Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre, after sheaves had be invented byJean Leray. Grothendieck took them to a higher level, changing the tools and the level of abstraction.

Amongst his insights, he shifted attention from the study of individual varieties to the *relative point of view* (pairs of varieties related by a morphism), allowing a broad generalization of many classical theorems. This he applied first to the Riemann-Roch theorem, around 1954, which had already recently been generalized to any dimension by Hirzebruch).

He adapted the use of non-closed (*generic*) points, which led to the theory of schemes. He also pioneered the systematic use of nilpotents. As 'functions' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings.

His work is at a higher level of abstraction than prior versions of algebraic geometry. His theory of schemes has become established as the best universal foundation for this major field, due to its great power. In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

Its influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules. (It also provoked adverse reactions, with many mathematicians seeking out more concrete areas and problems.)

The bulk of Grothendieck's published work is collected in the monumental, and yet incomplete, *Elements de geometrie algebrique* (EGA) and *Séminaire de géométrie algébrique* (SGA). This is considered to have gone a long way to answering Kronecker's wish for a foundational theory based on the integers alone, by using the relative point of view based on finitely-generated commutative rings. The *style* of the mathematics is very distant from Kronecker's, though. On the EGA project Grothendieck collaborated with Jean Dieudonné. It is axiomatic, and claims descent (according to Dieudonné) from David Hilbert's approach; as interpreted by Nicolas Bourbaki. On the other hand Grothendieck himself applied an intuitive approach, as well as a generalising one. There were many other contributors to the SGA series.

Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers. Weil realized that to prove such a connection one needed a new cohomology theory, but neither he nor any other expert saw how to do this until such a theory was found by Grothendieck. This program culminated in the proofs of the Weil conjectures by Grothendieck's student Pierre Deligne in the early 1970's after Grothendieck had withdrawn from mathematics.

- Topological tensor products and nuclear spaces
- "Continual" and "discrete" duality (derived categories and "six operations").
*Yoga*of the Grothendieck-Riemann-Roch theorem (K-theory, relation with intersection theory).- Schemes.
- Toposes.
- Étale cohomology including l-adic cohomology.
- Motivess and the motivic Galois group (and Grothendieck categories)
- Crystalss and crystalline cohomology,
*yoga*of De Rham and Hodge coefficients. - Topological algebra, infinity-fields (?), 'dérivateurs', cohomological formalism of toposes as an inspiration for a new homotopic algebra
- Tame topology.
*Yoga*of anabelian geometry and Galois-Teichmüller theory.- Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.

Here the usage of *yoga* means a kind of 'meta-theory' that can be used heuristically.

Born to Jewish parents, he was a displaced person during much of his childhood due to the upheavals of World War II. Alexander lived with his father, a revolutionary named Alexander Shapiro, and his mother, Hanka Grothendieck, in Berlin until 1933. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Alexander with a family in Hamburg where he went to school. During this time, his parents fought in the Spanish Civil War. In 1939 Alexander came to France and lived in various camps for displaced persons with his mother. His father was sent to Auschwitz where he died in 1942. After the war, young Grothendieck studied mathematics in France, initially at Montpellier; he came to Paris in 1948. He wrote his dissertation under Laurent Schwartz in functional analysis, from 1950. At this time he was a leading expert in the theory of topological vector spaces. However he set this subject aside by 1957 in order to work in algebraic geometry and homological algebra.

Grothendieck's radical left-wing and pacifist politics were doubtless born of his wartime experiences. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam war. He retired from scientific life around 1970, after having discovered the partly military funding of IHES (see pp. xii and xiii of SGA1, Springer Lecture Notes 224). He returned to academics a few years later as a professor at the University of Montpellier, where he stayed until his retirement in 1988. His criticisms of the scientific community are also contained in a letter written in 1988, in which he states the reasons for his refusal of the Crafoord Prize.

While not publishing mathematical research in conventional ways during the 1980s, he produced several influential manuscripts with limited distribution, with both mathematical and biographical content.
The 2000 page autobiographical manuscript *Récoltes et Semailles* is now available on the internet in the French original, and an English translation is underway.
His *Esquisse d'un programme* (1984)is a proposal for a position at the Centre National de la Recherche Scientifique, which he held from 1984 to his retirement in 1988. Ideas from it have proved influential, and have been developed by others, in particular in a new field emerging as anabelian geometry. In *La Clef des Songes* he explains how the reality of dreams convinced him of God's existence.

In 1991, he left his home and disappeared. He is said to now live in the Pyrenees, a Buddhist, and to entertain no visitors. Other rumors have him live in Ardèche (in the Massif Central mountains), to be herding goats and entertaining radical ecological theories. Though he has been inactive for many years, he remains one of the greatest and most influential mathematicians of modern times.

- MacTutor biography of Grothendieck
- Grothendieck Circle, collection of mathematical and biographical information, photos, links to his writings
- Institut des Hautes Études Scientifiques
- Récoltes et Semailles (original text)