Algebraic K-theory

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

K

0

Let

R

be a ring and denote by

M

∞

⁢

(

R

)

the algebraic

direct limit

of matrix algebras

M

n

⁢

(

R

)

under the

embeddings

M

n

⁢

(

R

)

→

M

n

+

1

⁢

(

R

)

:

a

↦

(

a

0

0

0

)

.
The zeroth K-group of

R

,

K

0

⁢

(

R

)

, is the

Grothendieck group

(

abelian group

of formal

differences

) of

idempotents

in

M

∞

⁢

(

R

)

up to similarity transformations.
Let

p

∈

M

m

⁢

(

R

)

and

q

∈

M

n

⁢

(

R

)

be two idempotents.
The sum of their

equivalence classes

[

p

]

and

[

q

]

is the equivalence class of their

direct sum

:

[

p

]

+

[

q

]

=

[

p

⊕

q

]

where

p

⊕

q

=

diag

⁢

(

p

,

q

)

∈

M

m

+

n

⁢

(

R

)

.
Equivalently, one can work with

finitely generated projective modules

over

R

.

The functor

K

1

Denote by

GL

∞

⁢

(

R

)

the direct limit of

general linear groups

GL

n

⁢

(

R

)

under the embeddings

GL

n

⁢

(

R

)

→

GL

n

+

1

⁢

(

R

)

:

g

↦

(

g

0

0

1

)

.
Give

GL

∞

⁢

(

R

)

the direct limit

topology

, i.e. a subset

U

of

GL

∞

⁢

(

R

)

is open if and only if

U

∩

GL

n

⁢

(

R

)

is an open subset of

GL

n

⁢

(

R

)

, for all

n

.
The first K-group of

R

,

K

1

⁢

(

R

)

, is the

abelianisation

of

GL

∞

⁢

(

R

)

, i.e.

K

1

⁢

(

R

)

=

GL

∞

⁢

(

R

)

/

[

GL

∞

⁢

(

R

)

,

GL

∞

⁢

(

R

)

]

.

Note that this is the same as

K

1

⁢

(

R

)

=

H

1

⁢

(

GL

∞

⁢

(

R

)

,

ℤ

)

,
the first

group homology

group (with integer

coefficients

).

The functor

K

2

Let

E

n

⁢

(

R

)

be the

elementary

subgroup

of

GL

n

⁢

(

R

)

.
That is, the group generated by the elementary

n

×

n

matrices

e

i

⁢

j

⁢

(

r

)

,

r

∈

R

,
where

e

i

⁢

j

⁢

(

r

)

is the matrix with ones on the diagonals, the value

r

in row

i

, column

j

and zeros elsewhere.
Denote by

E

∞

⁢

(

R

)

the direct limit of the

E

n

⁢

(

R

)

using the construction above (note

E

∞

⁢

(

R

)

is a subgroup of

GL

∞

⁢

(

R

)

).
The second K-group of

R

,

K

2

⁢

(

R

)

, is the second group homology group (with integer coefficients) of

E

∞

⁢

(

R

)

,

K

2

⁢

(

R

)

=

H

2

⁢

(

E

∞

⁢

(

R

)

,

ℤ

)

.

Higher K-functors

Higher K-groups are defined using the Quillen plus construction,

K

n

alg

⁢

(

R

)

=

π

n

⁢

(

B

⁢

GL

∞

⁢

(

R

)

+

)

,

(1)

where

B

⁢

GL

∞

⁢

(

R

)

is the

classifying space

of

GL

∞

⁢

(

R

)

.

Rough sketch of

suspension

:

Σ

⁢

R

=

Σ

⁢

ℤ

⊗

ℤ

R

(2)

where

Σ

⁢

ℤ

=

C

⁢

ℤ

/

J

⁢

ℤ

.
The cone,

C

⁢

ℤ

, 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

J

⁢

ℤ

consists of those matrices that have only finitely many
non-trivial coefficients.

K

i

⁢

(

R

)

≅

K

i

+

1

⁢

(

Σ

⁢

R

)

(3)

Algebraic K-theory has a

product structure

,

K

i

⁢

(

R

)

⊗

K

j

⁢

(

S

)

→

K

i

+

j

⁢

(

R

⊗

S

)

.

(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

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