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The Brauer group of a Stein algebra

Olivier Benoist, James Hotchkiss

Abstract

We investigate the Brauer group of the ring $\mathcal{O}(S)$ of holomorphic functions on a finite-dimensional Stein space S. We provide a purely topological computation of this group and deduce a comparison theorem between the étale cohomology of $\textrm{Spec}(\mathcal{O}(S))$ and the singular cohomology of S in degree 2. Furthermore, we prove a purity theorem when S is nonsingular and study the index of classes in the Brauer group of $\mathcal{O}(S)$.

The Brauer group of a Stein algebra

Abstract

We investigate the Brauer group of the ring of holomorphic functions on a finite-dimensional Stein space S. We provide a purely topological computation of this group and deduce a comparison theorem between the étale cohomology of and the singular cohomology of S in degree 2. Furthermore, we prove a purity theorem when S is nonsingular and study the index of classes in the Brauer group of .
Paper Structure (30 sections, 41 theorems, 56 equations)

This paper contains 30 sections, 41 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Stein spaces and their Stein algebras
  3. Algebraic and topological Brauer groups
  4. Cohomological interpretations of Brauer groups
  5. Degree 2 cohomology with finite coefficients
  6. Degree 2 cohomology with invertible coefficients
  7. The meromorphic function field
  8. Purity
  9. The index
  10. Preliminaries
  11. Brauer groups
  12. The analytification functor
  13. The Brauer group of a Stein algebra
  14. Stein algebras and Stein spaces
  15. The topological Brauer group of a finite-dimensional CW complex
  16. ...and 15 more sections

Key Result

Theorem 1.1

The natural morphism is an isomorphism for any finite-dimensional Stein space $S$.

Theorems & Definitions (73)

  • Theorem 1.1: Propositions \ref{['propcompa1']} and \ref{['propcompa2']}
  • Theorem 1.2: Theorem \ref{['th2+']}
  • Theorem 1.3: Theorem \ref{['cor1+']}
  • Theorem 1.4: Theorem \ref{['th6+']}
  • Theorem 1.5: Theorem \ref{['th7+']} and Corollary \ref{['th7++']}
  • Theorem 1.6: Corollary \ref{['th3+']}
  • Theorem 1.7: Theorem \ref{['th4+']}
  • Theorem 1.8: Theorem \ref{['th5+']}
  • Theorem 1.9: Theorem \ref{['th10+']}
  • Theorem 1.10: Theorem \ref{['th9+']}
  • ...and 63 more