nLab Hochschild cohomology

Idea

Hochschild (co)homology is a homological construction which makes sense for any associative algebra, or more generally any dg-algebra or ring spectrum. It has multiple interpretations in higher category theory. Presently, everything below pertains to Hochschild homology of commutative algebras; an exposition of the noncommutative case remains to be written.

Thus, for $A$ a commutative ∞-algebra, its Hochschild homology complex is its (∞,1)-tensoring $S^1 \cdot A$ with the ∞-groupoid incarnation of the circle. More generally, for $S$ any $\infty$-groupoid/simplicial set, $S \cdot A$ is the corresponding higher order Hochschild homology of $A$.

In the presence of function algebras on ∞-stacks it may happen that $A = \mathcal{O}(X)$ is the algebra of functions on some ∞-stack $X$ and that $\mathcal{O}(-)$ sends powerings of $X$ to tensorings of $\mathcal{O}(X)$. In that case it follows that the Hochschild homology complex of $\mathcal{O}(X)$ is the function complex $\mathcal{O}(\mathcal{L}(X))$ on the derived loop space $\mathcal{L}X$ of $X$.

The Hochschild complex

Originally the notion of Hochschild homology was introduced as the chain homology of a certain chain complex associated to any bimodule $N$ over some algebra $A$: its bar complex, written

$$C_\bullet(A,N) := N \otimes_{A \otimes A^{op}} \mathrm{B}_\bullet A \,,$$

where $N$ and $A$ are regarded as $A \otimes A^{op}$-bimodules in the obvious way.

Then it was understood that this complex is the result of tensoring the $A$-bimodules $N$ with $A$ over $A \otimes A^{op}$ but using the derived functor of the tensor product functor – the Tor functor – in the ambient model structure on chain complexes:

$$C_\bullet(A,N) = N \otimes^D_{A\otimes A^{op}} A = Tor^\bullet_{A\otimes A^{op}}(N,A) \,.$$

Then still a little later, it was understood that this is just the ordinary tensor product in the symmetric monoidal (∞,1)-category of chain complexes. If this is understood, we can just write again simply

$$C_\bullet(A,N) := N \otimes_{A \otimes A^{op}} A \,.$$

This, generally, is the definition of the Hochschild homology object of any bimodule over a monoid in a symmetric monoidal $(\infty,1)$-category (symmetry is needed to make sense of $A^{op}$). Dually, the Hochschild cohomology object is

$$C^\bullet(A,N) := Hom_{A\otimes A^{op}}(A,N).$$

Of special interest is the case where $N = A$. In this case the Hochschild cohomology object is also called the (“$(\infty,1)$-” or “derived-”)center of $A$:

$$Z(A) := Hom_{A\otimes A^{op}}(A,A).$$

Dually, the Hochschild homology object when $N=A$ is called the universal trace or shadow. In this case, if $A = \mathcal{O}(X)$ can be identified with an $\infty$-algebra of functions on an object $X$, which is therefore commutative so that $A^{op}= A$, and if taking functions commutes with $(\infty,1)$-pullbacks, then

$$Z(\mathcal{O}(X)) \simeq \mathcal{O}(X \times_{X \times X} X) \simeq \mathcal{O}(\mathcal{L}X)$$

is the $\infty$-algebra of functions on the free loop space object of $X$.

Properties

By the Hochschild-Kostant-Rosenberg theorem and its generalizations, the Hochschild homology $HH_\bullet(\mathcal{O}(X),\mathcal{O}(X))$ of an ordinary algebra tends to behave like the algebra of Kähler differentials of $\mathcal{O}(X)$. More generally, this computes the cotangent complex of the $\infty$-algebra $\mathcal{O}(X)$. The cup product gives the wedge product of forms and the $S^1$-action the de Rham differential.

Dually this means that in derived geometry the free loop space object $\mathcal{L} X$ consists of infinitesimal loops in $X$ (in ordinary geometry it would be equal to $Spec A$, consisting only of constant loops).

Analogously, Hochschild cohomology $HH^\bullet(\mathcal{O}(X), \mathcal{O}(X))$ of $\mathcal{O}(X)$ computes the multivector fields on $X$. There are pairing operations on HH homology and cohomology that make them support a general differential calculus on $X$, which makes sense even if $\mathcal{O}(X)$ is a noncommutative algebra.

Definition

We start with the general-abstract definition of Hochschild homology and then look at special and more traditional cases.

General abstract

We look at the very general abstract definition of Hochschild (co)homology and some important subcases.

Hochschild homology

We discuss Hochschild homology of commutative algebras for the case that these are related to function algebras on derived loop spaces.

Definition

Let $\mathbf{H}$ be an (∞,1)-topos that admits function algebras on ∞-stacks (see there for details)

$$Alg^{op} \stackrel{\overset{\mathcal{O}}{\longleftarrow}}{\underset{}{\longrightarrow}} \mathbf{H} \,.$$

In particular the (∞,1)-category of ∞-algebras $Alg^{op}$ is (∞,1)-tensored over ∞Grpd. Then for $A \in Alg$ and $K \in \infty Grpd$ we say that

$$K \cdot A \in Alg$$

is the Hochschild homology complex of $A$ over $K$.

We say a full sub-(∞,1)-category of $\mathbf{H}$ consists of $\mathcal{O}$-perfect objects if on these $\mathcal{O}$ commutes with (∞,1)-limits.

Then for $X$ an $\mathcal{O}$-perfect object we have

$$K \cdot \mathcal{O}(X) \simeq \mathcal{O}(X^{K}) \,.$$

For $K = S^1$ the circle, this is ordinary Hochschild homology, while for general $K$ it is called higher order Hochschild homology .

Example

For $\mathcal{O}$ the functor that forms symmetric monoidal (∞,1)-categories of quasicoherent ∞-stacks of modules over ∞-stacks over an (∞,1)-site of ∞-algebras for the ordinary theory of commutative $k$-algebras this has setup been considered in detail in (Ben-ZviFrancisNadler).

The following definition formalizes large classes of $\mathcal{O}$-perfect objects given by representables.

Definition

Let $T$ be an (∞,1)-algebraic theory and $T Alg_\infty$ its (∞,1)-category of $\infty$-algebras. Let $C$ with $T \hookrightarrow C \hookrightarrow T Alg_\infty^{op}$ be a small full sub-(∞,1)-category of $T Alg_\infty^{op}$ which is closed under (∞,1)-limits in $T Alg$ and equipped with the structure of a subcanonical (∞,1)-site.

Take $\mathbf{H} := Sh(C)$ the (∞,1)-category of (∞,1)-sheaves on $C$. This is an (∞,1)-topos for derived geometry modeled on $T Alg_\infty$. Write $C \hookrightarrow \mathbf{H}$ for the (∞,1)-Yoneda embedding.

For $X \in C\stackrel{}{\hookrightarrow} \mathbf{H}$ write $\mathcal{O}(X)$ for the same object regarded as an object of $T Alg_\infty$.

Proposition/Definition

In the context of the above definition we have

$$\mathcal{O} (\mathcal{L}X) \simeq S^1 \cdot \mathcal{O}(X) \in T Alg_\infty \,,$$

where on the right we have the (∞,1)-tensoring of $T Alg_\infty$ over $\infty Grpd$, which is the (∞,1)-colimit over the diagram $S^1$ of the (∞,1)-functor constant on $\mathcal{O}(X)$

$$S^1 \cdot \mathcal{O}(X) \simeq {\lim_{\to}}_{S^1} \mathcal{O}(X) \,.$$

This object we call the Hochschild homology complex of $\mathcal{O}X$.

Generally for higher order Hochschild homology we have

$$\mathcal{O}(X^K) \simeq K \cdot \mathcal{O}(X) \simeq {\lim_{\to}}_{K} \mathcal{O}(X) \in T Alg_\infty \,.$$

Proof

Because the (∞,1)-Yoneda embedding preserves (∞,1)-limits the limit $X^K$ may be computed in $C$. By assumption $C$ is closed under limits in $T Alg_\infty^{op}$. The limit $X^K$ in $T Alg^{op}$ is the colimit $K \cdot \mathcal{O}(X)$ in the opposite (∞,1)-category of $\infty$-algebras.

