3Hochschild homology and cohomology
III Algebras
3.2 Cohomology
As mentioned, the construction of Hochschild cohomology involves applying
Hom
A-A
(
· , M
) to the Hochschild chain complex, and looking at the terms
Hom
A-A
(
A
⊗(n+2)
, M
). This is usually not very convenient to manipulate, as
it involves talking about bimodule homomorphisms. However, we note that
A
⊗(n+2)
is a free
A
-
A
-bimodule generated by a basis of
A
⊗n
. Thus, there is a
canonical isomorphism
Hom
A-A
(A
⊗(n+2)
, M)
∼
=
Hom
k
(A
⊗n
, M),
and k-linear maps are much simpler to work with.
Definition
(Hochschild cochain complex)
.
The Hochschild cochain complex
of an
A
-
A
-bimodule
M
is what we obtain by applying
Hom
A-A
(
· , M
) to the
Hochschild chain complex of A. Explicitly, we can write it as
Hom
k
(k, M) Hom
k
(A, M) Hom
k
(A ⊗ A, M) · · · ,
δ
0
δ
1
where
(δ
0
f)(a) = af(1) − f(1)a
(δ
1
f)(a
1
⊗ a
2
) = a
1
f(a
2
) − f(a
1
a
2
) + f(a
1
)a
2
(δ
2
f)(a
1
⊗ a
2
⊗ a
3
) = a
1
f(a
2
⊗ a
3
) − f(a
1
a
2
⊗ a
3
)
+ f(a
1
⊗ a
2
a
3
) − f(a
1
⊗ a
2
)a
3
(δ
n−1
f)(a
1
⊗ · · · ⊗ a
n
) = a
1
f(a
2
⊗ · · · ⊗ a
n
)
+
n
X
i=1
(−1)
i
f(a
1
⊗ · · · ⊗ a
i
a
i+1
⊗ · · · ⊗ a
n
)
+ (−1)
n+1
f(a
1
⊗ · · · ⊗ a
n−1
)a
n
The reason the end ones look different is that we replaced
Hom
A-A
(
A
⊗(n+2)
, M
)
with Hom
k
(A
⊗n
, M).
The crucial observation is that the exactness of the Hochschild chain complex,
and in particular the fact that d
2
= 0, implies im δ
n−1
⊆ ker δ
n
.
Definition (Cocycles). The cocycles are the elements in ker δ
n
.
Definition (Coboundaries). The coboundaries are the elements in im δ
n
.
These names came from algebraic topology.
Definition (Hochschild cohomology groups). We define
HH
0
(A, M) = ker δ
0
HH
n
(A, N) =
ker δ
n
im δ
n−1
These are k-vector spaces.
If we do not want to single out
HH
0
, we can extend the Hochschild cochain
complex to the left with 0, and setting
δ
n
= 0 for
n <
0 (or equivalently extending
the Hochschild chain complex to the right similarly), Then
HH
0
(A, M) =
ker δ
0
im δ
−1
= ker δ
0
.
The first thing we should ask ourselves is when the cohomology groups van-
ish. There are two scenarios where we can immediately tell that the (higher)
cohomology groups vanish.
Lemma.
Let
M
be an injective bimodule. Then
HH
n
(
A, M
) = 0 for all
n ≥
1.
Proof. Hom
A-A
( · , M) is exact.
Lemma.
If
A
A
A
is a projective bimodule, then
HH
n
(
A, M
) = 0 for all
M
and
all n ≥ 1.
If we believed the previous remark that we could compute Hochschild coho-
mology with any projective resolution, then this result is immediate — indeed,
we can use
· · · →
0
→
0
→ A → A →
0 as the projective resolution. However,
since we don’t want to prove such general results, we shall provide an explicit
computation.
Proof.
If
A
A
A
is projective, then all
A
⊗n
are projective. At each degree
n
, we
can split up the Hochschild chain complex as the short exact sequence
0
A
⊗(n+3)
ker d
n
A
⊗(n+2)
im d
n−1
0
d
n
d
n−1
The im d is a submodule of A
⊗(n+1)
, and is hence projective. So we have
A
⊗(n+2)
∼
=
A
⊗(n+3)
ker d
n
⊕ im d
n−1
,
and we can write the Hochschild chain complex at n as
ker d
n
⊕
A
⊗(n+3)
ker d
n
A
⊗(n+3)
ker d
n
⊕ im d
n−1
A
⊗(n+1)
im d
n−1
⊕ im d
n−1
(a, b) (b, 0)
(c, d) (0, d)
d
n
d
n−1
Now
Hom
(
· , M
) certainly preserves the exactness of this, and so the Hochschild
cochain complex is also exact. So we have HH
n
(A, M) = 0 for n ≥ 1.
