Hochschild homology

Short description: Theory for associative algebras over rings

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 [math]\displaystyle{ A^e=A\otimes A^o }[/math] 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) defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by

[math]\displaystyle{ HH_n(A,M) = \operatorname{Tor}_n^{A^e}(A, M) }[/math]

[math]\displaystyle{ HH^n(A,M) = \operatorname{Ext}^n_{A^e}(A, M) }[/math]

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

[math]\displaystyle{ C_n(A,M) := M \otimes A^{\otimes n} }[/math]

with boundary operator [math]\displaystyle{ d_i }[/math] defined by

[math]\displaystyle{ \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} }[/math]

where [math]\displaystyle{ a_i }[/math] is in A for all [math]\displaystyle{ 1\le i\le n }[/math] and [math]\displaystyle{ m\in M }[/math]. If we let

[math]\displaystyle{ b_n=\sum_{i=0}^n (-1)^i d_i, }[/math]

then [math]\displaystyle{ b_{n-1} \circ b_{n} =0 }[/math], so [math]\displaystyle{ (C_n(A,M),b_n) }[/math] 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 [math]\displaystyle{ b_n }[/math] as simply [math]\displaystyle{ b }[/math].

Remark

The maps [math]\displaystyle{ d_i }[/math] are face maps making the family of modules [math]\displaystyle{ (C_n(A,M),b) }[/math] 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

[math]\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. }[/math]

Hochschild homology is the homology of this simplicial module.

Relation with the Bar complex

There is a similar looking complex [math]\displaystyle{ B(A/k) }[/math] called the Bar complex which formally looks very similar to the Hochschild complex^{pg 4-5}. In fact, the Hochschild complex [math]\displaystyle{ HH(A/k) }[/math] can be recovered from the Bar complex as[math]\displaystyle{ HH(A/k) \cong A\otimes_{A\otimes A^{op}} B(A/k) }[/math]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) [math]\displaystyle{ X }[/math] over some base scheme [math]\displaystyle{ S }[/math]. For example, we can form the derived fiber product[math]\displaystyle{ X\times^\mathbf{L}_SX }[/math]which has the sheaf of derived rings [math]\displaystyle{ \mathcal{O}_X\otimes_{\mathcal{O}_S}^\mathbf{L}\mathcal{O}_X }[/math]. Then, if embed [math]\displaystyle{ X }[/math] with the diagonal map[math]\displaystyle{ \Delta: X \to X\times^\mathbf{L}_SX }[/math]the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme[math]\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) }[/math]From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials [math]\displaystyle{ \Omega_{X/S} }[/math] since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex [math]\displaystyle{ \mathbf{L}_{X/S}^\bullet }[/math] since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative [math]\displaystyle{ k }[/math]-algebra [math]\displaystyle{ A }[/math] by setting[math]\displaystyle{ S = \text{Spec}(k) }[/math] and [math]\displaystyle{ X = \text{Spec}(A) }[/math]Then, the Hochschild complex is quasi-isomorphic to[math]\displaystyle{ HH(A/k) \simeq_{qiso} A\otimes_{A\otimes_{k}^\mathbf{L}A}^\mathbf{L}A }[/math]If [math]\displaystyle{ A }[/math] is a flat [math]\displaystyle{ k }[/math]-algebra, then there's the chain of isomorphism[math]\displaystyle{ A\otimes_k^\mathbf{L}A \cong A\otimes_kA \cong A\otimes_kA^{op} }[/math]giving an alternative but equivalent presentation of the Hochschild complex.

Hochschild homology of functors

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

[math]\displaystyle{ \Delta^o \overset{S^1}{\longrightarrow} \operatorname{Fin}_* \overset{F}{\longrightarrow} k\text{-mod}. }[/math]

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

[math]\displaystyle{ n_+ = \{0,1,\ldots,n\}, }[/math]

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 [math]\displaystyle{ L(A,M) }[/math] is given on objects in [math]\displaystyle{ \operatorname{Fin}_* }[/math] by

[math]\displaystyle{ n_+ \mapsto M \otimes A^{\otimes n}. }[/math]

