Algebraic K-Theory

Algebraic -theory associates Abelian groups to a unit ring to study its projective modules and their higher relations. The group is the Grothendieck group of the monoid of isomorphism classes of finitely generated projective modules, with direct sum as the operation. Thus in (Weibel 2013).

Writing under the inclusions , the group is its abelianization. Higher groups can be defined by homotopy groups of an associated space. For a field , via vector space dimension, and via the determinant.

The Gersten conjecture concerns the maps from to the -groups of the residue fields of a regular local ring.