This definition of general higher order Hochschild homology by $(\infty,1)$-copowering is

• explicit in ToënVezzosi, for ordinary Hochschild homology, hence $K = S^1$,

• almost explicit in (GinotTradlerZeinalian), for higher order Hochschild homology for dg-algebras. Details on that are below in the section Higher order Hochschild homology modeled on cdg-algebras

• explicit in Ben-Zvi/Francis/Nadler, corollary 4.12 for HH with values in quasicoherent ∞-stacks and over perfect $\infty$-stacks (see there for details).

Topological chiral homology

Notice that the tensoring that gives the Hochschild homology is given by the $\infty$-colimit over the constant functor

$$K \cdot A \simeq {\lim_\to}_K A \,.$$

This generalizes to $\infty$-colimits of functors constant on an algebra, but over a genuine (∞,1)-category diagram.

Specifically let $X$ be framed $n$-manifold, $A$ an En-algebra and $D_X$ the (∞,1)-category whose objects are framed embeddings of disjoint unions of open discs into $X$ and morphisms are inclusions of these. Let $F_A$ be the functor that assigns $A^{k}$ to an object corresponding to $k$ discs in $X$, and iterated products/units to morphisms

Then the (∞,1)-colimit

$${\lim_\to}_{D_X} F_A$$

is called the topological chiral homology of $X$.

For $A$ an ordinary associatve algebra, hence in particular an $E_1$-algebra, and $X$ the circle, this reproduces the ordinary Hochschild homology of $A$ (see below).

For more details see (GinotTradlerZeinalian).

Specific concrete

We unwind the above general abstract definition in special classes of examples and find more explicit and more traditional definitions of Hochschild homology.

Pirashvili’s higher order Hochschild homology

We demonstrate how the above $(\infty,1)$-category theoretic definition of higher order Hochschild homology reproduces the simplicial definition by (Pirashvili).

Proposition

Let $T$ be a Lawvere theory regarded as a 0-truncated (∞,1)-algebraic theory.

Consider a model structure on simplicial T-algebras/on homotopy T-algebras presenting $T Alg_\infty$ such that

1. it is a simplicial model category;

2. tensoring with simplicial sets preserves weak equivalences and hence cofibrant replacement.

Then for $\mathcal{O}(X) \in T Alg \hookrightarrow T Alg_\infty$ and $K \in \infty Grpd$ the higher order Hochschild homology complex $K \cdot \mathcal{O}(X)$ is presented by the ordinary tensoring $K \cdot \mathcal{O}(X)$ in the model category, for $K$ any simplicial set incarnation of the $\infty$-groupoid.

Proof

The $(\infty,1)$-tensoring in an $(\infty,1)$-category presented by a simplicial model category is modeled by the ordinary tensoring of the latter on a cofibrant resolution of the given object. This is discussed in the section ∞-tensoring – models.

Remark

We can always use the model structure on homotopy T-algebras to satisfy the assumption of the above proposition. That is a simplicial model category for every $T$ and every ordinary algebra is cofibrant in this structure.

Notice that in this model category even if $\mathcal{O}(X)$ is fibrant (which it is if $\mathcal{O}X$ is an ordinary algebra), then $K \cdot \mathcal{O}(X)$ is in general far from being fibant. Computing the simplicial homotopy groups of $K \cdot \mathcal{O}(X)$ and hence the Hochschild homology involves passing to a fibrant reolsution of $K \cdot \mathcal{O}(X)$ first, that will make it a homotopy T-algebra.

On the other hand, if we find a simplicial model structure on simplicial T-algebras (which are degreewise genuine $T$-algebras) then the coproducts involved degreewise in forming $K \cdot \mathcal{O}(X)$ will be tensor products of algebras, and hence in particular themselves again algebras. For such a model the tensoring $K \cdot \mathcal{O}(X)$ yields explicitly (under the Dold-Kan correspondence).

This is the case for the tensoring of dg-algebras over simplicial sets and leads to Teimuraz Pirashvili‘s formulation of higher order Hochschild homology for ordinary algebras (Pirashvili).

This we describe below

• for ordinary Hochschild homology in Examples – Simplicial algebra on the circle;

• for higher order Hochschild homology of dg-algebras in Higher order Hochschild homology modeled on cdg-algebras.

Examples

We first give a detailed discussion of the standard Hochschild complex of a commutative algebra, but from the general abstract $(\infty,1)$-category theoretic point of view.

Then we look in detail at higher order Hochschild homology in the $(\infty,1)$-topos over an (∞,1)-site of formal duals of dg-algebras. In this context the classical theorem by Jones on Hochschild homology and loop space cohomology is a natural consequence of the general machinery.

Gradings and conventions

In derived geometry two categorical gradings interact: a cohesive $\infty$-groupoid $X$ has a space of k-morphisms $X_k$ for all non-negative $k$, and each such has itself a simplicial T-algebra of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from function algebras on ∞-stacks.

Functions on a bare $\infty$-groupoid $K$, modeled as a simplicial set, form a cosimplicial algebra $\mathcal{O}(K)$, which under the monoidal Dold-Kan correspondence identifies with a cochain dg-algebra (meaning: with positively graded differential) in non-negative degree

$$\left( \array{ \vdots \\ \downarrow \downarrow \downarrow \downarrow \\ K_2 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \downarrow^{\partial_2} \\ K_1 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \\ K_0 } \right) \;\;\;\;\; \stackrel{\mathcal{O}}{\mapsto} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \uparrow \uparrow \uparrow \\ \mathcal{O}(K_2) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \uparrow^{\partial_2^*} \\ \mathcal{O}(K_1) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \\ \mathcal{O}(K_0) } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \cdots \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_2 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_1 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_0 \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \vdots } \right) \,.$$

On the other hand, a representable $X$ has itself a simplicial T-algebra of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write

$$\mathcal{O}(X) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \mathcal{O}(X)_0 \\ \uparrow \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \uparrow \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \uparrow \uparrow \uparrow \\ \vdots } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \mathcal{O}(X)_0 \\ \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \\ \vdots } \right) \,.$$

Taking this together, for $X_\bullet$ a general ∞-stack, its function algebra is generally an unbounded cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:

$$\mathcal{O}(X_\bullet) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ \bigoplus_{k-p = q} \mathcal{O}(X_k)_{-p} \\ \uparrow \\ \vdots \\ \uparrow^d \\ \mathcal{O}(X_1)_0 \oplus \mathcal{O}(X_2)_{-1} \oplus \mathcal{O}(X_3)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_0 \oplus \mathcal{O}(X_1)_{-1} \oplus \mathcal{O}(X_2)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_{-1} \oplus \mathcal{O}(X_1)_{-2} \oplus \mathcal{O}(X_2)_{-3}\oplus \cdots \\ \uparrow^{d} \\ \vdots } \right) \,.$$

The Hochschild chain complex of an associative algebra

We consider in detail the classical case of Hochschild (co)homology of an associative algebra approaching it from the general abstract perspective on Hochschild homology.

This section focuses on exposition. The formal context in which the constructions considered here follow from first principles is discussed below in Higher order Hochschild homology modeled on cdg-algebras

The simplicial circle

We shall use two different equivalent models of the circle in $\infty Grpd$ in terms of models in $sSet$:

1. the simplicial set $\Delta[1]/\partial \Delta[1]$

   This is not fibrant (not a Kan complex). On the contrary, this is the smallest simplicial model available for the circle, with the least number of horn fillers.

   In low degrees it looks as follows

   $$\array{ \vdots && \vdots \\ (\Delta[1]/\partial\Delta[1])_3 & = & (* * * *) \coprod (* \to * * *) \coprod (* * \to * *) \coprod (* * * \to * ) & \\ \\ (\Delta[1]/\partial\Delta[1])_2 & = & (* * *) \coprod (*\to* *) \coprod (* * \to *) \\ \\ (\Delta[1]/\partial\Delta[1])_1 & = & (* *) \coprod (* \to *) \\ \\ (\Delta[1]/\partial\Delta[1])_0 & = & (*) } \,.$$

   Here for instance the expression $(* * \to * )$ denotes the morphism of simplicial sets $\Delta[2] \to \Delta[1]/\partial \Delta[1]$ that sends the first edge (the 2-face) of the 2-simplex to the unique degenerate 1-cell and the second edge (the 0-face) to the unique non-degenerate 1-cell of $\Delta[1]/ \partial \Delta[1]$.

2. the nerve of the delooping groupoid $\mathbf{B}\mathbb{Z}$ of the additive group of integers.