This is a rather strong result. By knowing something about
A
itself, we
deduce that the Hochschild cohomology of any bimodule whatsoever must vanish.
Of course, it is not true that
HH
n
(
A, M
) vanishes in general for
n ≥
1, or
else we would have a rather boring theory. In general, we define
Definition (Dimension). The dimension of an algebra A is
Dim A = sup{n : HH
n
(A, M) 6= 0 for some A-A-bimodule M}.
This can be infinite if such a sup does not exist.
Thus, we showed that
A
A
A
embeds as a direct summand in
A ⊗ A
, then
Dim A = 0.
Definition
(
k
-separable)
.
An algebra
A
is
k
-separable if
A
A
A
embeds as a
direct summand of A ⊗ A.
Since A ⊗ A is a free A-A-bimodule, this condition is equivalent to A being
projective. However, there is some point in writing the definition like this.
Note that an
A
-
A
-bimodule is equivalently a left
A ⊗ A
op
-module. Then
A
A
A
is a direct summand of
A ⊗ A
if and only if there is a separating idempotent
e ∈ A ⊗ A
op
so that
A
A
A
viewed as A ⊗ A
op
-bimodule is (A ⊗ A
op
)e.
This is technically convenient, because it is often easier to write down a
separating idempotent than to prove directly A is projective.
Note that whether we write
A⊗A
op
or
A⊗A
is merely a matter of convention.
They have the same underlying set. The notation
A ⊗ A
is more convenient
when we take higher powers, but we can think of
A ⊗ A
op
as taking
A
as a
left-
A
right-
k
module and
A
op
as a left-
k
right-
A
, and tensoring them gives a
A-A-bimodule.
We just proved that separable algebras have dimension 0. Conversely, we
have
Lemma. If Dim A = 0, then A is separable.
Proof. Note that there is a short exact sequence
0 ker µ A ⊗ A A 0
µ
If we can show this splits, then
A
is a direct summand of
A ⊗ A
. To do so, we
need to find a map A ⊗ A → ker µ that restricts to the identity on ker µ.
To do so, we look at the first few terms of the Hochschild chain complex
· · · im d ⊕ ker µ A ⊗ A A 0
d
µ
.
By assumption, for any
M
, applying
Hom
A-A
(
· , M
) to the chain complex gives
an exact sequence. Omitting the annoying
A-A
subscript, this sequence looks like
0 −→ Hom(A, M)
µ
∗
−→ Hom(A ⊗ A, M)
(∗)
−→ Hom(ker µ, M) ⊕ Hom(im d, M)
d
∗
−→ · · ·
Now
d
∗
sends
Hom
(
ker µ, M
) to zero. So
Hom
(
ker µ, M
) must be in the image
of (∗). So the map
Hom(A ⊗ A, M) −→ Hom(ker µ, M)
must be surjective. This is true for any
M
. In particular, we can pick
M
=
ker µ
.
Then the identity map
id
ker µ
lifts to a map
A ⊗ A → ker µ
whose restriction to
ker µ is the identity. So we are done.
Example. M
n
(
k
) is separable. It suffices to write down the separating idem-
potent. We let
E
ij
be the elementary matrix with 1 in the (
i, j
)th slot and 0
otherwise. We fix j, and then
X
i
E
ij
⊗ E
ji
∈ A ⊗ A
op
is a separating idempotent.
Example.
Let
A
=
kG
with
char k - |G|
. Then
A ⊗ A
op
=
kG ⊗
(
kG
)
op
∼
=
kG ⊗ kG. But this is just isomorphic to k(G × G), which is again semi-simple.
Thus, as a bimodule,
A ⊗ A
is completely reducible. So the quotient of
A
A
A
is a direct summand (of bimodules). So we know that whenever
char k - |G|
,
then kG is k-separable.
The obvious question is — is this notion actually a generalization of separable
field extensions? This is indeed the case.
Fact.
Let
L/K
be a finite field extension. Then
L
is separable as a
K
-algebra
iff L/K is a separable field extension.
However
k
-separable algebras must be finite-dimensional
k
-vector spaces. So
this doesn’t pass on to the infinite case.
In the remaining of the chapter, we will study what Hochschild cohomology
in the low dimensions mean. We start with
HH
0
. The next two propositions
follow from just unwrapping the definitions:
Proposition. We have
HH
0
(A, M) = {m ∈ M : am − ma = 0 for all a ∈ A}.
In particular, HH
0
(A, A) is the center of A.
Proposition.
ker δ
1
= {f ∈ Hom
k
(A, M) : f(a
1
a
2
) = a
1
f(a
2
) + f(a
1
)a
2
}.
These are the derivations from A to M. We write this as Der(A, M).
