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 Ae=A⊗Ao 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
Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write A⊗n 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 di defined by
where ai is in A for all 1 ≤ i ≤ n and m ∈ M. If we let
then so (Cn(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.
The maps di are face maps making the family of modules Cn(A,M) 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.
Hochschild homology of functors
The simplicial circle S1 is a simplicial object in the category of finite pointed sets, i.e. a functor Thus, if F is a functor F: Fin → k-mod, we get a simplicial module by composing F with S1
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.
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 L(A,M) is given on objects in by
is sent to the morphism given by
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.
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) k-modules by an ∞-category (equipped with a tensor product) C, and A by an associative algebra in this category. Applying this to the category C = Sp of spectra, and A being the Eilenberg–MacLane spectrum associated to an ordinary ring R yields topological Hochschild homology, denoted THH(R). The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for C the derived category of Z-modules (as an ∞-category).
Replacing tensor products over the sphere spectrum by tensor products over Z (or the Eilenberg–MacLane-spectrum HZ) leads to a natural comparison map THH(R) → HH(R). It induces an isomorphism on homotopy groups in degrees 0, 1, and 2. In general, however, they are different, and THH 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.
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- Govorov, V.E.; Mikhalev, A.V. (2001) , "Cohomology of algebras", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Hesselholt, Lars (2016), Topological Hochschild homology and the Hasse-Weil zeta function, arXiv:1602.01980, Bibcode:2016arXiv160201980H
- Hochschild, G. (1945), "On the cohomology groups of an associative algebra", Annals of Mathematics, Second Series, 46: 58–67, doi:10.2307/1969145, ISSN 0003-486X, JSTOR 1969145, MR 0011076
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- Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology