# nLab Pierre Gabriel

Pierre Gabriel also often as Peter Gabriel (1933-2015) was a French and Swiss mathematician, professor at Zürich, was a president of Swiss Mathematical Society in 1980/1981.

A version of Gabriel’s 1960 PhD thesis has been published as Des catégories abéliennes in 1962. His thesis was a major breakthrough in the theory of localization, and the study of abelian categories, including categories of quasicoherent sheaves on schemes. In retrospective, it can be said that it was in its ideas and methods one of the starting points of modern noncommutative algebraic geometry as well.

Gabriel assisted Grothendieck in reformulating the pseudofunctor version of descent theory in invariant (property characterized way) i.e. as fibered categories which he wrote up under the guidance of Grothendieck in SGA I.6. Gabriel contributed to some other parts of SGA, namely in study of group schemes and formal schemes, e.g. in SGA III.2 (Exp. 7a, P. Gabriel, Étude infinitésimale des schémas en groupe et groupes formels; Exp. 7b, P. Gabriel, Groupes formels). Soon after with Demazure writes a first tome of an unfinished but monumental work on algebraic groups which, more than EGA, emphasised functor of points view.

• M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970

With Zisman he introduced a general localization method in homotopy theory (see also calculus of fractions and Gabriel-Zisman):

• P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Springer 1967. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35.

In later part of his mathematical career, Gabriel worked on representation theory of finite-dimensional associative algebras and quivers, where he found some of the basic theorems.

Some of the $n$lab entries related to Gabriel’s work include Gabriel filter, Gabriel composition of filters, Gabriel multiplication, Gabriel-Ulmer duality and we mention here and there Gabriel localization, Gabriel spectrum of indecomposable injectives, Gabriel–Popescu embedding theorem, Gabriel–Rosenberg reconstruction theorem, Gabriel–Zisman localization and Gabriel’s property (sup) in the theory of abelian categories.

Students of Gabriel include Bernhard Keller. Gabriel has also an important work in pure category theory on locally presentable categories:

• P. Gabriel, F. Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, Vol. 221. Springer-Verlag, Berlin-New York, 1971. v+200 pp. MR0327863

Last revised on March 20, 2016 at 21:10:10. See the history of this page for a list of all contributions to it.