flat cohomology

flat cohomology
Name	Flat cohomology
Field	Algebraic geometry
Introduced	20th century
Notable contributors	Jean-Pierre Serre, Alexander Grothendieck, Jean Giraud, Michel Raynaud, Barry Mazur

flat cohomology is a sheaf cohomology theory defined for the fppf (faithfully flat and locally of finite presentation) and fpqc (faithfully flat and quasi-compact) Grothendieck topologies on schemes. It plays a central role in the study of torsors, descent problems, group scheme classification, and obstruction theory, connecting work of Jean-Pierre Serre, Alexander Grothendieck, Jean Giraud, Michel Raynaud, and Barry Mazur with classical results associated to André Weil, Emil Artin, and Helmut Hasse.

Introduction

Flat cohomology arose in the development of modern algebraic geometry by Alexander Grothendieck in the period following the formulation of the Éléments de géométrie algébrique, influenced by foundational work of André Weil, Jean-Pierre Serre, Oscar Zariski, and Claude Chevalley. It extends earlier sheaf cohomology frameworks used by Jean Leray and Henri Cartan and interfaces with duality theorems of John Tate, Serge Lang, and Helmut Hasse in arithmetic geometry. The theory is closely intertwined with descent theory as formalized by Michel Demazure, Jean Giraud, and Jean-Pierre Serre and links to moduli problems studied by David Mumford, Pierre Deligne, and Gerd Faltings.

Definitions and Basic Properties

One defines flat cohomology by equipping a scheme with the fppf or fpqc topology introduced by Alexander Grothendieck and studying cohomology of sheaves of groups such as group schemes of Deligne, Pierre Cartier, and Michel Raynaud. The first cohomology set classifies torsors under group schemes following the perspectives of Jean Giraud and Grothendieck, while higher cohomology groups generalize obstructions studied by Barry Mazur, Kenkichi Iwasawa, and Ernst Kummer. Fundamental properties and exact sequences mirror results by Jean-Pierre Serre on Galois cohomology and are organized using spectral sequence techniques related to Jean Leray, Henri Cartan, and Claude Chevalley. Key structures involve fppf sheaves representing algebraic groups as in the work of Alexander Grothendieck, Michiel Hazewinkel, and Michel Raynaud.

Relationship with Other Cohomology Theories

Flat cohomology relates to étale cohomology developed by Alexander Grothendieck, Jean-Pierre Serre, and Michael Artin and to Zariski cohomology rooted in Oscar Zariski and Emil Artin’s methods. Comparisons with Galois cohomology as studied by Jean-Pierre Serre, John Tate, and Emil Artin arise in fields and local fields contexts influenced by Helmut Hasse and Richard Taylor. Flat cohomology also connects to crystalline cohomology as advanced by Pierre Berthelot, Christopher Illusie, and Jean-Marc Fontaine and to de Rham cohomology in the work of Alexander Grothendieck and Luc Illusie. Relations to étale fundamental group notions due to Alexander Grothendieck and foundational monodromy results of Pierre Deligne, Nicholas Katz, and David Mumford provide bridges to arithmetic applications pursued by Barry Mazur and Gerd Faltings.

Computations and Examples

Computations of flat cohomology groups for classical group schemes rely on explicit structures studied by Pierre Cartier, Michel Raynaud, and Jean-Pierre Serre. Examples include computations for finite flat group schemes such as group schemes of Ofer Gabber, Jean-Marc Fontaine, and Jean-Pierre Serre; for multiplicative-type group schemes linked to Claude Chevalley and Michel Demazure; and for abelian schemes connected to David Mumford, Gerd Faltings, and Barry Mazur. Calculations on Dedekind schemes and arithmetic surfaces draw on methods related to Helmut Hasse, André Weil, and John Tate. Explicit low-degree computations often invoke local duality theorems by John Tate and global duality frameworks developed by Alexander Grothendieck, Pierre Deligne, and Jean-Pierre Serre, with illustrative examples from work of Barry Mazur on torsion and of Jean-Marc Fontaine on p-adic Hodge structures.

Applications in Algebraic Geometry and Number Theory

Flat cohomology is applied to classification of torsors and principal homogeneous spaces in contexts studied by Alexander Grothendieck, Jean Giraud, and Jean-Pierre Serre, with arithmetic implications pursued by John Tate, Barry Mazur, and Gerd Faltings. It underpins obstruction theory for rational points and descent obstructions analyzed by Yuri Manin, Jean-Marc Fontaine, and Bjorn Poonen, and informs the study of Selmer groups and Shafarevich–Tate groups in the research traditions of John Tate, Vladimir Drinfeld, and Ken Ribet. Moduli problems for abelian varieties and level structures involve insights by David Mumford, Pierre Deligne, and Nicholas Katz, while deformation-theoretic applications connect to work by Michael Artin, Alexander Grothendieck, and Michel Raynaud. Connections to reciprocity laws and local-global principles echo research by Emil Artin, Helmut Hasse, and André Weil.

Advanced Topics and Generalizations

Advanced directions include nonabelian flat cohomology developed in the tradition of Jean Giraud and Grothendieck, higher stacky formulations influenced by Jacob Lurie, Bertrand Toën, and Gabriele Vezzosi, and derived analogues inspired by Maxim Kontsevich and Dennis Gaitsgory. Generalizations to logarithmic geometry engage Kazuya Kato and Luc Illusie; p-adic and perfectoid approaches draw on Peter Scholze, Jean-Marc Fontaine, and Bhargav Bhatt. Interactions with Tannakian duality reference Pierre Deligne and Saavedra Rivano, while categorical and homotopical frameworks trace to Alexander Grothendieck’s seminars, Grothendieck–Verdier duality, and modern developments by Jacob Lurie and Vladimir Drinfeld.

Category:Algebraic geometry