Hochschild homology explained

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

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 A^e-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

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 Δ^o → k-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δ^o 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$$HH(A/k)\; \backslash cong\; A\backslash otimes\_\; B(A/k)$$giving 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)

over some base scheme . For example, we can form the derived fiber product$$X\backslash times^\backslash mathbf\_SX$$which has the sheaf of derived rings . Then, if embed with the diagonal map$$\backslash Delta:\; X\; \backslash to\; X\backslash times^\backslash mathbf\_SX$$the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme$$HH(X/S)\; :=\; \backslash Delta^*(\backslash mathcal\_X\backslash otimes\_^\backslash mathbf\backslash mathcal\_X)$$From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials since the Kähler 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 Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative -algebra by setting$$S\; =\; \backslash text(k)$$ and $$X\; =\; \backslash text(A)$$Then, the Hochschild complex is quasi-isomorphic to$$HH(A/k)\; \backslash simeq\_\; A\backslash otimes\_^\backslash mathbfA$$If is a flat -algebra, then there's the chain of isomorphisms $$A\backslash otimes\_k^\backslash mathbfA\; \backslash cong\; A\backslash otimes\_kA\; \backslash cong\; A\backslash otimes\_kA^$$giving an alternative but equivalent presentation of the Hochschild complex.

Hochschild homology of functors

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. 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 $$\backslash Omega^n\_\; \backslash cong\; HH\_n(A/k)$$ for every . This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential -form has the map$$a\backslash ,db\_1\backslash wedge\; \backslash cdots\; \backslash wedge\; db\_n\; \backslash mapsto\backslash sum\_\backslash operatorname(\backslash sigma)\; a\backslash otimes\; b\_\backslash otimes\; \backslash cdots\; \backslash otimes\; b\_.$$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 $$\backslash frac\; \backslash cong\; \backslash mathbb^i\_[+i].$$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 rationals

One simple example is to compute the Hochschild homology of a polynomial ring of

with -generators. The HKR theorem gives the isomorphism $$HH\_*(\backslash mathbb[x\_1,\backslash ldots,\; x\_n])\; =\; \backslash mathbb[x\_1,\backslash ldots,\; x\_n]\backslash otimes\; \backslash Lambda(dx\_1,\backslash dotsc,\; dx\_n)$$ 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 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$$\backslash mathbb\backslash xrightarrow\; \backslash mathbb$$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$$\backslash mathbb\_p\backslash otimes^\backslash mathbb\_\backslash mathbb\_p$$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 } \to).We can perform this recursively to get the underlying module of the divided power algebra$$(\backslash mathbb\_p\backslash otimes^\backslash mathbf\_\backslash mathbb\backslash mathbb\_p)\backslash langle\; x\; \backslash rangle\; =\; \backslash frac$$with and the degree of is , namely . Tensoring this algebra with over gives$$HH\_*(\backslash mathbb\_p)\; =\; \backslash mathbb\_p\backslash langle\; x\; \backslash rangle$$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 homology

See main article: 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)

l{C}

, and by an associative algebra in this category. Applying this to the category

l{C}=bf{Spectra}

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

l{C}=D(Z)

the derived category of

\Z

-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

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.

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

• Cyclic homology

References

1. Web site: Morrow. Matthew. Topological Hochschild homology in arithmetic geometry. live. https://web.archive.org/web/20201224194152/https://www.math.arizona.edu/~swc/aws/2019/2019MorrowNotes.pdf. 24 Dec 2020.
2. Ginzburg. Victor. 2005-06-29. Lectures on Noncommutative Geometry. math/0506603.
3. Web site: Section 23.6 (09PF): Tate resolutions—The Stacks project. 2020-12-31. stacks.math.columbia.edu.

• • • • • Jean-Louis Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998)
• 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. 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).
• math/0506603. Ginzburg. Victor. Lectures on Noncommutative Geometry. 2005.
• Topological Hochschild homology in arithmetic geometry

Commutative case

• 1909.11437. Antieau. Benjamin. Bhatt. Bhargav. Mathew. Akhil. Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p. 2019. math.AG.

Noncommutative case

• 10.1016/S0022-4049(03)00146-4 . free. Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras. 2004. Richard. Lionel. Journal of Pure and Applied Algebra. 187. 1–3. 255–294. math/0207073.
• 2006.00495. Quddus. Safdar. Non-commutative Poisson Structures on quantum torus orbifolds. 2020. math.KT.
• 1210.4531. Yashinski. Allan. The Gauss-Manin connection and noncommutative tori. 2012. math.KT.