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 ${\displaystyle A^{e}=A\otimes A^{o}}$ 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

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

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

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 ${\displaystyle A^{\otimes n}}$ 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 ${\displaystyle d_{i}}$ 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 ${\displaystyle a_{i}}$ is in A for all ${\displaystyle 1\leq i\leq n}$ and ${\displaystyle m\in M}$. If we let

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

then ${\displaystyle b_{n-1}\circ b_{n}=0}$, so ${\displaystyle (C_{n}(A,M),b_{n})}$ 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 ${\displaystyle b_{n}}$ as simply ${\displaystyle b}$.

Remark

The maps ${\displaystyle d_{i}}$ are face maps making the family of modules ${\displaystyle (C_{n}(A,M),b)}$ 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.

Relation with the Bar complex

There is a similar looking complex ${\displaystyle B(A/k)}$ called the Bar complex which formally looks very similar to the Hochschild complex^{[1]pg 4-5}. In fact, the Hochschild complex ${\displaystyle HH(A/k)}$ can be recovered from the Bar complex as

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) ${\displaystyle X}$ over some base scheme ${\displaystyle S}$. For example, we can form the derived fiber product

which has the sheaf of derived rings ${\displaystyle {\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{S}}^{\mathbf {L} }{\mathcal {O}}_{X}}$. Then, if embed ${\displaystyle X}$ with the diagonal mapthe Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product schemeFrom this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials ${\displaystyle \Omega _{X/S}}$ since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex ${\displaystyle \mathbf {L} _{X/S}^{\bullet }}$ since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative ${\displaystyle k}$-algebra ${\displaystyle A}$ by setting and Then, the Hochschild complex is quasi-isomorphic toIf ${\displaystyle A}$ is a flat ${\displaystyle k}$-algebra, then there's the chain of isomorphismgiving an alternative but equivalent presentation of the Hochschild complex.

Hochschild homology of functors

The simplicial circle ${\displaystyle S^{1}}$ is a simplicial object in the category ${\displaystyle \operatorname {Fin} _{*}}$ of finite pointed sets, i.e., a functor ${\displaystyle \Delta ^{o}\to \operatorname {Fin} _{*}.}$ Thus, if F is a functor ${\displaystyle F\colon \operatorname {Fin} \to k-\mathrm {mod} }$, we get a simplicial module by composing F with ${\displaystyle S^{1}}$.

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

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^{[further explanation needed]}. The Loday functor ${\displaystyle L(A,M)}$ is given on objects in ${\displaystyle \operatorname {Fin} _{*}}$ 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

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

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

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

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 ${\displaystyle HH_{*}(A)}$ for an associative algebra ${\displaystyle A}$. 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 ${\displaystyle A/k}$ where ${\displaystyle \mathbb {Q} \subseteq k}$, the Hochschild homology has two main theorems concerning smooth algebras, and more general non-flat algebras ${\displaystyle A}$; but, the second is a direct generalization of the first. In the smooth case, i.e. for a smooth algebra ${\displaystyle A}$, the Hochschild-Kostant-Rosenberg theorem^{[2]pg 43-44} states there is an isomorphism

for every ${\displaystyle n\geq 0}$. This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential ${\displaystyle n}$-form has the map If the algebra ${\displaystyle A/k}$ isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex. For a simplicial resolution ${\displaystyle P_{\bullet }\to A}$, we set ${\displaystyle \mathbb {L} _{A/k}^{i}=\Omega _{P_{\bullet }/k}^{i}\otimes _{P_{\bullet }}A}$. Then, there exists a descending ${\displaystyle \mathbb {N} }$-filtration ${\displaystyle F_{\bullet }}$ on ${\displaystyle HH_{n}(A/k)}$ whose graded pieces are isomorphic to 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 ${\displaystyle A=R/I}$ for ${\displaystyle R=k[x_{1},\dotsc ,x_{n}]}$, the cotangent complex is the two-term complex ${\displaystyle I/I^{2}\to \Omega _{R/k}^{1}\otimes _{k}A}$.

