Hochschild homology

Template:Short description 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 Template:Harvs for algebras over a field, and extended to algebras over more general rings by Template:Harvs.

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 A^e-modules. Template:Harvtxt defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by

$H{H}_n(A,M)={\mathrm{Tor}}_n^{A^e}(A,M)$

$H{H}^n(A,M)={\mathrm{Ext}}_{A^e}^n(A,M)$

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

$\begin{array}{cc}{d}_0(m\otimes a_1\otimes \cdots \otimes a_n)& =m{a}_1\otimes a_2\cdots \otimes a_n\\ d_i(m\otimes a_1\otimes \cdots \otimes a_n)& =m\otimes a_1\otimes \cdots \otimes a_i{a}_{i+1}\otimes \cdots \otimes a_n\\ d_n(m\otimes a_1\otimes \cdots \otimes a_n)& =a_{n}m\otimes a_1\otimes \cdots \otimes a_{n-1}\end{array}$

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 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

$s_i(a_0\otimes \cdots \otimes a_n)=a_0\otimes \cdots \otimes a_i\otimes 1\otimes a_{i+1}\otimes \cdots \otimes a_n.$

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$$X\underset{S}{\overset{𝐋}{\times }}X$$which has the sheaf of derived rings ${𝒪}_X\underset{{𝒪}_S}{\overset{𝐋}{\otimes }}{𝒪}_X$. Then, if embed $X$ with the diagonal map$$\mathrm{\Delta }:X\to X\underset{S}{\overset{𝐋}{\times }}X$$the Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product scheme$$HH(X/S):={\mathrm{\Delta }}^{*}({𝒪}_X\underset{{𝒪}_X\underset{{𝒪}_S}{\overset{𝐋}{\otimes }}{𝒪}_X}{\overset{𝐋}{\otimes }}{𝒪}_X)$$From this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials ${\mathrm{\Omega }}_{X/S}$ since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex ${𝐋}_{X/S}^{\bullet }$ 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$$HH(A/k){\simeq }_{qiso}A\underset{A\underset{k}{\overset{𝐋}{\otimes }}A}{\overset{𝐋}{\otimes }}A$$If $A$ is a flat $k$-algebra, then there's the chain of isomorphisms $$A\underset{k}{\overset{𝐋}{\otimes }}A\cong A{\otimes }_{k}A\cong A{\otimes }_k{A}^{op}$$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 ${\mathrm{Fin}}_{*}$ of finite pointed sets, i.e., a functor ${\mathrm{\Delta }}^o\to {\mathrm{Fin}}_{*}.$ Thus, if F is a functor $F:\mathrm{Fin}\to k-\mathrm{m}\mathrm{o}\mathrm{d}$, we get a simplicial module by composing F with $S^1$.

${\mathrm{\Delta }}^o\stackrel{S^1}{⟶}{\mathrm{Fin}}_{*}\stackrel{F}{⟶}k\text{-mod}.$

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

$n_{+}=\{0,1,\dots ,n\},$

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-bimoduleTemplate:Elucidate. The Loday functor $L(A,M)$ is given on objects in ${\mathrm{Fin}}_{*}$ 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

$\mathrm{\forall }j\in \{0,\dots ,n\}:b_j=\{\begin{array}{cc}\prod _{i\in f^{-1}(j)}{a}_i& f^{-1}(j)\ne \mathrm{\varnothing }\\ 1& f^{-1}(j)=\mathrm{\varnothing }\end{array}$

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

${\mathrm{\Delta }}^o\stackrel{S^1}{⟶}{\mathrm{Fin}}_{*}\stackrel{ℒ(A,M)}{⟶}k\text{-mod},$

