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 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 A^e=A⊗A^o 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. 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

${\displaystyle HH_{n}(A,M)={\text{Tor}}_{n}^{A^{e}}(A,M)}$

${\displaystyle HH^{n}(A,M)={\text{Ext}}_{A^{e}}^{n}(A,M)}$

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 A^⊗n for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by

${\displaystyle C_{n}(A,M):=M\otimes A^{\otimes n}}$

with boundary operator d_i defined by

${\displaystyle {\begin{aligned}d_{0}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=ma_{1}\otimes a_{2}\cdots \otimes a_{n}\\d_{i}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=m\otimes a_{1}\otimes \cdots \otimes a_{i}a_{i+1}\otimes \cdots \otimes a_{n}\\d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=a_{n}m\otimes a_{1}\otimes \cdots \otimes a_{n-1}\end{aligned}}}$

where a_i is in A for all 1 ≤ i ≤ n and m ∈ M. If we let

${\displaystyle b=\sum _{i=0}^{n}(-1)^{i}d_{i},}$

then ${\displaystyle b\circ b=0,}$ so (C_n(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.

Remark

The maps d_i are face maps making the family of modules C_n(A,M) 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

${\displaystyle s_{i}(a_{0}\otimes \cdots \otimes a_{n})=a_{0}\otimes \cdots \otimes a_{i}\otimes 1\otimes a_{i+1}\otimes \cdots \otimes a_{n}.}$

Hochschild homology is the homology of this simplicial module.

Hochschild homology of functors

The simplicial circle S¹ is a simplicial object in the category ${\displaystyle {\text{Fin}}_{*}}$ of finite pointed sets, i.e. a functor ${\displaystyle \Delta ^{o}\to {\text{Fin}}_{*}.}$ Thus, if F is a functor F: Fin → k-mod, we get a simplicial module by composing F with S¹

${\displaystyle \Delta ^{o}{\overset {S^{1}}{\longrightarrow }}{\text{Fin}}_{*}{\overset {F}{\longrightarrow }}k{\text{-mod}}.}$

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

${\displaystyle n_{+}=\{0,1,\ldots ,n\},}$

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 L(A,M) is given on objects in ${\displaystyle {\text{Fin}}_{*}}$ by

${\displaystyle n_{+}\mapsto M\otimes A^{\otimes n}.}$

A morphism

${\displaystyle f:m_{+}\to n_{+}}$

is sent to the morphism ${\displaystyle f_{*}}$ given by

${\displaystyle f_{*}(a_{0}\otimes \cdots \otimes a_{n})=b_{0}\otimes \cdots \otimes b_{m}}$

where

${\displaystyle \forall j\in \{0,\ldots ,n\}:\qquad b_{j}={\begin{cases}\prod _{i\in f^{-1}(j)}a_{i}&f^{-1}(j)\neq \emptyset \\1&f^{-1}(j)=\emptyset \end{cases}}}$

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

${\displaystyle \Delta ^{o}{\overset {S^{1}}{\longrightarrow }}{\text{Fin}}_{*}{\overset {{\mathcal {L}}(A,M)}{\longrightarrow }}k{\text{-mod}},}$

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,

${\displaystyle THH(\mathbf {F} _{p})=\mathbf {F} _{p}[x],}$

${\displaystyle HH(\mathbf {F} _{p})=\mathbf {F} _{p}\langle x\rangle }$

is the polynomial ring (with x in degree 2), compared to the ring of divided powers in one variable.

Hesselholt (2016) showed that the Hasse-Weil zeta-function of a smooth proper variety over F_p can be expressed using regularized determinants involving topological Hochschild homology.

See also

• Cyclic homology

References

• Cartan, Henri; Eilenberg, Samuel (1956), Homological algebra, Princeton Mathematical Series, 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, 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
• 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.
• Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology

External links

• Dylan G.L. Allegretti, Differential Forms on Noncommutative Spaces. An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize differential forms).
• Hochschild cohomology in nLab