A morphism

[math]\displaystyle{ f:m_+ \to n_+ }[/math]

is sent to the morphism [math]\displaystyle{ f_* }[/math] given by

[math]\displaystyle{ f_*(a_0 \otimes \cdots \otimes a_m) = b_0 \otimes \cdots \otimes b_n }[/math]

where

[math]\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} }[/math]

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

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

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 [math]\displaystyle{ HH_*(A) }[/math] for an associative algebra [math]\displaystyle{ A }[/math]. 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 [math]\displaystyle{ A/k }[/math] where [math]\displaystyle{ \mathbb{Q}\subseteq k }[/math], the Hochschild homology has two main theorems concerning smooth algebras, and more general non-flat algebras [math]\displaystyle{ A }[/math]; but, the second is a direct generalization of the first. In the smooth case, i.e. for a smooth algebra [math]\displaystyle{ A }[/math], the Hochschild-Kostant-Rosenberg theorem^{pg 43-44} states there is an isomorphism [math]\displaystyle{ \Omega^n_{A/k} \cong HH_n(A/k) }[/math] for every [math]\displaystyle{ n \geq 0 }[/math]. This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential [math]\displaystyle{ n }[/math]-form has the map[math]\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)}. }[/math] If the algebra [math]\displaystyle{ A/k }[/math] isn't smooth, or even flat, then there is an analogous theorem using the cotangent complex. For a simplicial resolution [math]\displaystyle{ P_\bullet \to A }[/math], we set [math]\displaystyle{ \mathbb{L}^i_{A/k} = \Omega^i_{P_\bullet/k}\otimes_{P_\bullet} A }[/math]. Then, there exists a descending [math]\displaystyle{ \mathbb{N} }[/math]-filtration [math]\displaystyle{ F_\bullet }[/math] on [math]\displaystyle{ HH_n(A/k) }[/math] whose graded pieces are isomorphic to [math]\displaystyle{ \frac{F_i}{F_{i+1}} \cong \mathbb{L}^i_{A/k}[+i]. }[/math] 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 [math]\displaystyle{ A = R/I }[/math] for [math]\displaystyle{ R = k[x_1,\dotsc,x_n] }[/math], the cotangent complex is the two-term complex [math]\displaystyle{ I/I^2 \to \Omega^1_{R/k}\otimes_k A }[/math].

Polynomial rings over the rationals

One simple example is to compute the Hochschild homology of a polynomial ring of [math]\displaystyle{ \mathbb{Q} }[/math] with [math]\displaystyle{ n }[/math]-generators. The HKR theorem gives the isomorphism [math]\displaystyle{ HH_*(\mathbb{Q}[x_1,\ldots, x_n]) = \mathbb{Q}[x_1,\ldots, x_n]\otimes \Lambda(dx_1,\dotsc, dx_n) }[/math] where the algebra [math]\displaystyle{ \bigwedge(dx_1,\ldots, dx_n) }[/math] is the free antisymmetric algebra over [math]\displaystyle{ \mathbb{Q} }[/math] in [math]\displaystyle{ n }[/math]-generators. Its product structure is given by the wedge product of vectors, so [math]\displaystyle{ \begin{align} dx_i\cdot dx_j &= -dx_j\cdot dx_i \\ dx_i\cdot dx_i &= 0 \end{align} }[/math] for [math]\displaystyle{ i \neq j }[/math].

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 [math]\displaystyle{ \mathbb{Z} }[/math]-algebra [math]\displaystyle{ \mathbb{F}_p }[/math]. We can compute a resolution of [math]\displaystyle{ \mathbb{F}_p }[/math] as the free differential graded algebras[math]\displaystyle{ \mathbb{Z}\xrightarrow{\cdot p} \mathbb{Z} }[/math]giving the derived intersection [math]\displaystyle{ \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p \cong \mathbb{F}_p[\varepsilon]/(\varepsilon^2) }[/math] where [math]\displaystyle{ \text{deg}(\varepsilon) = 1 }[/math] and the differential is the zero map. This is because we just tensor the complex above by [math]\displaystyle{ \mathbb{F}_p }[/math], giving a formal complex with a generator in degree [math]\displaystyle{ 1 }[/math] which squares to [math]\displaystyle{ 0 }[/math]. Then, the Hochschild complex is given by[math]\displaystyle{ \mathbb{F}_p\otimes^\mathbb{L}_{\mathbb{F}_p\otimes^\mathbb{L}_\mathbb{Z} \mathbb{F}_p}\mathbb{F}_p }[/math]In order to compute this, we must resolve [math]\displaystyle{ \mathbb{F}_p }[/math] as an [math]\displaystyle{ \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p }[/math]-algebra. Observe that the algebra structure