On the other hand,
im δ
0
= {f ∈ Hom
k
(A, M) : f(a) = am − ma for some m ∈ M}.
These are called the inner derivations from A to M. So
HH
1
(A, M) =
Der(A, M)
InnDer(A, M)
.
Setting A = M, we get the derivations and inner derivations of A.
Example. If A = k[x], and char k = 0, then
DerA =
p(X)
d
dx
: p(x) ∈ k[X]
,
and there are no (non-trivial) inner derivations because
A
is commutative. So
we find
HH
1
(k[X], k[X])
∼
=
k[X].
In general, Der(A) form a Lie algebra, since
D
1
D
2
− D
2
D
1
∈ End
k
(A)
is in fact a derivation if D
1
and D
2
are.
There is another way we can think about derivations, which is via semi-direct
products.
Definition
(Semi-direct product)
.
Let
M
be an
A
-
A
-bimodule. We can form
the semi-direct product of
A
and
M
, written
A n M
, which is an algebra with
elements (a, m) ∈ A × M, and multiplication given by
(a
1
, m
1
) · (a
2
, m
2
) = (a
1
a
2
, a
1
m
2
+ m
1
a
2
).
Addition is given by the obvious one.
Alternatively, we can write
A n M
∼
=
A + Mε,
where
ε
commutes with everything and
ε
2
= 0. Then
Mε
forms an ideal with
(Mε)
2
= 0.
In particular, we can look at
A n A
∼
=
A
+
Aε
. This is often written as
A
[
ε
].
Previously, we saw first cohomology can be understood in terms of derivations.
We can formulate derivations in terms of this semi-direct product.
Lemma. We have
Der
k
(A, M)
∼
=
{algebra complements to M in A n M isomorphic to A}.
Proof. A complement to M is an embedded copy of A in A n M ,
A A n M
a (a, D
a
)
The function
A → M
given by
a 7→ D
a
is a derivation, since under the embedding,
we have
ab 7→ (ab, aD
b
+ D
a
b).
Conversely, a derivation
f
:
A → M
gives an embedding of
A
in
A n M
given by
a 7→ (a, f(a)).
We can further rewrite this characterization in terms of automorphisms of
the semi-direct product. This allows us to identify inner derivations as well.
Lemma. We have
Der(A, M)
∼
=
automorphisms of A n M of the form
a 7→ a + f(a)ε, mε 7→ mε
,
where we view A n M
∼
=
A + Mε.
Moreover, the inner derivations correspond to automorphisms achieved by
conjugation by 1 + mε, which is a unit with inverse 1 − mε.
The proof is a direct check.
This applies in particular when we pick
M
=
A
A
A
, and the Lie algebra of
derivations of A may be thought of as the set of infinitesimal automorphisms.
Let’s now consider
HH
2
(
A, M
). This is to be understood in terms of exten-
sions, of which the “trivial” example is the semi-direct product.
Definition
(Extension)
.
Let
A
be an algebra and
M
and
A
-
A
-bimodule. An
extension of
A
by
M
. An extension of
A
by
M
is a
k
-algebra
B
containing a
2-sided ideal I such that
– I
2
= 0;
– B/I
∼
=
A; and
– M
∼
=
I as an A-A-bimodule.
Note that since
I
2
= 0, the left and right multiplication in
B
induces an
A-A-bimodule structure on I, rather than just a B-B-bimodule.
We let
π
:
B → A
be the canonical quotient map. Then we have a short
exact sequence
0 I B A 0 .
Then two extensions
B
1
and
B
2
are isomorphic if there is a
k
-algebra isomorphism
θ : B
1
→ B
2
such that the following diagram commutes:
B
1
0 I A 0
B
2
θ
.
Note that the semi-direct product is such an extension, called the split extension.
Proposition.
There is a bijection between
HH
2
(
A, M
) with the isomorphism
classes of extensions of A by M.
This is something that should be familiar if we know about group cohomology.
Proof.
Let
B
be an extension with, as usual,
π
:
B → A
,
I
=
M
=
ker π
,
I
2
= 0.
We now try to produce a cocycle from this.
Let
ρ
be any
k
-linear map
A → B
such that
π
(
ρ
(
a
)) =
a
. This is possible
since
π
is surjective. Equivalently,
ρ
(
π
(
b
)) =
b mod I
. We define a
k
-linear map
f
ρ
: A ⊗ A → I
∼
=
M
by
a
1
⊗ a
2
7→ ρ(a
1
)ρ(a
2
) − ρ(a
1
a
2
).
Note that the image lies in I since
ρ(a
1
)ρ(a
2
) ≡ ρ(a
1
a
2
) (mod I).
It is a routine check that f
ρ
is a 2-cocycle, i.e. it lies in ker δ
2
.
