nLab
Hochschild cohomology

Idea

From the nPOV on cohomology, the notion of Hochschild cohomology is the following.

Hochschild cohomology is the cohomology $H(ℒX,C)$ of free loop space objects $ℒX$ in a derived stack (∞,1)-topos $H$ with coefficients in quasicoherent ∞-stacks of modules $C$.

This naturally inherits an $S^1$-action from the free loop space object. The $S^1$-equivariant cohomology refinement of Hochschild cohomology is cyclic cohomology.

After unwinding what this means in algebraic terms, one obtains the tradional way of conceiving Hochschild cohomology.

Hochschild homology and cohomology are characteristic objects associated to a bimodule $N$ over a monoid $A$, in a context where $A$ is really to be thought of as a monoid in an monoidal (∞,1)-category.

If the monoid in question plays the role of an algebra of functions on a space $X$, $A=C(X)$, then this has geometric interpretations. Notably

• the Hochschild homology of $A$ regarded as a bimodule over itself tends to be like the algebra of Kähler differentials ${\Omega }_K^{•}(A)$ on $A$, and hence is something like the algebra of differential forms on $X$;
• the Hochschild cohomology of $A$ regarded as a bimodule over itself tends to be like the algebra of derivations $\mathrm{Der}(A)$ of $A$, and hence something like the vector fields on $X$.

The seminal Hochschild-Kostant-Rosenberg theorem states that under suitable smoothness conditions on $A$, these statements become exactly true.

Notice that differential forms are objects in de Rham cohomology but are still computed by the Hochschild homology of $A$. The terminology reflects the dualization in passing from the space $X$ to the algebra of functions $A=C(X)$ on it: it is the Hochschild homology of algebra objects that relates to the cohomology of spaces.

Below in section

• The nPOV on Hochschild cohomology

we discuss a conceptual interpretation of Hochschild homology, that will explain why this is true, and what Hochschild homology is conceptually, from the nPOV on cohomology, as described there.

Complementary to that, in the section

• More traditional description of Hochschild cohomology

we describe the original definition of Hochschild cohomology and the evolution of its understanding approaching the nPOV.

Finally in the section titled Details the technical details are spelled out.

More traditional descriptions of Hochschild cohomology

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

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

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

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

$$C_{•}(N,A)=N{\otimes }_{A\otimes A^{\mathrm{op}}}^{D}A={\mathrm{Tor}}_{A\otimes A^{\mathrm{op}}}^{•}(N,A).$$C_\bullet(N,A) = 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

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

This, generally, is the definition of the Hochschild homology object of any bimodule over an monoid in a monoidal (∞,1)-category. Of special interest is the case where $N=A$. In this case this object is also called the (”$(\mathrm{\infty }\mathrm{,1})$-” or “derived-“)center of $A$:

$$Z(A):=A{\otimes }_{A\otimes A^{\mathrm{op}}}A$$Z(A) := A \otimes_{A \otimes A^{op}} A

In parallel to this formal understanding of Hochschild homology, its conceptual meaning has been better understood: from staring at the explicit description of $C_{•}(N,A)$ one sees that it has something to do with loop space objects: a chain in $C_n(N,A)$ is usefully thought of as a circle with $n$ marked points. One of these points is labeled $N$, the other are labeled $A$. The differential on $C_{•}(N,A)$ acts by taking tensor products over $A$ separately of all neighbour pairs of bimodules sitting on this circle, and taking the alternating sum of this as a collection of such circles with $(n-1)$ marked points.

A fully geometric understanding of these was given by Ben-Zvi/Nadler/Francis in their work on derived loop space objects and their geometric ∞-function theory. This we unify now with our nPOV-perspective on cohomology in order to give the following nPOV-perspective on Hochschild cohomology, proper.

The $n$POV on Hochschild cohomology

Let $H$ be an (infinity,1)-topos of (∞,1)-category of (∞,1)-sheaves and let $K$ the $(\mathrm{\infty }\mathrm{,2})$-topos of $(\mathrm{\infty }\mathrm{,2})$-sheaves on a site $C$, such that the quasicoherent ∞-stack

$$C:C^{\mathrm{op}}\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$$C : C^{op} \to (\infty,1)Cat

$$U\mapsto \mathrm{Stab}(C_{/U})$$U \mapsto Stab(C_{/U})

as described at ∞-vector bundle does indeed satisfy descent in that it is indeed an object in $K$.

Recall that for $X\in H\subset K$ any object, its free loop space object $ℒX$ is the $(\mathrm{\infty }\mathrm{,1})$-pullback

$$\begin{array}{ccc}& & ℒX\\ & \swarrow & & \searrow \\ X& & & & X\\ & {}_{(\mathrm{Id},\mathrm{Id})}\searrow & & {\swarrow }_{(\mathrm{Id},\mathrm{Id})}\\ & & X\times X\end{array}.$$\array{ && \mathcal{L} X \\ & \swarrow && \searrow \\ X &&&& X \\ & {}_{\mathllap{(Id,Id)}}\searrow && \swarrow_{\mathrlap{(Id,Id)}} \\ && X \times X } \,.

Notice that this is usefully thought of as the span trace of the identity span

$$ℒX=\mathrm{Tr}\left(\begin{array}{ccc}& & X\\ & {}^{\mathrm{Id}}\swarrow & & {\searrow }^{\mathrm{Id}}\\ X& & & & X\end{array}\right).$$\mathcal{L} X = Tr \left( \array{ && X \\ & {}^{\mathllap{Id}}\swarrow && \searrow^{\mathrlap{Id}} \\ X &&&& X } \right) \,.