[math]\displaystyle{ \mathbb{F}_p[\varepsilon]/(\varepsilon^2) \to \mathbb{F}_p }[/math]

forces [math]\displaystyle{ \varepsilon \mapsto 0 }[/math]. This gives the degree zero term of the complex. Then, because we have to resolve the kernel [math]\displaystyle{ \varepsilon \cdot \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p }[/math], we can take a copy of [math]\displaystyle{ \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p }[/math] shifted in degree [math]\displaystyle{ 2 }[/math] and have it map to [math]\displaystyle{ \varepsilon \cdot \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p }[/math], with kernel in degree [math]\displaystyle{ 3 }[/math][math]\displaystyle{ \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}}). }[/math]We can perform this recursively to get the underlying module of the divided power algebra[math]\displaystyle{ (\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}} }[/math]with [math]\displaystyle{ dx_i = \varepsilon\cdot x_{i-1} }[/math] and the degree of [math]\displaystyle{ x_i }[/math] is [math]\displaystyle{ 2i }[/math], namely [math]\displaystyle{ |x_i| = 2i }[/math]. Tensoring this algebra with [math]\displaystyle{ \mathbb{F}_p }[/math] over [math]\displaystyle{ \mathbb{F}_p\otimes^\mathbf{L}_\mathbb{Z}\mathbb{F}_p }[/math] gives[math]\displaystyle{ HH_*(\mathbb{F}_p) = \mathbb{F}_p\langle x \rangle }[/math]since [math]\displaystyle{ \varepsilon }[/math] multiplied with any element in [math]\displaystyle{ \mathbb{F}_p }[/math] 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 [math]\displaystyle{ \mathbb{F}_p\langle x \rangle }[/math] is not well behaved. For instance, [math]\displaystyle{ x^p = 0 }[/math]. One technical response to this problem is through Topological Hochschild homology, where the base ring [math]\displaystyle{ \mathbb{Z} }[/math] is replaced by the sphere spectrum [math]\displaystyle{ \mathbb{S} }[/math].

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) [math]\displaystyle{ k }[/math]-modules by an ∞-category (equipped with a tensor product) [math]\displaystyle{ \mathcal{C} }[/math], and [math]\displaystyle{ A }[/math] by an associative algebra in this category. Applying this to the category [math]\displaystyle{ \mathcal{C}=\textbf{Spectra} }[/math] of spectra, and [math]\displaystyle{ A }[/math] being the Eilenberg–MacLane spectrum associated to an ordinary ring [math]\displaystyle{ R }[/math] yields topological Hochschild homology, denoted [math]\displaystyle{ THH(R) }[/math]. The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for [math]\displaystyle{ \mathcal{C} = D(\mathbb{Z}) }[/math] the derived category of [math]\displaystyle{ \Z }[/math]-modules (as an ∞-category).

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

[math]\displaystyle{ THH(\mathbb{F}_p) = \mathbb{F}_p[x], }[/math]

[math]\displaystyle{ HH(\mathbb{F}_p) = \mathbb{F}_p\langle x \rangle }[/math]

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 [math]\displaystyle{ \mathbb{F}_p }[/math] 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, 19, Princeton University Press, ISBN 978-0-691-04991-5, https://books.google.com/books?id=0268b52ghcsC 
• Hazewinkel, Michiel, ed. (2001), "Cohomology of algebras", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=C/c023110 
• Hesselholt, Lars (2016), Topological Hochschild homology and the Hasse-Weil zeta function, Contemporary Mathematics, 708, pp. 157–180, doi:10.1090/conm/708/14264, ISBN 9781470429119 
• 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 
• 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. http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0. 

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 in 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. 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].