Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequence

K_n(R)

of functors from rings to abelian groups, for all integers n. For historical reasons, the lower K-groups K₀ and K₁ are thought of in somewhat different terms from the higher algebraic K-groups K_n for n ≥ 2. Indeed, the lower groups are more accessible, and have more applications, than the higher groups. The theory of the higher K-groups is noticeably deeper, and certainly much harder to compute (even when R is the ring of integers).

The group K₀(R) generalises the construction of the ideal class group of a ring, using projective modules. Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of Serre on projective modules that now is the Quillen-Suslin theorem; numerous other connections with classical algebraic problems were found in this era. Similarly, K₁(R) is a modification of the group of units in a ring, using elementary matrix theory. The group K₁(R) is important in topology, especially when R is a group ring, because its quotient the Whitehead group contains the Whitehead torsion used to study problems in simple homotopy theory and surgery theory; the group K₀(R) also contains other invariants such as the finiteness invariant. Since the 1980s, algebraic K-theory has increasingly had applications to algebraic geometry. For example, motivic cohomology is closely related to algebraic K-theory.