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 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 $$A^e=A\otimes A^o$$ 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 Ae-modules. defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by


 * $$ HH_n(A,M) = \operatorname{Tor}_n^{A^e}(A, M)$$
 * $$ HH^n(A,M) = \operatorname{Ext}^n_{A^e}(A, M)$$

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{align}

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{align}$$

where $$a_i$$ is in A for all $$1\le i\le n$$ 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 Δo → k-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δo 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$$X\times^\mathbf{L}_SX$$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$$\Delta: X \to X\times^\mathbf{L}_SX$$the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme$$HH(X/S) := \Delta^*(\mathcal{O}_X\otimes_{\mathcal{O}_X\otimes_{\mathcal{O}_S}^\mathbf{L}\mathcal{O}_X}^\mathbf{L}\mathcal{O}_X)$$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$$A\otimes_k^\mathbf{L}A \cong A\otimes_kA \cong A\otimes_kA^{op}$$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 $$\Omega^n_{A/k} \cong HH_n(A/k)$$ 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 $$\mathbb{L}^i_{A/k} = \Omega^i_{P_\bullet/k}\otimes_{P_\bullet} A$$. Then, there exists a descending $$\mathbb{N}$$-filtration $$F_\bullet$$ on $$HH_n(A/k)$$ whose graded pieces are isomorphic to $$\frac{F_i}{F_{i+1}} \cong \mathbb{L}^i_{A/k}[+i].$$ 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 $$I/I^2 \to \Omega^1_{R/k}\otimes_k A$$.

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 $$\begin{align} dx_i\cdot dx_j &= -dx_j\cdot dx_i \\ dx_i\cdot dx_i &= 0 \end{align}$$ 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 $$\mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p \cong \mathbb{F}_p[\varepsilon]/(\varepsilon^2)$$ 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$$\mathbb{F}_p\otimes^\mathbb{L}_{\mathbb{F}_p\otimes^\mathbb{L}_\mathbb{Z} \mathbb{F}_p}\mathbb{F}_p$$In order to compute this, we must resolve $$\mathbb{F}_p$$ as an $$\mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p$$-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 $$\varepsilon \cdot \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p$$, we can take a copy of $$\mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p$$ shifted in degree $$2$$ and have it map to $$\varepsilon \cdot \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p$$, with kernel in degree $$3$$$$\varepsilon \cdot \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p = \text{Ker}({\displaystyle \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}} \to {\displaystyle \varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}).$$We can perform this recursively to get the underlying module of the divided power algebra$$(\mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p)\langle x \rangle = \frac{ (\mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p)[x_1,x_2,\ldots] }{x_ix_j = \binom{i+j}{i}x_{i+j}}$$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 $$\mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p$$ 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
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 $$\Z$$-modules (as an ∞-category).

Replacing tensor products over the sphere spectrum by tensor products over $$\Z$$ (or the Eilenberg–MacLane-spectrum $$H\Z$$) 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.

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.

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).
 * Topological Hochschild homology in arithmetic geometry
 * Topological Hochschild homology in arithmetic geometry