Alexander Grothendieck

1928–2014, Berlin, Germany, Saint-Girons, France

GeometryAlgebraTopologyFoundations of mathematicsFields Medal 1966

German-born mathematician in France who rebuilt algebraic geometry on the language of schemes, sheaves, topoi, and étale cohomology, and transformed homological algebra and category theory.

Alexander Grothendieck was born in Berlin to anarchist parents; during the Second World War he was interned with his mother in France while his father perished at Auschwitz. After studying at the University of Montpellier, he went to Nancy to work under Laurent Schwartz and Jean Dieudonné, solving in less than a year a list of open problems on locally convex spaces and earning his doctorate in 1953 on topological tensor products and nuclear spaces.

Turning to algebraic geometry and homological algebra, Grothendieck published his 1957 'Tôhoku' paper placing sheaf cohomology in the setting of abelian categories and derived functors, and proved the Grothendieck–Riemann–Roch theorem using K-theory. From 1959 to 1970 at the newly founded Institut des Hautes Études Scientifiques (IHÉS), his 'Éléments de géométrie algébrique' (written with Dieudonné) and 'Séminaire de géométrie algébrique' rebuilt algebraic geometry around schemes, relative morphisms, Grothendieck topologies (sites), topoi, and étale, l-adic, and crystalline cohomology — the machinery that enabled the proof of the Weil conjectures.

Awarded the Fields Medal in 1966, Grothendieck refused to travel to Moscow in political protest; in 1970 he resigned from the IHÉS over military funding of its budget, later taught at Montpellier, declined the 1988 Crafoord Prize, and spent his last decades in seclusion in the Pyrenees.

Workplaces: Centre National de la Recherche Scientifique (CNRS), Institut des Hautes Études Scientifiques (IHÉS), University of Montpellier

France, Germany

Award-winning work

Fields Medal 1966Awarded for rebuilding algebraic geometry on entirely new foundations. In the 1960s Grothendieck replaced the classical notion of an algebraic variety with the far more general notion of a scheme, allowing geometric methods to be applied to arithmetic questions such as geometry over the integers. With Artin and Verdier he constructed étale cohomology, giving algebraic varieties over any field a cohomology theory analogous to singular cohomology and underpinning his program of motives. These ideas supplied the machinery later used in the proofs of the Weil conjectures and became the standard language of modern algebraic geometry.

Contributions, linked to the library

Library topicAlgebraic geometryRebuilt algebraic geometry on schemes, Grothendieck topologies, and étale cohomology.Library topicHomological algebraIntroduced abelian categories, derived functors on sheaf categories, and later derived categories.Library topicCategories and functorsIntroduced representable functors, fibred categories, descent, and Grothendieck topoi.Library topicArithmetic geometryCreated the scheme-theoretic and cohomological framework linking varieties over finite fields to number theory.