Assume that $X$ is such that on it $C$ satisfies the axioms of geometric function theory (see geometric ∞-function theory for more details).

Then:

The Hochschild cohomology of $X$ is the cohomology of the free loop space object $ℒX$ with coefficients in $C$:

$$\mathrm{HH}(X)=K(ℒX,C)=:C(ℒX).$$HH(X) = \mathbf{K}(\mathcal{L}X, C) =: C(\mathcal{L}X) \,.

$${\mathrm{HH}}^n(X):={\mathrm{HH}}_n(C(X)):={\pi }_0{\Omega }^{n}H(ℒX,C).$$HH^n(X) := HH_n(C(X)) := \pi_0\Omega^n\mathbf{H}(\mathcal{L}X, C) \,.

By the assumption that $C(-)$ is a geometric function object on $X$ we can rewrite

$$\mathrm{HH}(X)=H(ℒX,C)=:C(𝒦X)$$HH(X) = \mathbf{H}(\mathcal{L}X,C) =: C(\mathcal{K}X)

as

$$\cdots \simeq C(X){\otimes }_{C(X)\otimes C(X{)}^{\mathrm{op}}}\mathrm{\infty }\mathrm{Mod}(X).$$\cdots \simeq C(X) \otimes_{C(X)\otimes C(X)^{op}} \infty Mod(X) \,.

This is hence indeed the Hochschild homology object ${\mathrm{HH}}_{•}(A)=A{\otimes }_{A\otimes A^{\mathrm{op}}}$ of the algebra object $A=\mathrm{\infty }\mathrm{Mod}(X)$, regarded as a bimodule over itself.

More generally, let $f:Y\to X$ be a morphism in $H$ and consider the span trace

$${ℒ}_{X}Y:=\mathrm{Tr}\left(\begin{array}{ccc}& & Y\\ & {}^f\swarrow & & {\searrow }^f\\ X& & & & X\end{array}\right)$$\mathcal{L}_X Y := Tr \left( \array{ && Y \\ & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{f}} \\ X &&&& X } \right)

which is the $(\mathrm{\infty }\mathrm{,1})$-pullback

$$\begin{array}{ccc}& & {ℒ}_{X}Y\\ & \swarrow & & \searrow \\ Y& & & & X\\ & {}_{f,f}\searrow & & {\swarrow }_{\mathrm{Id},\mathrm{Id}}\\ & & X\times X\end{array}.$$\array{ && \mathcal{L}_X Y \\ & \swarrow && \searrow \\ Y &&&& X \\ & {}_{\mathllap{f,f}}\searrow && \swarrow_{\mathrlap{Id,Id}} \\ && X \times X } \,.

Then we take the Hochschild cohomology of $Y$ relative to $f:Y\to X$ to be

$$\mathrm{HH}(Y,X):=K({ℒ}_{X}Y,C)=:\mathrm{\infty }\mathrm{Mod}({ℒ}_{X}Y).$$HH(Y,X) := \mathbf{K}(\mathcal{L}_X Y, C) =: \infty Mod(\mathcal{L}_X Y) \,.

In the case that $f$ is such that $C(-)=K(-,C)$ satisfies geometric function theory on it, this is

$$\cdots =C(Y{\times }_{X\times X}Y)\simeq C(Y){\otimes }_{C(X)\otimes C(X{)}^{\mathrm{op}}}C(X).$$\cdots = C( Y \times_{X \times X} Y) \simeq C(Y) \otimes_{C(X) \otimes C(X)^{op}} C(X) \,.

This is indeed the Hochschild homology object of the $C(X)$-bimodule object structure on $C(Y)$ induced from $f^{*}$:

$$\cdots =:{\mathrm{HH}}_{•}(C(Y),C(X)).$$\cdots =: HH_\bullet(C(Y), C(X)) \,.

Differential forms and Hochschild (co)homolology

The above identifies Hochschild homology objects of function algebras with function algebras on a free loop space object. If one adds to this the observation that for a sufficiently wel behaved ordinary space regarded as derived stack its free loop space object is essentially the formal dual of the algebra of Kähler differential forms, one recovers from a higher geometry picture the stamenet of the Hochschild-Kostant-Rosenberg theorem mentioned above. Details are in

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

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

Notice that the function algebra on the derived loop space is just the differential forms as a graded algebra, without the differential. The differential itself comes from the $S^1$-action on the loop space.

…details later… but see the above references…

Details

…

The bar complex

The bar complex is called the bar complex because its inventors wrote it down using lots of bars. If you don’t believe this it shows that you have no idea how careless mathematicians can be about finding good terminology for the objects they hold in high esteem.

References

The traditional story of Hochschild (co)homology is exposed for instance in chapter 4 of

• Victor Ginzburg, Lectures on noncommutative geometry (arXiv:math/0506603)

An original paper on this is

• Tsygan and Feigin, Additive K-theory 1980-s (LNM 1289, editor Manin, pp 67-209, seminar 1984-1986 in Moscow)

The $(\mathrm{\infty }\mathrm{,1})$-categorical picture of derived free loop space objects and their geometric ∞-function theory is discussed in

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

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

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

• David Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv:0805.0157)