1Artinian algebras
III Algebras
1.5 K
0
We now briefly talk about the notion of K
0
.
Definition
(
K
0
)
.
For any associative
k
-algebra
A
, consider the free abelian
group with basis labelled by the isomorphism classes [
P
] of finitely-generated
projective A-modules. Then introduce relations
[P
1
] + [P
2
] = [P
1
⊕ P
2
],
This yields an abelian group which is the quotient of the free abelian group by
the subgroup generated by
[P
1
] + [P
2
] − [P
1
⊕ P
2
].
The abelian group is K
0
(A).
Example.
If
A
is an Artinian algebra, then we know that any finitely-generated
projective is a direct sum of indecomposable projectives, and this decomposition
is unique by Krull-Schmidt. So
K
0
(A) =
abelian group generated by the isomorphism
classes of indecomposable projectives
.
So
K
0
(
A
)
∼
=
Z
r
, where
r
is the number of isomorphism classes of indecomposable
projectives, which is the number of isomorphism classes of simple modules.
Here we’re using the fact that two indecomposable projectives are isomorphic
iff their simple tops are isomorphic.
It turns out there is a canonical map
K
0
(
A
)
→ A/
[
A, A
]. Recall we have
met
A/
[
A, A
] when we were talking about the number of simple modules. We
remarked that it was the 0th Hochschild homology group, and when
A
=
kG
,
there is a
k
-basis of
A/
[
A, A
] given by
g
i
+ [
A, A
], where
g
i
are conjugacy class
representatives.
To construct this canonical map, we first look at the trace map
M
n
(A) → A/[A, A].
This is a
k
-linear map, invariant under conjugation. We also note that the
canonical inclusion
M
n
(A) → M
n+1
(A)
X 7→
X 0
0 0
is compatible with the trace map. We observe that the trace induces an isomor-
phism
M
n
(A)
[M
n
(A), M
n
(A)]
→
A
[A, A]
,
by linear algebra.
Now if
P
is finitely generated projective. It is a direct summand of some
A
n
.
Thus we can write
A
n
= P ⊕ Q,
for
P, Q
projective. Moreover, projection onto
P
corresponds to an idempotent
e in M
n
(A) = End
A
(A
n
), and that
P = e(A
n
).
and we have
End
A
(P ) = eM
n
(A)e.
Any other choice of idempotent yields an idempotent
e
1
conjugate to
e
in
M
2n
(
A
).
Therefore the trace of an endomorphism of
P
is well-defined in
A/
[
A, A
],
independent of the choice of e. Thus we have a trace map
End
A
(P ) → A/[A, A].
In particular, the trace of the identity map on
P
is the trace of
e
. We call this
the trace of P .
Note that if we have finitely generated projectives
P
1
and
P
2
, then we have
P
1
⊕ Q
1
= A
n
P
2
⊕ Q
2
= A
m
Then we have
(P
1
⊕ P
2
) ⊕ (Q
1
⊕ Q
2
) = A
m+n
.
So we deduce that
tr(P
1
⊕ P
2
) = tr P
1
+ tr P
2
.
Definition
(Hattori-Stallings trace map)
.
The map
K
0
(
A
)
→ A/
[
A, A
] induced
by the trace is the Hattori–Stallings trace map.
Example.
Let
A
=
kG
, and
G
be finite. Then
A/
[
A, A
] is a
k
-vector space
with basis labelled by a set of conjugacy class representatives
{g
i
}
. Then we
know, for a finitely generated projective P , we can write
tr P =
X
r
P
(g
i
)g
i
,
where the
r
p
(
g
i
) may be regarded as class functions. However,
P
may be regarded
as a k-vector space.
So the there is a trace map
End
K
(P ) → k,
and also the “character”
χ
p
:
G → k
, where
χ
P
(
g
) =
tr g
. Hattori proved that if
C
G
(g) is the centralizer of g ∈ G, then
χ
p
(g) = |C
G
(G)|r
p
(g
−1
). (∗)
If
char k
= 0 and
k
is is algebraically closed, then we know
kG
is semi-simple.
So every finitely generated projective is a direct sum of simples, and
K
0
(kG)
∼
=
Z
r
with r the number of simples, and (∗) implies that the trace map
Z
r
∼
=
K
0
(kG) →
kG
[kG, kG]
∼
=
k
r
is the natural inclusion.
This is the start of the theory of algebraic
K
-theory, which is a homology
theory telling us about the endomorphisms of free
A
-modules. We can define
K
1
(A) to be the abelianization of
GL(A) = lim
n→∞
GL
n
(A).
K
2
(
A
) tells us something about the relations required if you express
GL
(
A
) in
terms of generators and relations. We’re being deliberately vague. These groups
are very hard to compute.
Just as we saw in the i = 0 case, there are canonical maps
K
i
(A) → HH
i
(A),
where
HH
∗
is the Hochschild homology. The
i
= 1 case is called the Dennis
trace map. These are analogous to the Chern maps in topology.