Algebraic K-theory
Algebraic K-theory is a series of functors
on the category of rings.
Broadly speaking, it classifies ring invariants, i.e. ring properties that are Morita invariant.
The functor
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 .
The functor
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.
Note that this is the same as , the first group homology group (with integer coefficients).
The functor
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-functors
Higher K-groups are defined using the Quillen plus construction,
(1) |
where is the classifying space of .
Rough sketch of suspension:
(2) |
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.
(3) |
Algebraic K-theory has a product structure,
(4) |
References
- 1 H. Inassaridze, Algebraic K-theory. Kluwer Academic Publishers, 1994.
- 2 Jean-Louis Loday, Cyclic Homology. Springer-Verlag, 1992.
Title | Algebraic K-theory |
---|---|
Canonical name | AlgebraicKtheory |
Date of creation | 2013-03-22 13:31:32 |
Last modified on | 2013-03-22 13:31:32 |
Owner | mhale (572) |
Last modified by | mhale (572) |
Numerical id | 10 |
Author | mhale (572) |
Entry type | Topic |
Classification | msc 19-00 |
Classification | msc 18F25 |
Related topic | KTheory |
Related topic | GrothendieckGroup |
Related topic | StableIsomorphism |