Today Alexander Grothendieck, probably best known for his significant development of the theory of schemes, died this morning at the Saint-Girons hospital in Ariège. He was 86 years old. Let me just list my four favourite Grothendieck inventions as a personal tribute:
- The functor of points is the perspective that a scheme (also invented by Grothendieck) can be viewed as a contravariant functor from affine schemes to sets. This perspective is immensely useful and clarifying in the theory of algebraic groups especially, and the representability of such functors is a fascinating field of study.
- The definition of $K_0$ was invented for his Riemann-Roch theorem, Grothendieck defined this simple functor of an essentially small abelian category to be the free abelian group whose generators $[A]$ correspond to isomorphism classes of objects $A$ and with relations $[B] = [A] + [C]$ for every short exact sequence $0\to A\to B\to C\to 0$. It's such a simple and elegant definition and today it forms part of the basis for algebraic $K$-theory.
- Grothendieck topologies are generalisations of open covers where the open inclusions are replaced with more general morphisms, so that things like etale cohomology can be defined. It's a fairly simple idea with enormous applications.
- Universal homological $\delta$-functors are a clean and crisp way to talk about derived functors and constructing natural transformations between them and they are used in homological algebra everywhere.