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 $H{H}_{*}(A)$ 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 $${\mathrm{\Omega }}_{A/k}^n\cong H{H}_n(A/k)$$ for every $n\ge 0$. This isomorphism can be described explicitly using the anti-symmetrization map. That is, a differential $n$-form has the map$$ad{b}_1\wedge \cdots \wedge d{b}_n\mapsto \sum _{\sigma \in S_n}\mathrm{sign}(\sigma )a\otimes b_{\sigma (1)}\otimes \cdots \otimes b_{\sigma (n)}.$$ 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 ${𝕃}_{A/k}^i={\mathrm{\Omega }}_{P_{\bullet }/k}^i{\otimes }_{P_{\bullet }}A$. Then, there exists a descending $\mathbb{N}$-filtration $F_{\bullet }$ on $H{H}_n(A/k)$ whose graded pieces are isomorphic to $$\frac{F_i}{F_{i+1}}\cong {𝕃}_{A/k}^i[+i].$$ 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 $R=k[x_1,\dots ,x_n]$, the cotangent complex is the two-term complex $I/I^2\to {\mathrm{\Omega }}_{R/k}^1{\otimes }_{k}A$.

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 $$H{H}_{*}(\mathbb{Q}[x_1,\dots ,x_n])=\mathbb{Q}[x_1,\dots ,x_n]\otimes \mathrm{\Lambda }(d{x}_1,\dots ,d{x}_n)$$ where the algebra $\bigwedge (d{x}_1,\dots ,d{x}_n)$ is the free antisymmetric algebra over $\mathbb{Q}$ in $n$-generators. Its product structure is given by the wedge product of vectors, so $$\begin{array}{cc}d{x}_i\cdot d{x}_j& =-d{x}_j\cdot d{x}_i\\ d{x}_i\cdot d{x}_i& =0\end{array}$$ for $i\ne j$.

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

${𝔽}_p[\epsilon ]/({\epsilon }^2)\to {𝔽}_p$

forces $\epsilon \mapsto 0$. This gives the degree zero term of the complex. Then, because we have to resolve the kernel $\epsilon \cdot {𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p$, we can take a copy of ${𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p$ shifted in degree $2$ and have it map to $\epsilon \cdot {𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p$, with kernel in degree $3$$\epsilon \cdot {𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p=\text{Ker}({𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p\to \epsilon \cdot {𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p).$We can perform this recursively to get the underlying module of the divided power algebra$$({𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p)⟨x⟩=\frac{({𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p)[x_1,x_2,\dots ]}{x_i{x}_j=(\genfrac{}{}{0ex}{}{i+j}{i})x_{i+j}}$$with $d{x}_i=\epsilon \cdot x_{i-1}$ and the degree of $x_i$ is $2i$, namely $|x_i|=2i$. Tensoring this algebra with ${𝔽}_p$ over ${𝔽}_p\underset{\mathbb{Z}}{\overset{𝐋}{\otimes }}{𝔽}_p$ gives$$H{H}_{*}({𝔽}_p)={𝔽}_p⟨x⟩$$since $\epsilon$ multiplied with any element in ${𝔽}_p$ 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 ${𝔽}_p⟨x⟩$ 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 $𝕊$.

Topological Hochschild homology

Template:Main 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) $𝒞$, and $A$ by an associative algebra in this category. Applying this to the category $𝒞=\mathbf{\text{Spectra}}$ 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 $𝒞=D(\mathbb{Z})$ 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,

$THH({𝔽}_p)={𝔽}_p[x],$

$HH({𝔽}_p)={𝔽}_p⟨x⟩$

is the polynomial ring (with x in degree 2), compared to the ring of divided powers in one variable.

Template:Harvs showed that the Hasse–Weil zeta function of a smooth proper variety over ${𝔽}_p$ can be expressed using regularized determinants involving topological Hochschild homology.

See also

• Cyclic homology

References

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• Jean-Louis Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) Template:ISBN
• Richard S. Pierce, Associative Algebras, Graduate Texts in Mathematics (88), Springer, 1982.
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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).
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• Topological Hochschild homology in arithmetic geometry
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Commutative case

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Noncommutative case

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