   This model is fibrant (is a Kan complex) and makes manifest the group structure on $S^1$, which is the strict 2-group structure on $\mathbf{B}\mathbb{Z}$ or equivalently the structure of a simplicial group on its nerve.

   $$\mathbf{B}\mathbb{Z} \times \mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}$$
   $$((* \stackrel{k}{\to} * ), (* \stackrel{l}{\to} * )) \mapsto (* \stackrel{k+l}{\to} * ) \,.$$

Tensoring with the simplicial circle

Let $A \in CAlg_k$ be a commutative associative algebra over a commutative ring $k$.

Above in the section on Higher order Hochschild homology we had discussed how the Hochschild homology of $A$ is given by the simplicial algebra $(\Delta[1]/\partial \Delta[1]) \cdot A \in CAlg_k^{\Delta^{op}}$ that is the tensoring of $A$ regarded as a constant simplicial algebra with the simplicial set $\Delta[1]/\partial \Delta[1]$ (the 1-simplex with its two 0-cells identified).

We describe now in detail what this simplicial circle algebra looks like. The proof that this construction is indeed homotopy-good is given below in As functions on the derived loop space

When forming the copowering of $A$ with the simplicial circle $S^1$, we get the same structure as displayed above, but with one copy of $A$ for each item in parenthesis.

To be very explicit, we recall and demonstrate the following elementary fact.

Proposition

In $CAlg_k$ the coproduct is given by the tensor product over $k$:

$$\left( \array{ A &\stackrel{i_A}{\to}& A \coprod B &\stackrel{i_B}{\leftarrow}& B } \right) \simeq \left( \array{ A &\stackrel{Id_A \otimes_k e_B}{\to}& A \otimes_k B & \stackrel{e_A \otimes Id_B}{\leftarrow}& B } \right)$$

Proof

We check the universal property of the coproduct: for $C \in CAlg_k$ and $f,g : A,B \to C$ two morphisms, we need to show that there is a unique morphism $(f,g) : A \otimes_k B \to C$ such that the diagram

$$\array{ A &\stackrel{Id_A \otimes e_B}{\to}& A \otimes_k B &\stackrel{e_A \otimes Id_B}{\leftarrow}& B \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{(f,g)}} & \swarrow_{\mathrlap{g}} \\ && C }$$

commutes. For the left triangle to commute we need that $(f,g)$ sends elements of the form $(a,e_B)$ to $f(a)$. For the right triangle to commute we need that $(f,g)$ sends elements of the form $(e_A, b)$ to $g(b)$. Since every element of $A \otimes_k B$ is a product of two elements of this form

$$(a,b) = (a,e_B) \cdot (e_A, b)$$

this already uniquely determines $(f,g)$ to be given on elements by the map

$$(a,b) \mapsto f(a) \cdot g(b) \,.$$

That this is indeed an $k$-algebra homomorphism follows from the fact that $f$ and $g$ are

Notice that for all this it is crucial that we are working with commutative algebras.

Corollary

We have that the tensoring of $A$ with the map of sets from two points to the single point

$$(* \coprod * \to *) \cdot A \simeq ( A \otimes_k A \stackrel{\mu}{\to} A )$$

is the product morphism on $A$. And that the tensoring with the map from the empty set to the point

$$(\emptyset \to *)\cdot A \simeq (k \stackrel{e_A}{\to} A)$$

is the unit morphism on $A$. Generally, for $f : S \to T$ any map of sets we have that the tensoring

$$(S \stackrel{f}{\to} T) \cdot A = A^{\otimes_k |S|} \to A^{\otimes_k |T|}$$

is the morphism between tensor powers of $A$ of the cardinalities of $S$ and $T$, respectively, whose component over a copy of $A$ on the right corresponding to $t \in T$ is the iterated product $A^{\otimes_k |f^{-1}\{t\}|} \to A$ on as many tensor powers of $A$ as there are elements in the preimage of $t$ under $f$.

We see that in low degree the simplicial algebra $(\Delta[1]/\partial \Delta[1]) \cdot A$ has the components

$$\array{ \vdots && \vdots \\ ((\Delta[1]/\partial\Delta[1]) \cdot A)_3 & = & A \otimes A \otimes A \otimes A & \\ \\ ((\Delta[1]/\partial\Delta[1]) \cdot A)_2 & = & A \otimes A \otimes A \\ \\ ((\Delta[1]/\partial\Delta[1]) \cdot A)_1 & = & A \otimes A \\ \\ ((\Delta[1]/\partial\Delta[1]) \cdot A)_0 & = & A } \,.$$

The two face maps from degree 1 to degree 0 both come from mapping two points to a single point, so they are both the product on $A$.

$$A \otimes_k A \stackrel{\overset{\mu}{\longrightarrow}}{\underset{\mu}{\longrightarrow}} A \,.$$

The three face maps from degree 3 to degree 2 are more interesting. We have

$$\partial^2_0 \;\;\; : \;\;\; \array{ (* * *) &\mapsto & (* *) \\ \coprod & \nearrow &\coprod& \\ (* \to * *) & & (* \to *) \\ \coprod &\nearrow&& \\ (* * \to *) }$$

and

$$\partial^2_1 \;\;\; : \;\;\; \array{ (* * *) &\mapsto & (* *) \\ \coprod & &\coprod& \\ (* \to * *) & \mapsto & (* \to *) \\ \coprod &\nearrow&& \\ (* * \to *) }$$

and

$$\partial^2_2 \;\;\; : \;\;\; \array{ (* * \to *) &\mapsto & (* *) \\ \coprod & \nearrow &\coprod& \\ (* * *) & & (* \to *) \\ \coprod &\nearrow&& \\ (* \to * *) } \,.$$

Notice that for the last one we had to cyclically permute the source in order to display the maps in this planar fashion.

So therefore we get the tensorings

$$(\partial^2_0) \cdot A = ( A \otimes_k A \otimes_k A \stackrel{\mu \otimes_k Id}{\to} A )$$

and

$$(\partial^2_1) \cdot A = ( A \otimes_k A \otimes_k A \stackrel{Id \otimes_k \mu}{\to} A )$$

and

$$(\partial^2_2) \cdot A = ( A \otimes_k A \otimes_k A \stackrel{\mu \otimes_k Id \circ \sigma_{2,3,1}}{\to} A ) \,.$$

In summary we have so far

$$(\Delta[1]/\partial \Delta[1])\cdot A = \left( \cdots A\otimes_k A \otimes_k A \stackrel{\overset{\mu \otimes_k Id \sigma_{2,3,1}}{\to}}{\stackrel{\overset{Id \otimes_k \mu}{\longrightarrow}}{\underset{\mu \otimes_k Id}{\longrightarrow}}} A \otimes_k A \stackrel{\overset{\mu}{\longrightarrow}}{\underset{\mu}{\longrightarrow}}A \right) \,.$$

The Moore complex of this simplicial algebra is the traditional Hochschild chain complex of $A$

$$C_\bullet(A,A) = C_\bullet((\Delta[1]/\partial \Delta[1]) \cdot A) \,.$$

This we describe in more detail in the section Explicit description of the Hochschild complex.

Generally, for $K$ any simplicial set, $K \cdot A$ is the simplicial algebra whose Moore complex is the complex that (Pirashvili) uses to define higher order Hochschild homology.

Identification with Kähler differential forms

We spell out in detail how in degree 0 and 1 the homology of the Hochschild complex of $A$ is that of its Kähler differential forms. Under mild conditions on $A$ this is also true in higher degrees, which is the statement of the Hochschild-Kostant-Rosenberg theorem.

Proposition

The homology of the Hochschild complex $S^1 \cdot A$ in degree 1 is the Kähler differential forms of $A$

$$HH_1(A,A) = H_\bullet(S^1\cdot A) \simeq \Omega^1_K(A/k) \,.$$

The isomorphism is induced by the identifications

$$\array{ \vdots \\ (f \in A_{(* * * )}, g \in A_{(*\to * *)}, h \in A_{(* * \to *)} & \mapsto & \frac{1}{2} f \wedge d g \wedge d h \\ (f \in A_{(* *)}, g \in A_{* \to *}) &\mapsto& f \wedge d g \\ (f \in A_{(*)}) & \mapsto & f } \,,$$

where on the left we display elements of $A^{\otimes_k}$ under the above identification of these tensor powers in $S^1 \cdot A$.

