Let be a ring and denote by the algebraic direct limit of matrix algebras under the embeddings . The zeroth K-group of , , is the Grothendieck group (abelian group of formal differences) of idempotents in up to similarity transformations. Let and be two idempotents. The sum of their equivalence classes and is the equivalence class of their direct sum: where . Equivalently, one can work with finitely generated projective modules over .
Denote by the direct limit of general linear groups under the embeddings . Give the direct limit topology, i.e. a subset of is open if and only if is an open subset of , for all . The first K-group of , , is the abelianisation of , i.e.
Let be the elementary subgroup of . That is, the group generated by the elementary matrices , , where is the matrix with ones on the diagonals, the value in row , column and zeros elsewhere. Denote by the direct limit of the using the construction above (note is a subgroup of ). The second K-group of , , is the second group homology group (with integer coefficients) of ,
Higher K-groups are defined using the Quillen plus construction,
where is the classifying space of .
Rough sketch of suspension:
where . The cone, , is the set of infinite matrices with integral coefficients that have a finite number of non-trivial elements on each row and column. The ideal consists of those matrices that have only finitely many non-trivial coefficients.
Algebraic K-theory has a product structure,
- 1 H. Inassaridze, Algebraic K-theory. Kluwer Academic Publishers, 1994.
- 2 Jean-Louis Loday, Cyclic Homology. Springer-Verlag, 1992.
|Date of creation||2013-03-22 13:31:32|
|Last modified on||2013-03-22 13:31:32|
|Last modified by||mhale (572)|