Descent theory

Descent theory
Name	Descent theory
Field	Algebraic geometry, Category theory
Introduced	1950s–1960s
Key people	Grothendieck, Galois, Eilenberg, Cartan, Serre, Weil, Grothendieck–Verdier, Deligne, Artin, Gabriel
Notable works	SGA1, EGA, "Faisceaux Algébriques Cohérents", "Séminaire de Géométrie Algébrique"

Descent theory Descent theory studies how objects defined locally on covers glue to give global objects, relating local-to-global problems in Galois-style symmetry, cohomology, and moduli. It connects methods from Grothendieck’s school in SGA1 and EGA with categorical formulations arising in the work of Eilenberg and Cartan-inspired approaches, and informs constructions appearing in Serre’s and Weil’s development of modern algebraic geometry.

Introduction

Descent theory formalizes descent data for objects over a cover in the language of Grothendieck topologies, relating to the notion of a stack in the sense of Deligne and Deligne’s work on moduli, and draws on notions from Grothendieck’s foundations alongside categorical ideas from Mac Lane and Eilenberg. It provides criteria—such as effective descent—for when local objects defined over covers coming from morphisms like those in étale or flat topologies glue to global objects, interacting with techniques from Serre cohomology, Artin approximation, and the descent properties exploited in Mumford’s geometric invariant theory.

Historical Development

The roots trace to Galois theory and extension theory in Galois’s era, matured through work by Weil on algebraic varieties and Serre on coherent sheaves. Grothendieck synthesized these ideas in SGA1 and EGA, formalizing descent in the language of fibred categories and stacks, influenced by categorical frameworks of Mac Lane and Eilenberg. Subsequent refinements by Artin, Giraud, Deligne, Kontsevich, and Lurie expanded descent to nonabelian and higher-categorical contexts, while applications in moduli problems linked to work of Mumford, Grothendieck’s school, and Katz clarified practical criteria.

Basic Concepts and Definitions

A morphism of schemes or objects in a fibred category poses a covering in a chosen topology such as the Zariski, étale, smooth, or flat topologies; descent formalizes when an object over the cover plus gluing isomorphisms satisfying cocycle conditions yields a unique global object. Key definitions include fibred categories over a base like Spec Z, categories fibered in groupoids as in Giraud’s theory, effective descent morphisms introduced in SGA1 and clarified by Grothendieck and Serre, and notions of descent for morphisms, quasi-coherent sheaves (per Grothendieck’s EGA), and principal bundles related to Galois torsors and Grothendieck’s torsors.

Types of Descent (Faithfully Flat, Galois, Étale, etc.)

Faithfully flat descent (fpqc, fppf) arises from work of Grothendieck and is central to descent for quasi-coherent sheaves and modules as in EGA. Galois descent connects to classical Galois theory and has been applied in contexts studied by Artin and Weil. Étale descent leverages étale covers central to SGA1 and Grothendieck’s theory of fundamental groups, while smooth and syntomic descent are employed in deformation and arithmetic contexts considered by Deligne collaborators and Katz. Descent in higher categories and homotopical settings appears in work by Lurie and Kontsevich-inspired developments.

Descent Data and Gluing Criteria

Descent data consist of an object over the cover together with isomorphisms on overlaps satisfying a cocycle condition; effectiveness requires existence and uniqueness of a global object. Criteria such as faithfully flat descent for quasi-coherent sheaves (EGA/SGA1), effective descent for morphisms and schemes (Grothendieck, Artin), and obstruction-theoretic viewpoints via nonabelian cohomology (Giraud, Serre) are central. These criteria intersect with representability results for functors in Artin’s criteria, stack conditions formulated by Giraud and Deligne, and cohomological obstructions as in Serre and Weil.

Applications in Algebraic Geometry and Category Theory

Descent underlies construction of moduli stacks like the Deligne–Mumford stack and techniques in Mumford for quotients, affects the theory of principal bundles and torsors used in Weil’s and Grothendieck’s approaches to arithmetic geometry, and informs descent for quasi-coherent sheaves essential in EGA-style foundations. In category theory, descent theory intersects with monadicity theorems of Mac Lane and Beck, higher descent in Lurie’s work on higher topos theory, and stack-theoretic formalisms of Giraud and Deligne.

Examples and Counterexamples

Classical examples include Galois descent for vector spaces and forms over Galois extensions in the spirit of Galois and Weil, faithfully flat descent for quasi-coherent sheaves as in EGA/SGA1, and étale descent for finite étale covers connected to the Grothendieck fundamental group. Counterexamples illustrating failure of naïve gluing occur for nonflat morphisms and for certain pathological topologies; these were exhibited in discussions following Grothendieck’s lectures and in analyses by Artin and Giraud demonstrating the necessity of effective descent hypotheses and stack conditions.

Category:Algebraic geometry