Proof

By the above discussion, the Moore complex-differential acts on $(f,g,h) \in A \otimes_k A \otimes_k A$ by

$$\begin{aligned} \partial (f,g,h) &= (f g, h) - (f, g h) + (h f, g) \\ & \sim f g \wedge d h - f \wedge d (g h) + f h \wedge d g \end{aligned} \,.$$

The last term on the right is precisely the term by which one has to quotient out the module of formal expressions $f \wedge d g$ to get the module of Kähler differentials: setting it to 0 is the derivation property of $d$

$$(\partial (f,g,h) = 0) \Leftrightarrow f \wedge ( d(g h) = h \wedge d g + g \wedge d h ) \,.$$

Therefore we have manifestly

$$\Omega^1_K(A) \simeq C_1(A,A)/im(\partial) \,.$$

Remark

We may also compute the $\partial$-homology on the normalized chain complex, which is in degree 1 the quotient of $A \otimes_k A$ by the image of the degeneracy map $\sigma : A \to A \otimes_k A$, which is

$$\left( \array{ (*) &\to& (* *) \\ \coprod && \coprod \\ \emptyset &\to& (* \to *) } \right) \cdot A$$

and thus maps

$$f \mapsto f \wedge d 1 \,.$$

So passage to the normalized chains imposes the condition

$$d 1 = 0 \,.$$

Proposition

Under the identification of $HH_\bullet(A,A) = H_\bullet(S^1 \cdot A)$ with Kähler differential forms, the cup product on homology identifies with the wedge product of differential 0- and 1-forms.

Proof

Under the monoidal Dold-Kan correspondence the product on the Moore complex $N_\bullet(S^1 \cdot A)$ is given by the Eilenberg-Zilber map $\nabla$

$$N_\bullet(S^1 \cdot A) \otimes_k N_\bullet(S^1 \cdots) \stackrel{\nabla}{\to} N_\bullet((S^1 \cdot A) \otimes (S^1 \cdot A)) \stackrel{N_\bullet(\cdot)}{\to} N_\bullet(S^1 \cdot A) \,,$$

where for $\omega \in (S^1\cdot A)_p$ and $\lambda \in (S^1 \cdot A)_q$ we have

$$\nabla : \omega \otimes \lambda \mapsto \sum_{(\mu,\nu)\in Shuff(p,q)} sign(\mu,\nu) s_\nu(\omega) \otimes s_\mu(\lambda) \,.$$

For instance for $\omega = f \wedge d g \in (S^1 \cdot A )_0$ we have

$$s_1 (f \wedge d g) = f \wedge d g \wedge d 1$$

and

$$s_2(h \wedge d q) = (h \wedge d 1 \wedge d q)$$

and the tensor product (in $(S^1 \cdot A)_2$!) is componentwise

$$s_1(f \wedge d g) \otimes s_2(h \wedge d q) = (f \otimes h) \wedge d(g \otimes 1) \wedge d(1 \otimes q) \,.$$

Therefore

$$\nabla(\omega, \lambda) = f h \wedge d q \wedge d q \,.$$

The simplicial circle action

We describe the canonical action of the automorphism 2-group of the circle $S^1$ on $S^1 \cdot A$ and how its degree-1 part induces under the above identification $H_\bullet(S^1 \cdot A) \simeq \Omega^\bullet_K(A)$ the action of the de Rham differential.

Proposition

The automorphism 2-group of the categorical circle is

$$Aut_{\infty Grpd}(\mathbf{B}\mathbb{Z}, \mathbf{B}\mathbb{Z}) \simeq \coprod_{\{+1,-1\}} \mathbf{B}\mathbb{Z} \,.$$

Proof

We may compute the automorphism 2-group in the full sub-(∞,1)-category Grpd $\subset$ ∞Grpd, whose morphisms are functors and 2-morphisms are natural isomorphisms (see the statement about homotopy 1-types at homotopy hypothesis for details). A functor between delooping groupoids $\mathbf{B}G \to \mathbf{B}H$ is precisely a group homomorphism $G \to H$. The additive group endomorphisms of $\mathbb{Z}$ are precisely given by multiplication with elements in $\mathbb{Z}$, the two automorphisms in there are $\pm 1$.

The natural transformations between such functors are

$$\left( \array{ & \nearrow \searrow^{\mathrlap{\pm 1}} \\ \mathbf{B}\mathbb{Z} & \Downarrow^{r}& \mathbf{B}\mathbb{Z} \\ & \searrow \nearrow_{\mathrlap{\pm 1}} } \right) \;\; : \;\; \left( \array{ * \\ \downarrow^{\mathrlap{1}} \\ * } \right) \;\; \mapsto \;\; \left( \array{ * &\stackrel{r}{\to}& * \\ {}^{\mathllap{\pm 1}}\downarrow && \downarrow^{\mathrlap{\pm 1}} \\ * &\stackrel{r}{\to}& * } \right) \,.$$

Now consider the right homotopy that exhibits the morphism 1 in $Aut(\mathbf{B}\mathbb{Z})_{Id}$.

$$\array{ && \mathbf{B}\mathbb{Z} \\ &{}^{\mathllap{Id}}\nearrow & \uparrow \\ \mathbf{B}\mathbb{Z} &\stackrel{\eta}{\to}& \mathbf{B}\mathbb{Z}^{I} \\ & {}_{\mathllap{Id}}\searrow \\ && \mathbf{B}\mathbb{Z} } \,.$$

This sends

$$\eta : * \mapsto (* \stackrel{1}{\to} * ) \,.$$

This means that under copowering this on $A$

$$(\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}^I)\cdot A$$

we get in degree 0 the morphism

$$A_{*} \stackrel{Id}{\to} A_{* \to *} \hookrightarrow \bigotimes_r A_{* \stackrel{r}{\to} *} \,.$$

Under the above identification of the homology of $\mathbf{B}\mathbb{Z} \cdot A$ with Kähler forms, this is on elements the map

$$f \mapsto d f \,.$$

Remark

(automorphisms of the odd line)

This means that under the identification of $(\mathbf{B}\mathbb{Z}) \cdot k \simeq C^\infty(k^{0|1})$ with functions on the odd line,in degree 0 this corresponds to the even vector field $\theta \partial/\partial \theta$ on the odd line, and in degree 1 to the odd vector field $\partial/\partial\that$.

(…)

Traditional description of the Hochschild complex

We spell out explicitly the Hochschild chain complex for an associative algebra (over some ring $k$) with coefficients in a bimodule.

Definition

The bar complex of $A$ is the connective chain complex

$$\mathrm{B}_\bullet A := ( \cdots \to A^{\otimes_k n} \stackrel{\partial}{\to} A^{\otimes_k n-1} \to \cdots \to A \otimes_k A \otimes_k A \stackrel{\partial}{\to} A \otimes_k A )$$

which in degree $n$ has the $(n+1)$ tensor power of $A$ with itself, and whose differential is given by

$$\partial(a_0, a_1, \cdots a_n) := (a_0 a_1, a_2, \cdots, a_n) - (a_0, a_1 a_2 , a_3, \cdots, a_n) + \cdots - (-1)^n (a_0, a_1, \cdots, a_{n-1} a_n) \,,$$

regarded as a chain complex in $A$-bimodules for the evident bimodule structure in each degree.

Definition

Let $N$ be an $A$-bimodule. The Hochschild chain complex $C_\bullet(A,N)$ of $A$ with coefficients in $N$ is the chain complex obtained by taking in the bar complex degreewise the tensor product of $A$-bimodules with $N$:

$$C_\bullet(A,N) := N_{A \otimes A^{op}}\mathrm{B}_\bullet A \,.$$

The Hochschild homology of $A$ with coefficients in $N$ is the homology of the Hochschild chain complex, written

$$HH_n(A,N) := H_n( C_\bullet(A,N)) \,.$$

Proposition

At the level of the underlying $k$-modules we have natural isomorphisms

$$N_{A \otimes_k A^{op}} A^{\otimes_k (n+2)} \simeq N \otimes_k \otimes A^{\otimes_k n}$$

given on elements by sending

$$(\nu, (a_0, a_1, \cdots, a_n, a_{n+1})) \sim (a_{n+1} \nu a_{0}, (1, a_1, \cdots, a_n, 1)) \mapsto (a_0 \nu a_{n+1}, a_1, \cdots, a_n) \,.$$

The action of the differential in $C_\bullet(A,N)$ on elements of the latter form is then

$$\partial(\nu, a_1, \cdots, a_n) = (\nu a_1, a_2, \cdots, a_n) - (\nu, a_1 a_2, a_3, \cdots) + \cdots + (-1)^n (\nu , a_1, \cdots, a_{n-1} a_n) - (-1)^{n} (a_n \nu, a_1, a_2, \cdots, a_{n-1}) \,.$$

Remark

In words this means that the Hochschild complex is obtained froms the bar complex by “gluing the two ends of a sequence of elements of $A$ to a circle by a bimodule”.