If we replace ρ by any other ρ
0
, we get f
ρ
0
, and we have
f
ρ
(a
1
⊗ a
2
) − f
ρ
0
(a
1
⊗ a
2
)
= ρ(a
1
)(ρ(a
2
) − ρ
0
(a
2
)) − (ρ(a
1
a
2
) − ρ
0
(a
1
a
2
)) + (ρ(a
1
) − ρ
0
(a
1
))ρ
0
(a
2
)
= a
1
· (ρ(a
2
) − ρ
0
(a
2
)) − (ρ(a
1
a
2
) − ρ
0
(a
1
a
2
)) + (ρ(a
1
) − ρ
0
(a
1
)) · a
2
,
where · denotes the A-A-bimodule action in I. Thus, we find
f
ρ
− f
ρ
0
= δ
1
(ρ − ρ
0
),
noting that ρ − ρ
0
actually maps to I.
So we obtain a map from the isomorphism classes of extensions to the second
cohomology group.
Conversely, given an
A
-
A
-bimodule
M
and a 2-cocycle
f
:
A ⊗ A → M
, we
let
B
f
= A ⊕ M
as k-vector spaces. We define the multiplication map
(a
1
, m
1
)(a
2
, m
2
) = (a
1
a
2
, a
1
m
2
+ m
1
a
2
+ f(a
1
⊗ a
2
)).
This is associative precisely because of the 2-cocycle condition. The map (
a, m
)
→
a
yields a homomorphism
π
:
B → A
, with kernel
I
being a two-sided ideal of
B
which has
I
2
= 0. Moreover,
I
∼
=
M
as an
A
-
A
-bimodule. Taking
ρ
:
A → B
by
ρ(a) = (a, 0) yields the 2-cocycle we started with.
Finally, let
f
0
be another 2 co-cycle cohomologous to
f
. Then there is a
linear map τ : A → M with
f − f
0
= δ
1
τ.
That is,
f(a
1
⊗ A
2
) = f
0
(a
1
⊗ a
2
) + a
1
· τ(a
2
) − τ(a
1
a
2
) + τ(a
1
) · a
2
.
Then consider the map B
f
→ B
0
f
given by
(a, m) 7→ (a, m + τ(a)).
One then checks this is an isomorphism of extensions. And then we are done.
In the proof, we see 0 corresponds to the semi-direct product.
Corollary. If HH
2
(A, M) = 0, then all extensions are split.
We now prove some theorems that appear to be trivialities. However, they
are trivial only because we now have the machinery of Hochschild cohomology.
When they were first proved, such machinery did not exist, and they were written
in a form that seemed less trivial.
Theorem (Wedderburn, Malcev). Let B be a k-algebra satisfying
– Dim(B/J(B)) ≤ 1.
– J(B)
2
= 0
Then there is an subalgebra A
∼
=
B/J(B) of B such that
B = A n J(B).
Furthermore, if
Dim
(
B/J
(
B
)) = 0, then any two such subalgebras
A, A
0
are
conjugate, i.e. there is some x ∈ J(B) such that
A
0
= (1 + x)A(1 + x)
−1
.
Notice that 1 + x is a unit in B.
In fact, the same result holds if we only require
J
(
B
) to be nilpotent. This
follows from an induction argument using this as a base case, which is messy
and not really interesting.
Proof. We have J(B)
2
= 0. Since we know Dim(B/J(B)) ≤ 1, we must have
HH
2
(A, J(B)) = 0
where
A
∼
=
B
J(B)
.
Note that we regard
J
(
B
) as an
A
-
A
-bimodule here. So we know that all
extension of A by J(B) are semi-direct, as required.
Furthermore, if
Dim
(
B/J
(
B
)) = 0, then we know
HH
1
(
A, J
(
A
)) = 0. So by
our older lemmas, we see that complements are all conjugate, as required.
Corollary.
If
k
is algebraically closed and
dim
k
B < ∞
, then there is a subal-
gebra A of B such that
A
∼
=
B
J(B)
,
and
B = A n J(B).
Moreover,
A
is unique up to conjugation by units of the form 1+
x
with
x ∈ J
(
B
).
Proof.
We need to show that
Dim
(
A
) = 0. But we know
B/J
(
B
) is a semi-
simple
k
-algebra of finite dimension, and in particular is Artinian. So by
Artin–Wedderburn, we know
B/J
(
B
) is a direct sum of matrix algebras over
k
(since k is algebraically closed and dim
k
(B/J(B))).
We have previously observed that
M
n
(
k
) is
k
-separable. Since
k
-separability
behaves well under direct sums, we know
B/J
(
B
) is
k
-separable, hence has
dimension zero.
It is a general fact that J(B) is nilpotent.