# What is the history behind the concept of "schemes" in algebraic geometry?

Let me start by saying that I've just started studying algebraic geometry. I have a habit of trying to find out something about the motivation behind new concepts that I'm studying, in this case that means that I'm trying to find out something about how the concept of schemes arose in algebraic geometry, and how to view schemes from a historical perspective.

• The question is answered here hsm.stackexchange.com/questions/1813/… Aug 15 '15 at 22:44
• Two recommended references: "The Rising Sea: Grothendieck on simplicity and generality" by Colin McLarty and an article (or the book if you're interested in the mathematical details) by Dieudonné, from 1972 (respectively 1985), called The Historical Development of Algebraic Geometry, The Amer. Math. Monthly, Vol. 79, No. 8, pp. 827-866. (The book is called History of algebraic geometry.)
– Ben
Aug 16 '15 at 7:33

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”

The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry.

The lecturer was Claude Chevalley, and the date was december 12th 1955.

http://www.neverendingbooks.org/grothendiecks-cafe

The rising sea, Mclarty

The history is simple:-) Schemes were invented by Grothendieck. The purpose was unification and simplification of the foundations of algebraic geometry.

The general idea is to shift from considering points of a manifold to certain ideals of a ring of functions on this manifold. This general idea evolved gradually. An early predecessor was Gelfand's theory of commutative Banach algebras, where he starts with an algebra and recovers a space so that this algebra is the algebra of functions on this space. The points of the space are defined as maximal ideals. Grothendieck did similar thing in algebraic geometry. He proposed to consider prime ideals of a ring as points of certain space.

For the reference on Grothendieck's original text see "Elements de geometrie algebrique" on Wikipedia. I believe it was never completely published but various versions were widely circulated. So there is no well-defined publication year. Grothendieck worked and reworked it for almost 10 years ending around 1970.

• I think that it would be a good idea to expand a bit on the historical aspect of this answer. The mathematical content is already nicely covered. Could you provide some references to papers, or review articles discussing this?
– Danu
Aug 15 '15 at 14:08
• I think this is mentioned in one of the Grothendieck biographies, but I have to check. This concerns the influence of Gelfand. Grothendieck's own publications are well known: EGA and SGA. Aug 16 '15 at 2:57