The fact that the circle appears here has in fact a deep significance: the Hochschild chain complex may be understood in higher geometry as encoding functions on a free loop space object of whatever $A$ behaves like being functions on.

As function algebra on the derived loop space

We give a formal derivation of the Hochschild complex of an ordinary commutative associative algebra $\mathcal{O}(X)$ as the function algebra on the derived loop space object $\mathcal{L}X$ in the context of derived geometry.

So let now $T$ be the Lawvere theory of ordiary commutative associative algebras over a field $k$, regard as a 0-truncated (∞,1)-algebraic theory.

Proposition

The (∞,1)-category $CAlg(k)_\infty$ of ∞-algebras over $T$ is presented by the model structure on simplicial commutative k-algebras $(CAlg_k^{\Delta^{op}})_{proj}$.

This is Quillen equivalent to the standard model structure on connected dg-chain algebras.

$$(CAlg_k^{\Delta^{op}})_{proj} \simeq dgAlg_k^+ \,.$$

Proof

The first statement is discussed at (∞,1)-algebraic theory and homotopy T-algebra. The second statement is discussed at monoidal Dold-Kan correspondence.

Let

$$T \subset C \subset T Alg_\infty^{op}$$

be a subcanonical (∞,1)-site that is a full sub-(∞,1)-category of formal duals of $\infty$-$T$-algebras, closed under (∞,1)-limits in $T Alg_\infty^{op}$.

Let

$$\mathbf{H} \coloneqq Sh_{(\infty,1)}(C)$$

be the (∞,1)-sheaf (∞,1)-topos over $C$.

Following the notation at Isbell duality and function algebras on ∞-stacks we write $\mathcal{O}(X) \in T Alg_\infty$ for an object that under the (∞,1)-Yoneda embedding $C \hookrightarrow T Alg_\infty^{op} \to \mathbf{H}$ maps to an object called $X$ in $\mathbf{H}$.

Definition

For $\mathcal{O}(X) \in T Alg \hookrightarrow T Alg_\infty$ an ordinary $T$-algebra, we say that the free loop space object

$$\mathcal{L}X \coloneq [S^1,X]$$

of $X$ formed in $\mathbf{H}$ is the derived loop space of $X$.

Remark

The term derived is just to emphasize that we do not form the free loop space object in an (∞,1)-topos of $(\infty,1)$-sheaves over a 1-site inside the 1-category $Alg_k^{op}$. These “underived” (not embedded into (∞,1)-category theory) free loop space objects would just be equivalent to $X$. The derived loop space instead has rich interesting structure.

But if the ambient context of higher geometry over the genuine (∞,1)-site of formal duals to $\infty$-algebras is clear, we can just speak of free loop space objects . They are canonically given.

Proposition

We have that $\mathcal{O}(\mathcal{L}X)$ is given by the (∞,1)-pushout in $CAlg_\infty$

$$\mathcal{O}\mathcal{L}X \simeq \mathcal{O}(X) \coprod_{\mathcal{O}(X)\otimes \mathcal{O}(X) } \mathcal{O}(X)$$

hence by the universal cocone

$$\array{ \mathcal{O}\mathcal{L}X &\leftarrow& \mathcal{O}(X) \\ \uparrow && \uparrow \\ \mathcal{O}(X) &\leftarrow& \mathcal{O}(X) \otimes \mathcal{O}(X) }$$

Proof

Since ∞-stackification $L : PSh_{(\infty,1)}(C) \to \mathbf{H}$ is a left exact (∞,1)-functor and hence preserves finite (∞,1)-limits, we have that the defining pullback for $\mathcal{L}X$ may be computed in the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$. Since the (∞,1)-Yoneda embedding preserves all (∞,1)-limits this in turn may be computed in the (∞,1)-site $C$, hence by assumption in $T Alg_\infty$. The relevant $(\infty,1)$-pullback there is the claimed $(\infty,1)$-pushout in the opposite (∞,1)-category $T Alg_\infty$.

Proposition

The $\infty$-algebra $\mathcal{O} \mathcal{L}X$ of functions on the derived loop space of $X$ is when modeled by a simplicial algebra in $CAlg_k^{\Delta^{op}}$ under the monoidal Dold-Kan correspondence equivalent to the Hochschild chain complex of $\mathcal{O}X$ with coefficients in itself:

$$\mathcal{O} \mathcal{L}X \simeq C_\bullet(\mathcal{O}(X), \mathcal{O}(X)) \,.$$

Proof

First observe that the coproduct in $CAlg_k$ is the tensor product of commutative algebras over $k$

$$A \coprod B = A \otimes_k B \,.$$

By the discussion at homotopy T-algebra we may model $T Alg_\infty$ by the injective model structure on simplicial presheaves on $T^{op}$, left Bousfield localized at the morphisms $T[k] \otimes T[l] \to T[k+l]$. This localized model structure we write $[T, sSet]_{inj,prod}$.

By the above proposition we have that $\mathcal{O}\mathcal{L}X$ is given by the homotopy pushout in $[T, sSet]_{inj,prod}$ of

$$\mathcal{O}X \leftarrow \mathcal{O}(X)\otimes_k \mathcal{O}(X) \to \mathcal{O}(X) \,,$$

where both morphism are simple the product on $\mathcal{O}(X) \in CAlg_k$. By general properties of homotopy pushouts and the injective model structure on simplicial presheaves we have that this homotopy pushout is computed by an ordinary pushout once we pass to a weakly equivalent diagram in which one of the two morphism is a cofibration of simplicial algebras.

$$\array{ \mathcal{O}X &\leftarrow& \mathcal{O}(X)\otimes_k \mathcal{O}(X) &\hookrightarrow& \mathrm{B} \mathcal{O}(X) \\ \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{=}} && \downarrow^{\mathrlap{\simeq}} \\ \mathcal{O}X &\leftarrow& \mathcal{O}(X)\otimes_k \mathcal{O}(X) &\to& \mathcal{O}(X) } \,.$$

It is sufficient to find a resolution $\mathrm{B} \mathcal{O}(X)$ in the global model structure $[T, sSet]_{inj}$ because left Bousfield localization strictly increases the class of weak equivalences, so that every gloabl weak equivalence is also a local weak equivalence.

Since we are in the injective model structure this just means that this morphism $\mathcal{O}(X) \otimes_k \mathcal{O}(X) \to \mathrm{B} \mathcal{O}X$ needs to be over each $x^n$ in $T$ a monomorphism of simplicial sets. If we find $\mathrm{B} \mathcal{O}X$ also as a strictly product-preserving functor (notice that the general functor in our model category need not even preserve products weakly, it will do so after fibrant replacement) then it being monomorphism over $x^1$ implies that it is monic over every $x^n$.

There is a standard resolution of the kind we need called the bar complex, see for intance (Ginzburg, page 16) for an explicit description. This is usually discussed as a chain complex in the category of $\mathcal{O}(X)$-modules. But in fact after applying the Dold-Kan correspondence to regard it as a simplicial module it is naturally even a simplicial object in $CAlg_k$:

$$\mathrm{B} \mathcal{O}(X) := \left( \cdots \mathcal{O}(X) \otimes_k \mathcal{O}(X) \otimes_k \mathcal{O}(X) \otimes_k \mathcal{O}(X) \stackrel{\overset{\mu \otimes Id \otimes Id}{\longrightarrow}}{\stackrel{\overset{Id \otimes \mu Id}{\longrightarrow}}{\underset{Id \otimes Id \otimes \mu}{\longrightarrow}}} \mathcal{O}(X) \otimes_k \mathcal{O}(X) \otimes_k \mathcal{O}(X) \stackrel{\overset{\mu \otimes Id}{\longrightarrow}}{\underset{Id \otimes \mu}{\longrightarrow}} \mathcal{O}(X) \otimes_k \mathcal{O}(X) \right) \in CAlg_k^{\Delta^{op}} \,,$$

with the evident face and degeneracy maps given by binary product operation in the algebra and insertion of units.

