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 for algebras over a field, and extended to algebras over more general rings by .
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 formula_1 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. 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 formula_4 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 formula_6 defined by
where formula_8 is in "A" for all formula_9 and formula_10. If we let
then formula_12, so formula_13 is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of "A" with coefficients in "M".
Remark.
The maps formula_6 are face maps making the family of modules formula_13 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 formula_17 called the Bar complex which formally looks very similar to the Hochschild complexpg 4-5. In fact, the Hochschild complex formula_18 can be recovered from the Bar complex asformula_19giving an explicit isomorphism.
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) formula_20 over some base scheme formula_21. For example, we can form the derived fiber productformula_22which has the sheaf of derived rings formula_23. Then, if embed formula_20 with the diagonal mapformula_25the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product schemeformula_26From this interpretation, it should be clear the Hochschild homology should have some relation to the Kahler differentials formula_27 since the Kahler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex formula_28 since this is the derived replacement for the Kahler differentials. We can recover the original definition of the Hochschild complex of a commutative formula_29-algebra formula_30 by settingformula_31 and formula_32Then, the Hochschild complex is quasi-isomorphic toformula_33If formula_30 is a flat formula_29-algebra, then there's the chain of isomorphismformula_36giving an alternative but equivalent presentation of the Hochschild complex.
Hochschild homology of functors.
The simplicial circle formula_37 is a simplicial object in the category formula_38 of finite pointed sets, i.e., a functor formula_39 Thus, if "F" is a functor formula_40, we get a simplicial module by composing "F" with formula_37.
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. The Loday functor formula_44 is given on objects in formula_38 by
A morphism
is sent to the morphism formula_48 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 formula_52 for an associative algebra formula_30. 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 formula_54 where formula_55, the Hochschild homology has two main theorems concerning smooth algebras, and more general non-flat algebras formula_30; but, the second is a direct generalization of the first. In the smooth case, i.e. for a smooth algebra formula_30, the Hochschild-Kostant-Rosenberg theorempg 43-44 states there is an isomorphismformula_58for every formula_59. This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential n-form has the mapformula_60If the algebra formula_54 isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex. For a simplicial resolution formula_62, we set formula_63. Then, there exists a descending formula_64-filtration formula_65 on formula_66 whose graded pieces are isomorphic toformula_67Note 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 formula_68 for formula_69, the cotangent complex is the two-term complex formula_70.
Polynomial rings over the rationals.
One simple example is to compute the Hochschild homology of a polynomial ring of formula_71 with formula_72-generators. The HKR theorem gives the isomorphismformula_73where the algebra formula_74 is the free antisymmetric algebra over formula_71 in formula_72-generators. Its product structure is given by the Wedge product of vectors, soformula_77for formula_78.
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 formula_79-algebra formula_80. We can compute a resolution of formula_80 as the free differential graded algebrasformula_82giving the derived intersection formula_83 where formula_84 and the differential is the zero map. This is because we just tensor the complex above by formula_80, giving a formal complex with a generator in degree formula_86 which squares to formula_87. Then, the Hochschild complex is given byformula_88In order to compute this, we must resolve formula_80 as an formula_90-algebra. Observe that the algebra structure
formula_91
forces formula_92. This gives the degree zero term of the complex. Then, because we have to resolve the kernel formula_93, we can take a copy of formula_90 shifted in degree formula_95 and have it map to formula_93, with kernel in degree formula_97formula_98We can perform this recursively to get the underlying module of the divided power algebraformula_99with formula_100 and the degree of formula_101 is formula_102, namely formula_103. Tensoring this algebra with formula_80 over formula_90 givesformula_106since formula_107 multiplied with any element in formula_80 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 formula_109 is not well behaved. For instance, formula_110. One technical response to this problem is through Topological Hochschild homology, where the base ring formula_79 is replaced by the sphere spectrum formula_112.
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) formula_29-modules by an ∞-category (equipped with a tensor product) formula_114, and formula_30 by an associative algebra in this category. Applying this to the category formula_116 of spectra, and formula_30 being the Eilenberg–MacLane spectrum associated to an ordinary ring formula_118 yields topological Hochschild homology, denoted formula_119. The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for "formula_120" the derived category of formula_121-modules (as an ∞-category).
Replacing tensor products over the sphere spectrum by tensor products over formula_121 (or the Eilenberg–MacLane-spectrum formula_123) leads to a natural comparison map formula_124. It induces an isomorphism on homotopy groups in degrees 0, 1, and 2. In general, however, they are different, and formula_125 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.
 showed that the Hasse–Weil zeta function of a smooth proper variety over formula_80 can be expressed using regularized determinants involving topological Hochschild homology.