a Nisnevich analogue of the Bloch-Ogus theorem for étale cohomology over a henselian discrete valuation ring with infinite residue field.