Gabber's presentation lemma over noetherian domains

Nisnevich local Good compactifications

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The above theorem is different from [DHKY, Theorem 2.1] in the respect that the hypothesis on the residue fields is relaxed, and we also extend the argument to local complete intersection morphisms.Remark 1.2. Theorem 1.1 compares with [Kai, Theorem 4.1] as follows:(1) The above theorem is a weaker statement than [Kai, Theorem 4.1] (see also [Lev, Theorem 10.2.2]) in the sense that we prove the denseness of the fibre only in the n-dimensional components whereas in [Kai] X ′x is proved to be dense in (X ′ ) x .(2) On the other hand, [Kai, Theorem 4.1] only deals with base a Dedekind domain while Theorem 1.1 is proved for Noetherian rings of finite dimension.

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“…The above theorem is different from [DHKY, Theorem 2.1] in the respect that the hypothesis on the residue fields is relaxed, and we also extend the argument to local complete intersection morphisms.…”

“…In this note, we explore the good compactification constructed in [DHKY,Theorem 2.1]. We attempt to extend it to the arithmetic setting by removing the hypothesis on the residue fields.…”

A remark on Gersten complex for Milnor $K$-theory

“…s ⊂ Z is a non-empty, dense, open subset of Z. We partition the codimension 1 points of Z as Z (1) = Z where Z (1) η := {w} codimension 1 in Z such that w ∈ Z − U ′ s and Z (1) s := Z (1) − Z (1)…”

Nisnevich local good compactifications

“…We also want X to have additional nice properties, like projectivity, or that the open immersion i : X → X behaves well with respect to fibres over B (see Theorem 1.2 for the precise conditions that we impose). The existence of such "good compactifications" 1 for X is a useful technical tool for intersection theory and A 1 -algebraic topology.…”