Polynomial rings over the rationals

One simple example is to compute the Hochschild homology of a polynomial ring of ${\displaystyle \mathbb {Q} }$ with ${\displaystyle n}$-generators. The HKR theorem gives the isomorphism

where the algebra ${\displaystyle \bigwedge (dx_{1},\ldots ,dx_{n})}$ is the free antisymmetric algebra over ${\displaystyle \mathbb {Q} }$ in ${\displaystyle n}$-generators. Its product structure is given by the wedge product of vectors, so for ${\displaystyle i\neq j}$.

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 ${\displaystyle \mathbb {Z} }$-algebra ${\displaystyle \mathbb {F} _{p}}$. We can compute a resolution of ${\displaystyle \mathbb {F} _{p}}$ as the free differential graded algebras

giving the derived intersection ${\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}\cong \mathbb {F} _{p}[\varepsilon ]/(\varepsilon ^{2})}$ where ${\displaystyle {\text{deg}}(\varepsilon )=1}$ and the differential is the zero map. This is because we just tensor the complex above by ${\displaystyle \mathbb {F} _{p}}$, giving a formal complex with a generator in degree ${\displaystyle 1}$ which squares to ${\displaystyle 0}$. Then, the Hochschild complex is given byIn order to compute this, we must resolve ${\displaystyle \mathbb {F} _{p}}$ as an ${\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}$-algebra. Observe that the algebra structure

${\displaystyle \mathbb {F} _{p}[\varepsilon ]/(\varepsilon ^{2})\to \mathbb {F} _{p}}$

forces ${\displaystyle \varepsilon \mapsto 0}$. This gives the degree zero term of the complex. Then, because we have to resolve the kernel ${\displaystyle \varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}$, we can take a copy of ${\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}$ shifted in degree ${\displaystyle 2}$ and have it map to ${\displaystyle \varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}$, with kernel in degree ${\displaystyle 3}$${\displaystyle \varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}={\text{Ker}}({\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}\to {\displaystyle \varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}).}$We can perform this recursively to get the underlying module of the divided power algebra

with ${\displaystyle dx_{i}=\varepsilon \cdot x_{i-1}}$ and the degree of ${\displaystyle x_{i}}$ is ${\displaystyle 2i}$, namely ${\displaystyle |x_{i}|=2i}$. Tensoring this algebra with ${\displaystyle \mathbb {F} _{p}}$ over ${\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}$ givessince ${\displaystyle \varepsilon }$ multiplied with any element in ${\displaystyle \mathbb {F} _{p}}$ 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 ${\displaystyle \mathbb {F} _{p}\langle x\rangle }$ is not well behaved. For instance, ${\displaystyle x^{p}=0}$. One technical response to this problem is through Topological Hochschild homology, where the base ring ${\displaystyle \mathbb {Z} }$ is replaced by the sphere spectrum ${\displaystyle \mathbb {S} }$.

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) ${\displaystyle k}$-modules by an ∞-category (equipped with a tensor product) ${\displaystyle {\mathcal {C}}}$, and ${\displaystyle A}$ by an associative algebra in this category. Applying this to the category ${\displaystyle {\mathcal {C}}={\textbf {Spectra}}}$ of spectra, and ${\displaystyle A}$ being the Eilenberg–MacLane spectrum associated to an ordinary ring ${\displaystyle R}$ yields topological Hochschild homology, denoted ${\displaystyle THH(R)}$. The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for ${\displaystyle {\mathcal {C}}=D(\mathbb {Z} )}$ the derived category of ${\displaystyle \mathbb {Z} }$-modules (as an ∞-category).

Replacing tensor products over the sphere spectrum by tensor products over ${\displaystyle \mathbb {Z} }$ (or the Eilenberg–MacLane-spectrum ${\displaystyle H\mathbb {Z} }$) leads to a natural comparison map ${\displaystyle THH(R)\to HH(R)}$. It induces an isomorphism on homotopy groups in degrees 0, 1, and 2. In general, however, they are different, and ${\displaystyle THH}$ tends to yield simpler groups than HH. For example,

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

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

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 ${\displaystyle \mathbb {F} _{p}}$ 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, 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].