A Bloch-Ogus Theorem for henselian local rings in mixed characteristic

Abstract

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.

A Nisnevich Local Bloch-Ogus Theorem over a General Base

“…We will briefly review the set-up required to prove Theorem 1.1. There is no claim at originality of content or presentation and most of the material can be found in [SS2]. We reproduce it here for the sake of clarity of exposition.…”

“…Lemma 2.4. [SS2,Lemma 3.11] An objectwise fibrant spectrum E ∈ Spt S 1 (Sm S ) is Nisnevich local fibrant if and only if for all Nisnevich distinguished squares as above , the induced morphism…”

“…However, it turns out to be sufficient in the present context. In this note, we extend the theorems in [SS2] to an arbitrary base using the presentation lemma as in [Dru]. Our main result is the following (see also [SS2,Theorem 5.12]):…”

“…The essence of their methods lies in a geometric presentation lemma due to Gabber [CTHK,Theorem 3.1.1]. In [SS2], Strunk and Schmidt prove a Nisnevich local analogue of the Bloch-Ogus theorem for discrete valuation rings with only infinite residue fields. They adapt the results in [CTHK] to the mixed characteristic setting using a Nisnevich local version of the geometric presentation lemma for discrete valuation rings with only infinite residue fields (see [SS1,Theorem 2.1]).…”

“…To prove the above theorem, we follow very closely the methods in [SS2]. The important distinction being that we replace the presentation lemma [SS1,Theorem 2.1] with the more general result [Dru,Remark 3].…”