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## Summary

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 Ae-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

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

then $b\circ b=0$ , 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 $(C_{n}(A,M),b)$  a simplicial object in the category of k-modules, i.e., a functor Δok-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 complexpg 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 Kahler differentials $\Omega _{X/S}$  since the Kahler 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 Kahler 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[further explanation needed]. 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

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

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 theorempg 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 $3$ $\varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}={\text{Ker}}(\mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}\to \varepsilon \cdot \mathbb {F} _{p}\otimes _{\mathbb {Z} }^{\mathbf {L} }\mathbb {F} _{p}}).$ 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

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.