Reference Work Entry

Handbook of K-Theory

pp 71-110

K-Theory of Truncated Polynomial Algebras

  • Lars HesselholtAffiliated withDepartment of Mathematics, Massachusetts Institute of Technology

Abstract

In general, if A is a ring and IA a two-sided ideal, one defines the K-theory of A relative to I to be the mapping fiber of the map of K-theory spectra induced by the canonical projection from A to A/I. Hence, there is anatural exact triangle of spectra

$$ K(A,I) \rightarrow K(A) \rightarrow K(A/I) \xrightarrow{\partial} K(A,I)[-1] $$

and an induced natural long-exact sequence of K-groups

$$ ... \rightarrow K_{q}(A,I) \rightarrow K_{q}(A) \rightarrow K_{q}(A/I) \xrightarrow{\partial} K_{q-1}(A,I) \rightarrow ... $$
.