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Tame and curve-tame cohomology of $\mathbb A^1$-invariant étale sheaves

Sandeep S, Anand Sawant

Abstract

We extend the definition of the unramified curve-tame cohomology groups to $\mathbb{A}^1$-invariant étale sheaves under some additional hypotheses. We define a pairing of this group with the Suslin homology satisfying desirable properties and using this, we show that the unramified curve-tame cohomology of a smooth geometrically connected variety over a field of positive characteristic agrees with the cohomology of the base field.

Tame and curve-tame cohomology of $\mathbb A^1$-invariant étale sheaves

Abstract

We extend the definition of the unramified curve-tame cohomology groups to -invariant étale sheaves under some additional hypotheses. We define a pairing of this group with the Suslin homology satisfying desirable properties and using this, we show that the unramified curve-tame cohomology of a smooth geometrically connected variety over a field of positive characteristic agrees with the cohomology of the base field.
Paper Structure (6 sections, 26 theorems, 49 equations)

This paper contains 6 sections, 26 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Logarithmic de Rham-Witt sheaves
  3. The tame cohomology group
  4. Unramified cohomology
  5. Unramified curve-tame cohomology
  6. Pairing with Suslin homology

Key Result

Theorem 1.1

(see Theorem theorem existence_Kato_complex for a precise version) Let $k$ be a field of characteristic $p>0$ and let $\mathcal{G}$ be an $\mathbb{A}^1$-invariant étale sheaf on $Sm_k$ satisfying Conventions additional_assumptions_on_G. Then the groups $M_i(K)=H^1_{\text{\'et}}(K,\mathcal{G}_i)$ for

Theorems & Definitions (61)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Lemma 3.5
  • ...and 51 more