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

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

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

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

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

with boundary operator $d_{i}$ defined by

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

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

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

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

Remark

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

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

Hochschild homology is the homology of this simplicial module.

Relation with the Bar complex

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

$$HH(A/k)\cong A\otimes _{A\otimes A^{op}}B(A/k)$$

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

$$X\times _{S}^{\mathbf {L} }X$$

which has the sheaf of derived rings ${\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{S}}^{\mathbf {L} }{\mathcal {O}}_{X}$. Then, if embed $X$ with the diagonal map

$$\Delta :X\to X\times _{S}^{\mathbf {L} }X$$

the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme

$$HH(X/S):=\Delta ^{*}({\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{S}}^{\mathbf {L} }{\mathcal {O}}_{X}}^{\mathbf {L} }{\mathcal {O}}_{X})$$

From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials $\Omega _{X/S}$ since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex $\mathbf {L} _{X/S}^{\bullet }$ since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative $k$-algebra $A$ by setting

$$S={\text{Spec}}(k)$$

and

$$X={\text{Spec}}(A)$$

Then, the Hochschild complex is quasi-isomorphic to

$$HH(A/k)\simeq _{qiso}A\otimes _{A\otimes _{k}^{\mathbf {L} }A}^{\mathbf {L} }A$$

If $A$ is a flat $k$-algebra, then there's the chain of isomorphism

$$A\otimes _{k}^{\mathbf {L} }A\cong A\otimes _{k}A\cong A\otimes _{k}A^{op}$$

giving an alternative but equivalent presentation of the Hochschild complex.

Hochschild homology of functors

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

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

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

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

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 $\operatorname {Fin} _{*}$ by

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

A morphism

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

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

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

where

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

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

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

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

$$\Omega _{A/k}^{n}\cong HH_{n}(A/k)$$

for every $n\geq 0$. This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential $n$-form has the map

$$a\,db_{1}\wedge \cdots \wedge db_{n}\mapsto \sum _{\sigma \in S_{n}}\operatorname {sign} (\sigma )a\otimes b_{\sigma (1)}\otimes \cdots \otimes b_{\sigma (n)}.$$

If the algebra $A/k$ isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex. For a simplicial resolution $P_{\bullet }\to A$, we set $\mathbb {L} _{A/k}^{i}=\Omega _{P_{\bullet }/k}^{i}\otimes _{P_{\bullet }}A$. Then, there exists a descending $\mathbb {N}$-filtration $F_{\bullet }$ on $HH_{n}(A/k)$ whose graded pieces are isomorphic to

$${\frac {F_{i}}{F_{i+1}}}\cong \mathbb {L} _{A/k}^{i}[+i].$$

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 $A=R/I$ for $R=k[x_{1},\dotsc ,x_{n}]$, the cotangent complex is the two-term complex $I/I^{2}\to \Omega _{R/k}^{1}\otimes _{k}A$.

Polynomial rings over the rationals

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

$$HH_{*}(\mathbb {Q} [x_{1},\ldots ,x_{n}])=\mathbb {Q} [x_{1},\ldots ,x_{n}]\otimes \Lambda (dx_{1},\dotsc ,dx_{n})$$

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

$${\begin{aligned}dx_{i}\cdot dx_{j}&=-dx_{j}\cdot dx_{i}\\dx_{i}\cdot dx_{i}&=0\end{aligned}}$$

for $i\neq j$.

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

$$\mathbb {Z} \xrightarrow {\cdot p} \mathbb {Z}$$

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

$$\mathbb {F} _{p}\otimes _{\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbb {L} }\mathbb {F} _{p}}^{\mathbb {L} }\mathbb {F} _{p}$$

In order to compute this, we must resolve $\mathbb {F} _{p}$ as an $\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}$-algebra. Observe that the algebra structure

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

forces $\varepsilon \mapsto 0$. This gives the degree zero term of the complex. Then, because we have to resolve the kernel $\varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}$, we can take a copy of $\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}$ shifted in degree $2$ and have it map to $\varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}$, with kernel in degree $3$$\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

$$(\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p})\langle x\rangle ={\frac {(\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p})[x_{1},x_{2},\ldots ]}{x_{i}x_{j}={\binom {i+j}{i}}x_{i+j}}}$$

with $dx_{i}=\varepsilon \cdot x_{i-1}$ and the degree of $x_{i}$ is $2i$, namely $|x_{i}|=2i$. Tensoring this algebra with $\mathbb {F} _{p}$ over $\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}$ gives

$$HH_{*}(\mathbb {F} _{p})=\mathbb {F} _{p}\langle x\rangle$$

since $\varepsilon$ multiplied with any element in $\mathbb {F} _{p}$ 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 $\mathbb {F} _{p}\langle x\rangle$ is not well behaved. For instance, $x^{p}=0$. One technical response to this problem is through Topological Hochschild homology, where the base ring $\mathbb {Z}$ is replaced by the sphere spectrum $\mathbb {S}$.

Topological Hochschild homology

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) $k$-modules by an ∞-category (equipped with a tensor product) ${\mathcal {C}}$, and $A$ by an associative algebra in this category. Applying this to the category ${\mathcal {C}}={\textbf {Spectra}}$ 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 ${\mathcal {C}}=D(\mathbb {Z} )$ the derived category of $\mathbb {Z}$-modules (as an ∞-category).

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

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

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

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