Take the morphism $\mathcal{O}(X) \otimes \mathcal{O}(X) \to \mathrm{B} \mathcal{O}(X)$ degreewise to be the inclusion of $\mathcal{O}(X) \otimes \mathcal{O}(X)$ as the two outer direct summands

$$\mathcal{O}(X) \otimes_k \mathcal{O}(X) \stackrel{Id \otimes e \otimes e \otimes \cdots \otimes e \otimes Id}{\longrightarrow} \mathcal{O}(X) \otimes_k \mathcal{O}(X) \otimes_k \cdots \otimes_k \mathcal{O}(X) \,,$$

where $e : k \to \mathcal{O}(X)$ is the monoid unit.

This is clearly degreewise a monomorphism, hence is a monomorphism. Under the Moore complex functor $N : Ab^{\Delta^{op}} \to Ch_\bullet^+$ it maps to the standard bar complex resolution as found in the traditional literature (as reviewed for instance in Ginzburg). This morphism of chain complexes is an isomorphism in homology. Since under the Dold-Kan correspondence simplicial homotopy groups are identified with homology groups, we find that indeed $\mu : \mathrm{B}\mathcal{O}(X) \to \mathcal{O}(X)$ is a weak equivalence in $[T,sSet]_{inj}$ and hence in $[T, sSet]_{inj,prod}$.

We may now compute the pushout in $[T, sSet]$ and this will compute the desired homotopy pushout. Notice that this pushout indeed takes place just in simplicial copresheaves, not in product-preserving copresheaves!

But this ordinary pushout it manifestly the claimed one.

Remark

This derivation

• crucially uses the assumption that $A$ is a commutative algebra;

• curiously does not make use of any specific property of the set of morphisms $\{T[k] \coprod T[l] \to T[k+l] \}$ at which we are considering the left Bousfield localization. The entire construction proceeds entirely at the underlying simplicial sets of our simplicial algebras. In fact, the resulting homotopy pushout $\mathcal{O}(X) \coprod_{\mathcal{O}(X) \otimes \mathcal{O}(X)} \mathrm{B}\mathcal{O}(X)$ is a simplicial copresheaf on $T$ that no longer preserves any products: there is no manifest algebra structure.

  But also, this object is far from being fibant in the localized model structure $[T, sSet]_{proj,prod}$. The Bousfield localization, hence the information about the set of maps at which we are localizing, hence the algebra structure, kicks in only once we pass now to the fibrant resolution of our pushout. That fibrant replacement equips the Hochschild chain complex with the structure of an $\infty$-algebra.

Higher order Hochschild homology modeled on cdg-algebras

We discuss details of Hochschild homology in the context of dg-geometry: the (∞,1)-topos over an (∞,1)-site of formal duals of commutative dg-algebras over a field, presented by the model structure on dg-algebras.

Fix a field $k$ of characteristic 0. We consider now the context of dg-geometry with its function algebras on ∞-stacks taking values in unbounded dg-algebras, exhibited by the adjoint (∞,1)-functors

$$(\mathcal{O} \dashv Spec) : (cdgAlg_k^{op})^\circ \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\longrightarrow}} \mathbf{H} := Sh_\infty((cdgAlg_k^-)^{op}) \,.$$

For the discussion of Hochschild homology in this $\mathbf{H}$, the main fact about the model structure on dg-algebras that we need is this:

Proposition

In the projective model structure on unbounded commutative dg-algebras over $k$ we have that

1. the derived copowering of $cdgAlg_k$ over sSet is presented by the ordinary powering of $cdgAlg_k$ over $sSet$;

2. the derived powering of $cdgAlg_k$ over degreewise finite simplicial sets is presented by forming polynomial differential forms on simplices, $S \mapsto \Omega^\bullet_{poly}(S)$.

This is discussed in detail at model structure on dg-algebras in the sections Derived copowering and Derived powering.

Higher order Hochschild complexes

By the above fact Pirashvili’s copowering definition of higher order Hochschild homology holds true in dg-geometry. For $X$ a manifold regarded as a topological space and then as a constant ∞-stack in $\mathbf{H}$ we have for any $A \in cdgAlg_k$

$$\mathcal{O}[X, Spec A] \simeq X \cdot A$$

in $cdgAlg_k$.

Jones’ theorem

Jones' theorem asserts that the Hochschild homology of the dgc-algebra of differential forms on a smooth manifold computes the ordinary cohomology of the corresponding free loop space. We discuss now how this result follows using derived loop spaces of constant ∞-stacks

See also at Sullivan model for free loop spaces the section on Relation to Hochschild homology.

Fact

For $X$ a smooth manifold, $\Omega^\bullet(X)$ its de Rham dg-algebra and $\mathcal{L} X$ its free loop space, we have

$$H^\bullet(\mathcal{L} X, \mathbb{R}) \simeq HH_\bullet(\Omega^\bullet(X), \Omega^\bullet(X)) \,.$$

We sketch the proof in terms of the above derived loop space technology.

Proof

Set $k = \mathbb{R}$. Write $LConst X \in \mathbf{H}$ for the constant ∞-stack on the homotopy type of $X$, regarded as a topological space $\simeq$ ∞Grpd. Then

$$\mathcal{O} LConst X \simeq C^\bullet(X,k) \simeq \Omega^\bullet(X) \in dgcAlg_{\mathbb{R}}$$

is (…) the $k$-valued singular cochain complex of $X$, which by the de Rham theorem is equivalent to the de Rham dg-algebra.

Since $LConst$ is a left exact (∞,1)-functor it commutes with forming free loop space objects and therefore

$$\mathcal{L} LConst X \simeq LConst (\mathcal{L} X) \,.$$

Since $LConst X$ is $\mathcal{O}$-perfect (…) we have by the above copowering-description of the Hochschild complexes that the cohomology of the loop space of $X$

$$\mathcal{O} ((LConst X)^{S^1}) \simeq C^\bullet(\mathcal{L} X, k)$$

is given by the Hochschild complex of the dg-algebra $\Omega^\bullet(X)$

$$\mathcal{O} ((LConst X)^{S^1}) \simeq S^1 \cdot \Omega^\bullet(X) \,.$$

The circle and the odd line

Consider as before the categorical circle $S^1$ as the corresponding constant ∞-stack in $\mathbf{H}$. We describe the function $\infty$-algebra on $S^1$. Below this will serve to explain the nature of the canonical circle action on the Hochschild complex of a cdg-algebra.

Proposition

We have an equivalence

$$\mathcal{O}(S^1) \simeq (k \oplus k[-1]) \,,$$

where on the right we have the ring of dual numbers over $k$, regarded as a dg-algebra with the odd generator in degree 1 and trivial differential.

Proof

Every ∞-groupoid is the (∞,1)-colimit over itself (as described there) of the (∞,1)-functor constant on the point. This (∞,1)-colimit is preserved by the left adjoint (∞,1)-functor $LConst : \infty Grpd \to \mathbf{H}$, so that we have

$$S^1 \simeq {\lim_{\to}}_{S^1} *$$

in $\mathcal{H}$. The (∞,1)-functor $\mathcal{O}$ is also left adjoint, so that

$$\mathcal{O}(S^1) \simeq {\lim_{\leftarrow}}_{S^1} \mathcal{O}(*)$$

in $cdgAlg_k^\circ$. Since the point is representable, we have by the definition of $\mathcal{O}$ as the left $(\infty,1)$-Kan extension of the inclusion $(cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}$ that this is

$$\cdots \simeq {\lim_{\leftarrow}}_{S^1} k \,.$$

This is the formula for the $(\infty,1)$-power of the cdg-algebra $k$ by by $\infty$-groupoid $S^1$. By the above fact, using that the circle is a finite $(\infty,1)$-groupoid, this is given by the cdg-algebra of polynomial differential forms on simplices of $S^1$

$$\cdots \simeq \Omega^\bullet_{poly}(S^1) \,.$$

By a central theorem of rational homotopy theory (recalled at differential forms on simplices) this is equivalent to the singular cochains on the circle

$$\cdots \simeq C^\bullet(S^1, k) \,.$$

But $S^1 \simeq \mathcal{B}\mathbb{Z}$ is a classifying space of a Lie algebra, so that this is a formal dg-algebra, equivalent to its cochain cohomology. Over the field $k$ of characteristic 0 this is

$$H^n(S^1, k) = \left\{ \array{ k & for\; n = 0 \\ k & for \; n = 1 \\ 0 & otherwise } \right. \,.$$

Therefore

$$\cdots \simeq k \oplus k[-1] \,.$$

Remark

This means that $Spec \mathcal{O}(S^1)$ is no longer the circle itself, but the odd line, regarded with its canonical $\mathbb{Z}$-grading.

