|
|
|
|
Algebraic K-theory
|
(Topic)
|
|
|
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 ![$ [p]$](http://images.planetmath.org:8080/cache/objects/4117/l2h/img11.png "$ [p]$") and ![$ [q]$](http://images.planetmath.org:8080/cache/objects/4117/l2h/img12.png "$ [q]$") is the equivalence class of their direct sum:
![$ [p]+[q] = [p \oplus q]$](http://images.planetmath.org:8080/cache/objects/4117/l2h/img13.png "$ [p]+[q] = [p \oplus q]$") 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) |
- 1
- H. Inassaridze, Algebraic K-theory.
Kluwer Academic Publishers, 1994.
- 2
- Jean-Louis Loday, Cyclic Homology.
Springer-Verlag, 1992.
|
"Algebraic K-theory" is owned by mhale.
|
|
(view preamble)
Cross-references: structure, product, ideal, non-trivial elements, number, finite, integral, infinite, cone, suspension, classifying space, plus, column, row, diagonals, group generated by, subgroup, coefficients, integer, homology group, group, abelianisation, open subset, open, subset, topology, general linear groups, finitely generated projective modules, direct sum, equivalence classes, sum, similarity transformations, idempotents, differences, abelian group, Grothendieck group, embeddings, algebras, matrix, direct limit, algebraic, Morita invariant, properties, invariants, rings, category, functors, series
There are 3 references to this entry.
This is version 7 of Algebraic K-theory, born on 2003-03-20, modified 2006-02-24.
Object id is 4117, canonical name is AlgebraicKTheory.
Accessed 6210 times total.
Classification:
| AMS MSC: | 19-00 ($K$-theory :: General reference works ) | | | 18F25 (Category theory; homological algebra :: Categories and geometry :: Algebraic $K$-theory and $L$-theory) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle K_1(R) = \mathrm{GL}_\infty(R)/[\mathrm{GL}_\infty(R),\mathrm{GL}_\infty(R)]. $](http://images.planetmath.org:8080/cache/objects/4117/l2h/img29.png "$\displaystyle K_1(R) = \mathrm{GL}_\infty(R)/[\mathrm{GL}_\infty(R),\mathrm{GL}_\infty(R)]. $"){kind=small}
