Personal Reminiscences of the Birth of Algebraic K-theory

Abstract

These informal reminiscences, presented at the ICTP 2002 Conference on algebraic K- theory, recount the trajectory in the author's early research, from work on the Serre Conjecture (on projective modules over polynomial algebras), via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic K-theory. The solution of the Congruence Subgroup Problem is presented as a pivotal event. Mathematics Subject Classifications (2000): 19-xx, 20G30. The organizers, Remi Kuku, Claudio Pedrini, and Max Karoubi, of the ICTP 2002 Conference on Algebraic K-theory graciously used the occasion to celebrate my 70th birthday. During the ceremonial session there I offered some impromptu per- sonal reminiscences about how my own research led me into what came to be called algebraic K-theory, and about the early development of the subject. The interest shown there led the organizers to request that I record those reminiscences for the conference proceedings. That is the genesis of the still somewhat informal account that follows. In the late 1950s and early 1960s, a period of great mathematical ferment, there were several interwoven mathematical themes in play from which algebraic K-theory gradually took shape. • Various motifs in algebraic and geometric topology - generalized cohomology theories, fiber bundles, simple homotopy theory. • The precocious birth of homological algebra, and the development of category theory, in part for an axiomatic treatment of homological algebra. In particu- lar, the paradigm shifting intrusion of homological methods into commutative algebra. • Grothendieck's radical and visionary re-founding and expansion of algebraic geometry.