This point is amplified in (Ben-ZivNadler).

The cotangent complex as functions on the derived loop space

Corollary

We have that

$$[S^1, Spec A] : U \mapsto cdgAlg_k(A, \mathcal{O}(U) \oplus \mathcal{O}(U)[-1]) \,.$$

Proof

$$\begin{aligned} [S^1, Spec A](U) & \simeq \mathbf{H}(S^1 \times U , Spec A) \\ & \simeq cdgAlg_k(A, \mathcal{O}(S^1 \times U)) \\ & \simeq cdgAlg_k(A, \mathcal{O}(U) \oplus \mathcal{O}(U)[-1]) \end{aligned} \,.$$

(…)

Properties

Algebra structure on $(HH^\bullet(A,A), HH_\bullet(A,A))$

There is rich algebraic structure on Hochschild homology and cohomology itself, and on the pairing of the to. We describe various aspects of this.

Differential calculus

It turns out that

• Hochschild homology of $\mathcal{O}(X)$ encodes Kähler differential forms on $X$;

• Hochschild cohomology of $\mathcal{O}(X)$ encodes multivector fields on $X$;

• there are natural pairings between $HH_\bullet(\mathcal{O}(X), \mathcal{O}(X))$ and $HH^\bullet(\mathcal{O}(X), \mathcal{O}(X))$ that mimic the structure of the natural pairings between vector fields and differential forms on smooth manifold.

See (Tamarkin-Tsygan) and see at Kontsevich formality for more. This equivalence enters the construction of formal deformation quantization of Poisson manifolds.

One way to understand or interpret this conceptually is to regard the derived loop space object of a 0-truncated object $X$ to consist of infinitesimal loops in $X$.

Hochschild-Kostant-Rosenberg theorem

The Hochschild-Kostant-Rosenberg theorem states that under suitable conditions, the Hochschild homology of an algebra (with coefficients in itself) computes the wedge powers of its Kähler differentials.

Let $A$ be an associative algebra over $k$. Recall the natural identification

$$HH_1(A,A) \simeq \Omega^1(A)$$

of the first Hochschild homology of $A$ with coefficients in itself and degree-1 Kähler differential forms of $A$.

Write $\Omega^0(R/k) := R \simeq HH_0(R,R)$.

For $n \geq 2$ Write $\Omega^n(R/k) = \wedge^n_R \Omega(R/k)$ for the $n$-fold wedge product of $\Omega(R/k)$ with itself: the degree $n$-Kähler-differentials.

Theorem

The isomorphism $\Omega^1(R/k) \simeq H_1(R,R)$ extends to a graded ring morphism

$$\psi : \Omega^\bullet(R/k) \to H_\bullet(R,R) \,.$$

If the $k$-algebra $R$ is sufficiently well-behaved, then this morphism is an isomorphism that identifies the Hochschild homology of $R$ in degree $n$ with $\Omega^n(R/k)$ for all $n$:

Theorem

(Hochschild-Kostant-Rosenberg theorem)

If $k$ is a field and $R$ a commutative $k$-algebra which is

• essentially of finite type

• smooth over $k$

then there is an isomorphism of graded $R$-algebras

$$\psi : \Omega^\bullet(R/k) \stackrel{\simeq}{\to} HH_\bullet(R,R) \,.$$

Moreover, dually, there is an isomorphism of Hochschild cohomology with wedge products of derivations:

$$\wedge^\bullet_R Der(R,R) \simeq HH^\bullet(R,R)$$

This is reviewed for instance as (Weibel, theorem 9.4.7) or as (Ginzburg, theorem 9.1.3).

$E_n$-algebra structure: Deligne-Kontsevich conjecture/theorem

The next statement is known as the Deligne conjecture.

Proposition

The higher order Hochschild homology $\mathcal{O} (X^{S^d})$ of an object $X$ with respect to the $d$-sphere $S^d$ and with coefficients in a geometric function object is naturally an E(d+1)-algebra): an algebra over an operad over the little k-cubes operad for $k = d+1$ .

For let $\Sigma^{d+1} = D^{d+1}\setminus \coprod_r D^{d+1}$ be the $(d+1)$-ball with $r$ small $d+1$-balls taken out. We have a cospan of boundary inclusions

$$\array{ && \Sigma^{d+1} \\ & \nearrow && \nwarrow \\ \coprod_r S^d &&&& S^d }$$

in ∞Grpd and under $LConst : \infty Grpd \to \mathbf{H}$ then also in our (∞,1)-topos.

Applying the (∞,1)-topos internal hom $[-,X]$ or equivalent the (∞,1)-powering $X^{(-)}$ into a given object $X \in \mathbf{H}$ to this cospan produces the span

$$\array{ && X^{\Sigma^{d+1}} \\ & {}^{\mathllap{i_r}}\swarrow && \searrow^{\mathrlap{o}} \\ \prod_r X^{S^d} &&&& X^{S^d} }$$

in $\mathbf{H}$. Then the integral transforms on sheaves

$$o_1 i_r^* : \prod_r \mathbf{H}/X^{S^d} \to \mathbf{H}/X^{d}$$

induced by these spans constitute the $E_n$-action on the function objects on $X^{S^d}$.

This was observed in (Ben-ZviFrancisNadler, corollary 6.8).

For $d = 1$, under the identification of the HKR theorem above (when it applies), the Gerstenhaber bracket is identified with the Schouten bracket (Tsyagin, theorem 2.2.2)

Remark

(Deligne conjecture)

Some historical comments on the Deligne conjecture.

Historically it was first found that there is the structure of a Gerstenhaber algebra on $HH^\bullet(A,A)$. By (Cohen) it was known that Gerstenhaber algebras arise as the homology of E2-algebras in chain complexs. In a letter in 1993 Deligne wondered whether the Gerstenhaber structure on the Hochschild cohomology $HH^\bullet(A,A)$ lifts to an E2-algebra-structure on the cochain complex $C^\bullet(A,A)$.

In GerstenhaberVoronov (1994) a resolution of the Gerstenhaber algebra structure was given, but the relationship to $E_2$-algebras remained unclear.

In (Tamarkin (1998)) a genuine resolution in the model structure on operads of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on $C^\bullet(A,A)$. This proved Deligne’s conjecture.

Various authors later further refined this result. A summary of this history can be found in (Hess).

In Hu-Kriz-Voronov (2003) it was further shown that for $A$ an En-algebra, $C^\bullet(A,A)$ is an $E_{n+1}$-algebra.

Notice that the identification of Hochschild (co)homology as coming from higher order free loop spaces makes all this structure manifest.

$HH$ of constant $\infty$-stacks: String topology BV-algebra

Let $T$ be the algebraic theory of ordinary associative algebras over a field $k$, regarded as an (∞,1)-algebraic theory and let $\mathbf{H}$ be the (∞,1)-topos of $(\infty,1)$-sheaves over a small site in $T Alg_\infty^{op}$.

Under the inverse image of the global section (∞,1)-geometric morphism and the homotopy hypothesis-equivalence

$$\mathbf{H} \stackrel{\overset{LConst}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}} \infty Grpd \stackrel{\overset{\Pi}{\longleftarrow}}{\underset{|-|}{\longrightarrow}} Top$$

we may regard every topological space $X$ as a constant ∞-stack $LConst X$, an object in $\mathbf{H}$.

Proposition

The function algebra on $LConst X$ is the cosimplicial algebra of singular cochains on $X$. Under the monoidal Dold-Kan correspondence it identifies with the cochain dg-algebra $C^\bullet(X)$ that computes the singular cohomology of $X$.

This has maybe been first made explicit by Bertrand Toën. Details are at function algebras on ∞-stacks.

Corollary

The Hochschild homology of $C^\bullet(X)$ is the singular cohomology of the free loop space $L X$.

Proof

Apply the central identification $\mathcal{O} \mathcal{L}(LConst X) \simeq S^1 \cdot \mathcal{O}(LConst X)$. Then observe that the free loop space object $\mathcal{L} LConst X$ of the constant $\infty$-stack is the constant $\infty$-stack on the ordinary free loop space, because $LConst$ is a left exact (∞,1)-functor and because $\mathcal{L}X \simeq L X$ in Top. Then use by the above remark that $\mathcal{O} LConst L X$ is singular cochains on $L X$.

