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An introduction to field extensions and Galois descent for sheaves of vector spaces

Andreas Hohl

TL;DR

The article provides an accessible, self-contained account of extension of scalars and Galois descent for sheaves of vector spaces, extending classical linear-algebra results to the sheaf-theoretic setting and clarifying when descent is possible. It develops a categorical framework for $G$-structures and extension of scalars, analyzes compatibility with the six Grothendieck operations, and studies descent for sheaves, derived categories, and perverse sheaves. Using Beilinson’s gluing construction, it establishes a robust descent theory for perverse sheaves over finite Galois extensions, linking invariants and tensor extension to Beilinson’s data. The results illuminate how descent behaves in constructible contexts and provide practical tools for applications in areas like the Riemann–Hilbert correspondence and mixed Hodge theory, while highlighting limitations in the derived-category setting for general descent. Overall, the work offers a comprehensive, elementary pathway to Galois descent in sheaf theory with explicit constructions and explicit functorial compatibilities.

Abstract

We study extension of scalars for sheaves of vector spaces, assembling results that follow from well-known statements about vector spaces, but also developing some complements. In particular, we formulate Galois descent in this context, and we also discuss the case of derived categories and perverse sheaves. Most of the results are not new, but our aim is to give an accessible introduction to this subject relying only on techniques from basic sheaf theory. Our proofs also illustrate some applications of results about the structure of constructible and perverse sheaves.

An introduction to field extensions and Galois descent for sheaves of vector spaces

TL;DR

The article provides an accessible, self-contained account of extension of scalars and Galois descent for sheaves of vector spaces, extending classical linear-algebra results to the sheaf-theoretic setting and clarifying when descent is possible. It develops a categorical framework for -structures and extension of scalars, analyzes compatibility with the six Grothendieck operations, and studies descent for sheaves, derived categories, and perverse sheaves. Using Beilinson’s gluing construction, it establishes a robust descent theory for perverse sheaves over finite Galois extensions, linking invariants and tensor extension to Beilinson’s data. The results illuminate how descent behaves in constructible contexts and provide practical tools for applications in areas like the Riemann–Hilbert correspondence and mixed Hodge theory, while highlighting limitations in the derived-category setting for general descent. Overall, the work offers a comprehensive, elementary pathway to Galois descent in sheaf theory with explicit constructions and explicit functorial compatibilities.

Abstract

We study extension of scalars for sheaves of vector spaces, assembling results that follow from well-known statements about vector spaces, but also developing some complements. In particular, we formulate Galois descent in this context, and we also discuss the case of derived categories and perverse sheaves. Most of the results are not new, but our aim is to give an accessible introduction to this subject relying only on techniques from basic sheaf theory. Our proofs also illustrate some applications of results about the structure of constructible and perverse sheaves.
Paper Structure (23 sections, 22 theorems, 69 equations)

This paper contains 23 sections, 22 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. A very short review of sheaf theory
  5. Presheaves and sheaves
  6. (Locally) constant sheaves
  7. Constructibility and perversity
  8. A remark on operations and sheafification
  9. Linear categories and field extensions
  10. $G$-structures
  11. Extension of scalars
  12. Galois descent
  13. Scalar extensions and operations on sheaves
  14. Functorialities for sheaf operations
  15. Direct and inverse images
  16. ...and 8 more sections

Key Result

Lemma 3.3

Let $L/K$ be a field extension and $G\vcentcolon=\mathrm{Aut}(L/K)$. Let $g\in G$. Let $f\colon X\to Y$ be a continuous map between topological spaces.

Theorems & Definitions (57)

  • Definition 3.1
  • Example 3.2
  • Lemma 3.3: cf. BHHS22
  • proof
  • Definition 3.4
  • Example 3.5
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • Definition 3.8
  • ...and 47 more