Hochschild homology

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

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

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

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 isomorphisms $${\displaystyle A\otimes _{k}^{\mathbf {L} }A\cong A\otimes _{k}A\cong A\otimes _{k}A^{op}}$$ giving an alternative but equivalent presentation of the Hochschild complex.

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.

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

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

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.

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.

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 ${\displaystyle \mathbb {S} }$ .

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.

• Cyclic homology

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• Topological Hochschild homology in arithmetic geometry
• Hochschild cohomology at the nLab

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