Hochschild homology

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. 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).

Definition of Hochschild homology of algebrasEdit

Let k be a field, A an associative k-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 complexEdit

Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write   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   defined by


where   is in A for all   and  . If we let


then  , so   is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.


The maps   are face maps making the family of modules   a simplicial object in the category of k-modules, i.e., a functor Δok-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δo is the opposite category of Δ. The degeneracy maps are defined 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 complex[1]pg 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
Then, the Hochschild complex is quasi-isomorphic to
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 functor F. The above definition of Hochschild homology of commutative algebras is the special case where F is the Loday functor.

Loday functorEdit

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


A morphism


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 theorem[2]pg 43-44 states there is an isomorphism

for every  . This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential n-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 for not just smooth algebras, but also 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
for  .

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.[3] 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.

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.

See alsoEdit


  1. ^ Morrow, Matthew. "Topological Hochschild homology in arithmetic geometry" (PDF). Archived (PDF) from the original on 24 Dec 2020.
  2. ^ Ginzburg, Victor (2005-06-29). "Lectures on Noncommutative Geometry". arXiv:math/0506603.
  3. ^ "Section 23.6 (09PF): Tate resolutions—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-12-31.

External linksEdit

Introductory articlesEdit

Commutative caseEdit

  • Antieau, Benjamin; Bhatt, Bhargav; Mathew, Akhil (2019). "Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p". arXiv:1909.11437 [math.AG].

Noncommutative caseEdit