Classical algebraic K-theory: The functors K0, K1, K2

Abstract

This chapter discusses the functors of classical algebraic K-theory. The algebraic K-theory started with Grothendieck's construction of an Abelian group K (A) (now denoted K 0 (A)) associated to a suitable subcategory of an Abelian category. The Grothendieck group associated to a semigroup A and the ring associated to a semi-ring leading to discussions on K 0 of rings and K 0 of symmetric monoidal categories with examples such as Burnside rings or representation rings. The K 0 of the category of nilpotent endomorphisms with consequent fundamental theorems for K 0 , G 0 of rings, and schemes are also discussed. The K 1 (A) with the observation that the definition due to Bass coincides with Quillen's K 1 (Ρ (A)) or π 1 (BGL (A) + ). The local rings of K 1 and skew fields; Mennicke symbols and some stability results for K 1 are discussed. Some K 1 – K 0 exact sequences– Mayer–Vietoris, localization sequences and the exact sequence associated to an ideal are reviewed. The localization sequence leads to the introduction of the fundamental theorem for K l . In the context of the Bloch–Kato, conjecture for higher-dimensional K-theory of fields with a brief review of the current status of the conjecture is discussed. The applications of K 2 to pseudo-isotopy of manifolds and Bloch's formula for Chow groups are presented.