3Hochschild homology and cohomology
III Algebras
3.1 Introduction
We now move on to talk about Hochschild (co)homology. We will mostly talk
about Hochschild cohomology, as that is the one that is interesting. Roughly
speaking, given a
k
-algebra
A
and an
A
-
A
-bimodule
M
, Hochschild cohomology is
an infinite sequence of
k
-vector spaces
HH
n
(
A, M
) indexed by
n ∈ N
associated
to the data. While there is in theory an infinite number of such vector spaces,
we are mostly going to focus on the cases of
n
= 0
,
1
,
2, and we will see that
these groups can be interpreted as things we are already familiar with.
The construction of these Hochschild cohomology groups might seem a bit
arbitrary. It is possible to justify these a priori using the general theory of
homological algebra and/or model categories. On the other hand, Hochschild
cohomology is sometimes used as motivation for the general theory of homological
algebra and/or model categories. Either way, we are not going to develop these
general frameworks, but are going to justify Hochschild cohomology in terms of
its practical utility.
Unsurprisingly, Hochschild (co)homology was first developed by Hochschild
in 1945, albeit only working with algebras of finite (vector space) dimension. It
was introduced to give a cohomological interpretation and generalization of some
results of Wedderburn. Later in 1962/1963, Gerstenhaber saw how Hochschild
cohomology was relevant to the deformations of algebras. More recently, it’s
been realized that that the Hochschild cochain complex has additional algebraic
structure, which allows yet more information about deformation.
As mentioned, we will work with
A
-
A
-bimodules over an algebra
A
. If our
algebra has an augmentation, i.e. a ring map to
k
, then we can have a decent
theory that works with left or right modules. However, for the sake of simplicity,
we shall just work with bimodules to make our lives easier.
Recall that a
A
-
A
-bimodule is an algebra with both left and right
A
actions
that are compatible. For example,
A
is an
A
-
A
-bimodule, and we sometimes write
it as
A
A
A
to emphasize this. More generally, we can view
A
⊗(n+2)
=
A⊗
k
· · ·⊗
k
A
as an A-A-bimodule by
x(a
0
⊗ a
1
⊗ · · · ⊗ a
n+1
)y = (xa
0
) ⊗ a
1
⊗ · · · ⊗ (a
n+1
y).
The crucial property of this is that for any
n ≥
0, the bimodule
A
⊗(n+2)
is a
free
A
-
A
-bimodule. For example,
A ⊗
k
A
is free on a single generator 1
⊗
k
1,
whereas if {x
i
} is a k-basis of A, then A ⊗
k
A ⊗
k
A is free on {1 ⊗
k
x
i
⊗
k
1}.
The general theory of homological algebra says we should be interested in
such free things.
Definition
(Free resolution)
.
Let
A
be an algebra and
M
an
A
-
A
-bimodule. A
free resolution of M is an exact sequence of the form
· · · F
2
F
1
F
0
M
d
2
d
1
d
0
,
where each F
n
is a free A-A-bimodule.
More generally, we can consider a projective resolution instead, where we
allow the bimodules to be projective. In this course, we are only interested in
one particular free resolution:
Definition
(Hochschild chain complex)
.
Let
A
be a
k
-algebra with multiplication
map µ : A ⊗ A. The Hochschild chain complex is
· · · A ⊗
k
A ⊗
k
A A ⊗
k
A A 0.
d
1
d
0
µ
We refer to
A
⊗
k
(n+2)
as the degree
n
term. The differential
d
:
A
⊗
k
(n+3)
→
A
⊗
k
(n+2)
is given by
d(a
0
⊗
k
· · · ⊗
k
a
n+1
) =
n+1
X
i=0
(−1)
i
a
0
⊗
k
· · · ⊗
k
a
i
a
i+1
⊗
k
· · · ⊗
k
a
n+2
.
This is a free resolution of
A
A
A
(the exactness is merely a computation, and
we shall leave that as an exercise to the reader). In a nutshell, given an
A
-
A
-
bimodule
M
, its Hochschild homology and cohomology is obtained by applying
· ⊗
A-A
M
and
Hom
A-A
(
· , M
) to the Hochschild chain complex, and then taking
the homology and cohomology of the resulting chain complex. We shall explore
in more detail what this means.
It is a general theorem that we could have applied the functors
· ⊗
A-A
M
and
Hom
A-A
(
· , M
) to any projective resolution of
A
A
A
and take the (co)homology,
and the resulting vector spaces will be the same. However, we will not prove
that, and will just always stick to this standard free resolution all the time.