Hochschild homology

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 ${\displaystyle b}$.

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.

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

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

giving an explicit isomorphism.

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

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

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 map

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

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

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

From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials ${\displaystyle \Omega _{X/S}}$ 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

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

and

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

Then, the Hochschild complex is quasi-isomorphic to

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

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

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

giving an alternative but equivalent presentation of the Hochschild complex.

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 ${\displaystyle L(A,M)}$ 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

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

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.

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

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

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

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

If the algebra ${\displaystyle A/k}$ 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

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

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

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

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

where the algebra ${\displaystyle \bigwedge (dx_{1},\ldots ,dx_{n})}$ 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

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

for ${\displaystyle i\neq j}$.

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

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

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 by

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

In order to compute this, we must resolve ${\displaystyle \mathbb {F} _{p}}$ 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

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

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

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

since ${\displaystyle \varepsilon }$ 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 ${\displaystyle \mathbb {S} }$.

• 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

• Antieau, Benjamin; Bhatt, Bhargav; Mathew, Akhil (2019). "Counterexamples to Hochschild–Kostant–Rosenberg in characteristic p". arXiv:1909.11437 [math.AG].

• 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. 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].