3Hochschild homology and cohomology
III Algebras
3.5 Hochschild homology
We don’t really have much to say about Hochschild homology, but we are morally
obliged to at least write down the definition.
To do Hochschild homology, we apply
· ⊗
A-A
M
for an
A
-
A
-bimodule
M
to
the Hochschild chain complex.
· · · A ⊗
k
A ⊗
k
A A ⊗
k
A A 0.
d d
µ
,
We will ignore the
→ A →
0 bit. We need to consider what
· ⊗
A-A
·
means. If
we have bimodules
V
and
W
, we can regard
V
as a right
A ⊗ A
op
-module. We
can also think of W as a left A ⊗ A
op
module. We let
B = A ⊗ A
op
,
and then we just consider
V ⊗
B
W =
V ⊗
k
W
hvx ⊗ w − v ⊗ xw : w ∈ Bi
=
V ⊗
k
W
hava
0
⊗ w − v ⊗ a
0
wai
.
Thus we have
· · · (A ⊗
k
A ⊗
k
A) ⊗
A-A
M (A ⊗
k
A) ⊗
A-A
M
∼
=
M
b
1
b
0
,
Definition (Hochschild homology). The Hochschild homology groups are
HH
0
(A, M) =
M
im b
0
HH
i
(A, M) =
ker b
i−1
im b
i
for i > 0.
A long time ago, we counted the number of simple
kG
-modules for
k
alge-
braically closed of characteristic
p
when
G
is finite. In the proof, we used
A
[A,A]
,
and we pointed out this is HH
0
(A, A).
Lemma.
HH
0
(A, M) =
M
hxm − mx : m ∈ M, x ∈ Ai
.
In particular,
HH
0
(A, A) =
A
[A, A]
.
Proof. Exercise.