Let k be a field, A an associativek-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product of A with its opposite algebra. 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
Hochschild homology is the homology of this simplicial module.
Relation with the Bar complexEdit
There is a similar looking complex called the Bar complex which formally looks very similar to the Hochschild complexpg 4-5. In fact, the Hochschild complex can be recovered from the Bar complex as
giving an explicit isomorphism.
As a derived self-intersectionEdit
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) over some base scheme . For example, we can form the derived fiber product
which has the sheaf of derived rings . Then, if embed with the diagonal map
the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme
From this interpretation, it should be clear the Hochschild homology should have some relation to the Kahler differentials since the Kahler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex since this is the derived replacement for the Kahler differentials. We can recover the original definition of the Hochschild complex of a commutative -algebra by setting
If is a flat -algebra, then there's the chain of isomorphism
giving an alternative but equivalent presentation of the Hochschild complex.
Hochschild homology of functorsEdit
The simplicial circle is a simplicial object in the category of finite pointed sets, i.e., a functor Thus, if F is a functor , we get a simplicial module by composing F with .
The homology of this simplicial module is the Hochschild homology of the functorF. The above definition of Hochschild homology of commutative algebras is the special case where F is the Loday functor.
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]. The Loday functor is given on objects in by
is sent to the morphism given by
Another description of Hochschild homology of algebrasEdit
The Hochschild homology of a commutative algebra A with coefficients in a symmetric A-bimodule M is the homology associated to the composition
and this definition agrees with the one above.
The examples of Hochschild homology computations can be stratified into a number of distinct cases with fairly general theorems describing the structure of the homology groups and the homology ring for an associative algebra . 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.
Commutative characteristic 0 caseEdit
In the case of commutative algebras where , the Hochschild homology has two main theorems concerning smooth algebras, and more general non-flat algebras ; but, the second is a direct generalization of the first. In the smooth case, i.e. for a smooth algebra , the Hochschild-Kostant-Rosenberg theorempg 43-44 states there is an isomorphism
for every . This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential -form has the map
If the algebra isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex. For a simplicial resolution , we set . Then, there exists a descending -filtration on whose graded pieces are isomorphic to
Note this theorem makes it accessible to compute the Hochschild homology not just for smooth algebras, but also for local complete intersection algebras. In this case, given a presentation for , the cotangent complex is the two-term complex .
Polynomial rings over the rationalsEdit
One simple example is to compute the Hochschild homology of a polynomial ring of with -generators. The HKR theorem gives the isomorphism
where the algebra is the free antisymmetric algebra over in -generators. Its product structure is given by the wedge product of vectors, so
Commutative characteristic p caseEdit
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. Consider the -algebra . We can compute a resolution of as the free differential graded algebras
giving the derived intersection where and the differential is the zero map. This is because we just tensor the complex above by , giving a formal complex with a generator in degree which squares to . Then, the Hochschild complex is given by
In order to compute this, we must resolve as an -algebra. Observe that the algebra structure
forces . This gives the degree zero term of the complex. Then, because we have to resolve the kernel , we can take a copy of shifted in degree and have it map to , with kernel in degree We can perform this recursively to get the underlying module of the divided power algebra
with and the degree of is , namely . Tensoring this algebra with over gives
since multiplied with any element in is zero. The algebra structure comes from general theory on divided power algebras and differential graded algebras. Note this computation is seen as a technical artifact because the ring is not well behaved. For instance, . One technical response to this problem is through Topological Hochschild homology, where the base ring is replaced by the sphere spectrum.
Topological Hochschild homologyEdit
The above construction of the Hochschild complex can be adapted to more general situations, namely by replacing the category of (complexes of) -modules by an ∞-category (equipped with a tensor product) , and by an associative algebra in this category. Applying this to the category of spectra, and being the Eilenberg–MacLane spectrum associated to an ordinary ring yields topological Hochschild homology, denoted . The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for the derived category of -modules (as an ∞-category).
Replacing tensor products over the sphere spectrum by tensor products over (or the Eilenberg–MacLane-spectrum ) leads to a natural comparison map . It induces an isomorphism on homotopy groups in degrees 0, 1, and 2. In general, however, they are different, and tends to yield simpler groups than HH. For example,
is the polynomial ring (with x in degree 2), compared to the ring of divided powers in one variable.
Hesselholt, Lars (2016), Topological Hochschild homology and the Hasse-Weil zeta function, Contemporary Mathematics, vol. 708, pp. 157–180, arXiv:1602.01980, doi:10.1090/conm/708/14264, ISBN 9781470429119, S2CID 119145574
Antieau, Benjamin; Bhatt, Bhargav; Mathew, Akhil (2019). "Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p". arXiv:1909.11437 [math.AG].
Richard, Lionel (2004). "Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras". Journal of Pure and Applied Algebra. 187 (1–3): 255–294. doi:10.1016/S0022-4049(03)00146-4.