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Absolute Poincaré duality in étale cohomology

Adeel A. Khan

Abstract

We extend Poincaré duality in étale cohomology from smooth schemes to regular ones. This is achieved via a formalism of trace maps for local complete intersection morphisms.

Absolute Poincaré duality in étale cohomology

Abstract

We extend Poincaré duality in étale cohomology from smooth schemes to regular ones. This is achieved via a formalism of trace maps for local complete intersection morphisms.

Paper Structure

This paper contains 6 sections, 5 theorems, 33 equations.

Table of Contents

  1. Deformation to the normal stack
  2. Borel--Moore homology of stacks
  3. The construction
  4. Proofs of the asserted properties
  5. Remarks
  6. Acknowledgments

Key Result

Theorem 1

Let $X$ be a smooth $S$-scheme of relative dimension $d$. Then there is a canonical isomorphism in the derived category $\operatorname{\mathbf{D}\xspace}(X_\mathrm{\acute{e}t}\xspace, \Lambda)$ of étale sheaves of $\Lambda$-modules on $X$, where $f : X \to S$ is the structural morphism. In particular, there is a canonical isomorphism

Theorems & Definitions (9)

  • Theorem 1: Poincaré duality
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Proposition 1: Localization
  • Proposition 2
  • proof