# nLab Alexander Grothendieck

The European mathematician Alexander Grothendieck (in French sometimes Alexandre Grothendieck), created a very influential body of work foundational for (algebraic) geometry but also for modern mathematics more generally. He is widely regarded as a singularly important figure of 20th century mathematics and his ideas continue to be highly influential in the 21st century.

Initially working on topological vector spaces and analysis, Grothendieck then made revolutionary advances in algebraic geometry by developing sheaf and topos theory and abelian sheaf cohomology and formulating algebraic geometry in these terms (locally ringed spaces, schemes). Later topos theory further developed independently and today serves as the foundation also for other kinds of geometry. Notably its homotopy theoretic refinement to higher topos theory serves as the foundation for modern derived algebraic geometry.

# Contents

## Texts by Grothendieck

Grothendieck’s geometric work is documented in texts known as EGA (with Dieudonné), an early account FGA, and the many volume account SGA of the seminars at l’IHÉS, Bures-sur-Yvette, where he was based at the time. (See the wikipedia article for some indication of the story from there until the early 1980s.)

In the late 1970s and early 1980s Grothendieck wrote several documents that have been of outstanding importance in the origins of the theory that underlies the nPOV. These include

• La Longue Marche à travers la Théorie de Galois (1600 manuscript pages written between January and June 1981, plus addenda etc. which double its length!) (see Long March for some discussion of the ideas.)

• Esquisse d'un programme, (January 1984), in which Grothendieck sketches out a vaste programme of research, incorporating many of the ideas from Long March. A copy is available here. It is discussed in brief at Grothendieck's Esquisse.

• À la poursuite des Champs (also entitled ‘’Pursuing Stacks‘’).
It starts with a short (12 page) letter to Quillen, dated 19 Feb. 1983, but then discusses a wide ranging vision of homotopy theory and its applicability to problems in algebraic and arithmetic geometry.

• Les Dérivateurs (another 2000 page manuscript taking up some of the themes in Pursuing Stacks, section 69) Dating from the end of 1990 and the start of 1991.

In the same time he also wrote voluminous intellectual memoirs Récoltes et Semailles.

En Guise de Programme p I p II, a text written by Grothendieck as a course description while teaching in Montpellier “Introduction à la recherche”.

A chronological bibliography of Grothendieck’s published mathematical writings (pdf).

## Texts about Grothendieck

For an account of his work, including some of the work published in the 1980s, see the English Wikipedia entry.

The video of a talk by W. Scharlau on his life can be seen here.

A recent article in French on Grothendieck is to be found here.

There were two articles on Grothendieck’s life and work in the Notices AMS in 2004:

• Allyn Jackson, Comme Appelé du Néant, As If Summoned from the Void: The Life of Alexandre Grothendieck, Part 1, Notices AMS

• Allyn Jackson, Comme Appelé du Néant, As If Summoned from the Void: The Life of Alexandre Grothendieck, Part 2, Notices AMS

Grothendieck obituary in the Notices AMS (Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors):

• The obituary begins here with a brief sketch of Grothendieck’s life, followed by a description of some of his most outstanding work in mathematics.” Alexandre Grothendieck 1928–2014, (Part 1)

• set of reminiscences by some of the many mathematicians who knew Grothendieck and were influenced by him.” Alexandre Grothendieck 1928–2014, (Part 2).

## Correspondence

The Grothendieck-Serre correspondence

The Grothendieck-Mumford correspondence

Grothendieck circle

Grothendieck’s Angle by G. Aiello

A. Grothendieck by J. A. Navarro

A. Grothendieck by M. Carmona

A. Grothendieck, una guía a la obra matemática y filosófica (pdf) por F. Zalamea

## Quotes

On K-theory:

The way I first visualized a K-group was as a group of “classes of objects” of an abelian (or more generally, additive) category, such as coherent sheaves on an algebraic variety, or vector bundles, etc. I would presumably have called this group $C(X)$ ($X$ being a variety or any other kind of “space”), $C$ the initial letter of ‘class’, but my past in functional analysis may have prevented this, as $C(X)$ designates also the space of continuous functions on $X$ (when $X$ is a topological space). Thus, I reverted to $K$ instead of $C$, since my mother tongue is German, Class = Klasse (in German), and the sounds corresponding to $C$ and $K$ are the same.

from Grothendieck’s letter to Bruce Magurn, on 9th February 1985, quoted after:

category: people