This result, which follows directly from the general abstract desciption of Hichschild homology is known as Jones’ theorem. We now review the results in the literature on this point.

Let $X$ be a compact manifold oriented smooth manifold of dimension $d$. Write $C^\bullet(X)$ for the dg-algebra of cochains for singular cohomology of $X$. Write $L X$ for the topological free loop space of $X$ and $H_\bullet(L X)$ for its singular homology.

Theorem

There is a linear isomorphism of degree $d$

$$\mathbb{D} : HH^{-p-q}(C^\bullet(X), C^\bullet(X)^{\vee}) \simeq HH^{-p}(C^\bullet(X), C^\bullet(X)) \,.$$

This is due to (FelixThomasVigue-Poirrier, section 7)).

Theorem

(Jones’ theorem)

There is an isomorphism

$$J : H_{p+q}(L X) \stackrel{\simeq}{\to} HH^{-p-d}(C^\bullet(X), C^\bullet(X)^{\vee})$$

such that the canonical string topology BV-operator $\Delta$ of the BV-algebra $H_{\bullet + d}(L X)$ and the Connes coboundary $B^\vee$ on $HH^{\bullet-d}(C^\bullet(X), C^\bullet(X)^{\vee})$ satisfy

$$J \circ \Delta = B^{\vee} \circ J \,.$$

This is due to (Jones).

Theorem

The Connes coboundary defines via the isomorphism $\mathbb{D}$ from above the structure of a BV-algebra on $HH^\bullet(C^\bullet(X), C^\bullet(X))$.

This is (Menichi, theorem 3).

Relation to cyclic (co)homology

There is an intrinsic circle action on Hochschild (co)chains. Passing to the cyclically invariant (co)chains yields cyclic (co)homology.

Further

Hochschild cohomology and extensions

Definition

An exact sequence $0 \to N \to E \to R$ of $k$-modules where $E \to R$ is a surjective morphism of $k$-algebras is called a $k$-split extension or a Hochschild extension of $R$ by $E$ if the sequence is a split sequence as a sequence of $k$-modules.

Two extensions are equivalent if there is an isomorphism or $k$-algebra $E \stackrel{\simeq}{\to} E'$ that makes

$$\array{ N &\to& E &\to& R \\ \downarrow^{\mathrlap{=}} && \downarrow && \downarrow^{\mathrlap{=}} \\ N &\to& E' &\to& R }$$

commute.

Remark

Due to the $k$-splitness assumption there is an isomorphism of $k$-modules $E \simeq R \oplus N$ and this is equipped with a $k$-algebra structure such that the product on the $R$ direct summand is that of $R$. From this we find that the product on $E$ is of the form

$$(r_1, n_1) \cdot (r_2, n_2) = (r_1 r_2 , r_1 n_2 + n_1 r_2 + f(r_1, r_2)) \,,$$

where $f : R \otimes_k R \to N$ is some $k$-linear map. Since the product on $E$ is (by definition) associative, it follows that for $f$ that this satisfies for all $r_0, r_1, r_2 \in R$ the cocycle equation

$$r_0 f(r_1, r_2) - f(r_0 r_1, r_2) + f(r_0 , r_1 r_2) - f(r_0, r_1) r_2 = 0$$

as an equation in $N$. This says that $f$ must be a Hochschild cocycle

$$f \in HH^2(R,N) \,.$$

Conversely, every such cocycle yields a $k$-split extension of $R$ by $N$ this way:

Theorem

For $R$ a $k$-algebra and $N$ an $R$-bimodule, equivalence classes of Hochschild extensions of $R$ by $N$ are in bijection with degree 2 Hochschild cohomology $HH^2(R,N)$.

See for instance Weibel, theorem 9.3.1.

Hochschild cohomology and deformations

As a special case of the above statement about extensions of $R$, we obtain a statement about deformation of $R$.

A standard problem is to deform a $k$-algebra $R$ by introducing a new “parameter” $t$ that squares to 0 – $t \cdot t = 0$ and a new product

$$r_1 \cdot_t r_2 = r_1 r_2 + t f(r_1, r_2) \,.$$

From the above we see that this is the same as finding an $k$-split extension of $R$ by itself. So in particular such extensions are given by Hochschild cocycles $f \in HH^2(R,R)$.

See for instance Ginzburg, section 7 and for more see deformation quantization.

Related entries

• Sullivan model of free loop space

• cyclic homology, dihedral homology

• topological Hochschild homology

References

Hochschild cohomology of ordinary algebras was introduced in

• Gerhard Hochschild, On the cohomology groups of an associative algebra, The Annals of Mathematics, 2nd ser., 46 1 (1945) 58-6 [jstor:1969145]

Early further discussion:

• Michael Barr, Cohomology of commutative algebra I, Ph.D. Thesis, University of Pennsylvania (1962), Retyped with a few corections and notes (2003) [pdf, pdf]

and in relation to monadic cohomology:

• Michael Barr, Harrison homology, Hochschild homology and triples J. Algebra 8 3 (1968) 314–323 [doi:10.1016/0021-8693(68)90062-8, pdf, pdf]

Textbook discussions:

• Charles Weibel, chapter 9 of: An Introduction to Homological Algebra

• Victor Ginzburg, chapter 4 of: Lectures on noncommutative geometry [arXiv:math/0506603]

The definition of the higher order Hochschild complex as (implicitly) the tensoring of an algebra with a simplicial set:

• Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology Annales Scientifiques de l’École Normale Supérieure Volume 33, Issue 2, March 2000, Pages 151-179 (ps)

A survey of traditional higher order Hochschild (co)homology and further developments and results are described in

• Grégory Ginot, Higher order Hochschild cohomology

  (pdf)

A considerably refined discussion of this which almost makes the construction of Hochschild complexes as an $(\infty,1)$-copowering operation manifest is in

• Grégory Ginot, Thomas Tradler, Mahmoud Zeinalian, Derived higher Hochschild homology, topological chiral homology and factorization algebras, arxiv/1011.6483

• Nathalie Wahl, Craig Westerland, Hochschild homology of structured algebras, arxiv/1110.0651

The full $(\infty,1)$-categorical picture of Hochschild homology as the cohomology of derived free loop space objects is due to

• David Ben-Zvi, John Francis, David Nadler,

  Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv:0805.0157)

based on

• David Ben-Zvi, David Nadler, Loop Spaces and Langlands Parameters (arXiv:0706.0322) .

Specifically the dicussion of differential forms via such an $\infty$-category theoretic perspective of the HKR-theorem is discussed in

• David Ben-Zvi, David Nadler, Loop Spaces and Connections (arXiv:1002.3636)

• Bertrand ToënGabriele Vezzosi, $S^1$-Equivariant simplicial algebras and de Rham theory (arXiv:0904.3256)

General homotopy-theoretic setups and results for contexts in which this makes sense are discussed in

• Bertrand Toën, Gabriele Vezzosi, HAG II, geometric stacks and applicatons (arXiv:math/0404373v4)

Jones’s theorem is due to

• J. D. S. Jones, Cyclic homology and equivariant homology , Invent. Math. 87 (1987), no. 2, 403{423.

The BV-algebra structure on Hochschild cohomology of singular cochain algebras is discussed in

• Y. Félix, J.-C. Thomas, M. Vigué-Poirrier, The Hochschild cohomology of a closed manifold Publ. Math. IHÉS Sci. (2004) no 99, 235-252

• Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology (pdf)

The abstract differential caclulus on $(HH^\bullet(A,A), HH_\bullet(A,A))$ is discussed for instance in

• Dmitry Tamarkin, Boris Tsygan, Cyclic Formality and Index Theorems , Letters in Mathematical Physics

  Volume 56, Number 2, 85-97 (journal)

A review of Deligne’s conjecture and its solutions is in

• Kathryn Hess, Deligne’s Hochschild cohomology conjecture (pdf)

More developments are in

• Grégory Ginot, Thomas Tradler, Mahmoud Zeinalian, Higher Hochschild cohomology, Brane topology and centralizers of $E_n$-algebra maps, (arXiv:1205.7056)
• Nathalie Wahl, Universal operations in Hochschild homology, arxiv/1212.6498

Relation to factorization homology is discussed in

• Geoffroy Horel, Factorization homology and calculus à la Kontsevich Soibelman (arXiv:1307.0322)

For more references on the relation to topological chiral homology see there.

Interesting wishlists for treatments of Hochschild cohomology are in this MO discussion.