Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).
Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as Ae-modules.
Cartan & Eilenberg (1956) defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule.
for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by with boundary operator
is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M. Henceforth, we will write
are face maps making the family of modules
There's another useful interpretation of the Hochschild complex in the case of commutative rings, and more generally, for sheaves of commutative rings: it is constructed from the derived self-intersection of a scheme (or even derived scheme)
For example, we can form the derived fiber product
From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials
since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex
We can recover the original definition of the Hochschild complex of a commutative
giving an alternative but equivalent presentation of the Hochschild complex.
A skeleton for the category of finite pointed sets is given by the objects where 0 is the basepoint, and the morphisms are the basepoint preserving set maps.
Let A be a commutative k-algebra and M be a symmetric A-bimodule[further explanation needed].
For the case of commutative algebras, there are a number of theorems describing the computations over characteristic 0 yielding a straightforward understanding of what the homology and cohomology compute.
, the Hochschild-Kostant-Rosenberg theorem[2]pg 43-44 states there is an isomorphism
This isomorphism can be described explicitly using the anti-symmetrization map.
isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex.
Note this theorem makes it accessible to compute the Hochschild homology not just for smooth algebras, but also for local complete intersection algebras.
One simple example is to compute the Hochschild homology of a polynomial ring of
In the characteristic p case, there is a userful counter-example to the Hochschild-Kostant-Rosenberg theorem which elucidates for the need of a theory beyond simplicial algebras for defining Hochschild homology.
, giving a formal complex with a generator in degree
We can perform this recursively to get the underlying module of the divided power algebra
[3] Note this computation is seen as a technical artifact because the ring
One technical response to this problem is through Topological Hochschild homology, where the base ring
yields topological Hochschild homology, denoted
The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for
It induces an isomorphism on homotopy groups in degrees 0, 1, and 2.
Lars Hesselholt (2016) showed that the Hasse–Weil zeta function of a smooth proper variety over
can be expressed using regularized determinants involving topological Hochschild homology.