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 algebras
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 complex
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. Henceforth, we will write as simply .
Remark
The maps are face maps making the family of modules a simplicial object in the category of k-modules, i.e., a functor Δo → k-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 complex
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
As a derived self-intersection
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
Hochschild homology of functors
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 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
A morphism
is sent to the morphism given by
where
Another description of Hochschild homology of algebras
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.
Examples
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 case
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
Polynomial rings over the rationals
One simple example is to compute the Hochschild homology of a polynomial ring of with -generators. The HKR theorem gives the isomorphism
Commutative characteristic p case
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
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
Topological Hochschild homology
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 also
References
- ^ Morrow, Matthew. "Topological Hochschild homology in arithmetic geometry" (PDF). Archived (PDF) from the original on 24 Dec 2020.
- ^ Ginzburg, Victor (2005-06-29). "Lectures on Noncommutative Geometry". arXiv:math/0506603.
- ^ "Section 23.6 (09PF): Tate resolutions—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-12-31.
- Cartan, Henri; Eilenberg, Samuel (1956), Homological algebra, Princeton Mathematical Series, vol. 19, Princeton University Press, ISBN 978-0-691-04991-5, MR 0077480
- Govorov, V.E.; Mikhalev, A.V. (2001) [1994], "Cohomology of algebras", Encyclopedia of Mathematics, EMS Press
- 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
- Hochschild, Gerhard (1945), "On the cohomology groups of an associative algebra", Annals of Mathematics, Second Series, 46 (1): 58–67, doi:10.2307/1969145, ISSN 0003-486X, JSTOR 1969145, MR 0011076
- Jean-Louis Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0
- Richard S. Pierce, Associative Algebras, Graduate Texts in Mathematics (88), Springer, 1982.
- Pirashvili, Teimuraz (2000). "Hodge decomposition for higher order Hochschild homology". Annales Scientifiques de l'École Normale Supérieure. 33 (2): 151–179. doi:10.1016/S0012-9593(00)00107-5.
External links
Introductory articles
- Dylan G.L. Allegretti, Differential Forms on Noncommutative Spaces. An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize differential forms).
- Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math/0506603.
- Topological Hochschild homology in arithmetic geometry
- Hochschild cohomology at the nLab
Commutative case
- Antieau, Benjamin; Bhatt, Bhargav; Mathew, Akhil (2019). "Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p". arXiv:1909.11437 [math.AG].
Noncommutative case
- 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. arXiv:math/0207073. doi:10.1016/S0022-4049(03)00146-4.
- Quddus, Safdar (2020). "Non-commutative Poisson Structures on quantum torus orbifolds". arXiv:2006.00495 [math.KT].
- Yashinski, Allan (2012). "The Gauss-Manin connection and noncommutative tori". arXiv:1210.4531 [math.KT].