Hochschild Homology

Introduction

An algebra $A$ over a commutative ring $k$ has a multiplication $A\otimes_kA\to A$, and the associativity of that multiplication is a statement about a two-step complex, the beginning of a resolution of $A$ by free $A$-bimodules. The bar resolution packages this resolution once and for all; applying a bimodule $M$ and taking homology produces the Hochschild homology $HH_n(A,M)$, dually, the Hochschild cohomology $HH^n(A,M)$. The construction is the natural homology theory of an algebra: it is defined for every algebra and every bimodule, it is the derived functor of the coinvariants and of the invariants of a bimodule under the associative multiplication, and its low-dimensional groups carry the classical interpretations — $HH_0$ is the abelianisation of the algebra relative to its centre, $HH_1$ is generated by the derivations and the Kähler differentials, $HH^1$ is the module of derivations, $HH^2$ classifies infinitesimal deformations. On the cohomology there is, in addition, the Gerstenhaber bracket, which makes $HH^{\bullet}(A,A)$ a graded Lie algebra and gives the deformation theory its obstruction calculus.

This article develops the Hochschild complex and the bar resolution, the identification of Hochschild homology and cohomology as derived functors over the enveloping algebra, the low-dimensional interpretations and the relation to Kähler differentials and derivations, the Hochschild–Kostant–Rosenberg theorem identifying the Hochschild homology of a smooth commutative algebra with its differential forms, the cup and bracket products making the cohomology a Gerstenhaber algebra, and the Morita invariance of the theory. It follows Homological Algebra, Derived Functors, Ext and Tor and Derived Categories, and it prepares.

Throughout, $k$ is a commutative ring with $1\neq0$, $A$ is an associative $k$-algebra with unit, not necessarily commutative, and $M$ is an $A$-bimodule; the enveloping algebra is $A^{\mathrm{e}}=A\otimes_kA^{\mathrm{op}}$, so that $A$-bimodules are left $A^{\mathrm{e}}$-modules. The tensor products are over $k$ unless a subscript names another ring, and $\otimes$ without a subscript in a Hochschild complex means $\otimes_k$. No distance, norm, open set or completion occurs; the word smooth is used in its algebraic sense of formal smoothness and the word limit only in the algebraic sense of a filtered direct limit of modules. The cyclic cohomology of Part III is a different invariant constructed from the same cyclic object, and is not treated here; this article supplies its input.

The Hochschild Complex

The Bar Resolution

Definition. The bar resolution of $A$ over $A^{\mathrm{e}}$ is the complex $B_\bullet(A)$ with

$$ B_n(A)=A^{\otimes(n+2)}=A\otimes_kA^{\otimes n}\otimes_kA, \qquad n\ge0, $$

with the left and right $A$-actions on the outer factors and with the $A^{\mathrm{e}}$-linear differential $b:B_n(A)\to B_{n-1}(A)$

$$ b(a_0\otimes a_1\otimes\cdots\otimes a_{n+1})=\sum_{i=0}^{n}(-1)^ia_0\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_{n+1}. $$

The augmentation $B_0(A)=A\otimes_kA\to A$, $a\otimes a'\mapsto aa'$, is a morphism of $A$-bimodules.

Proposition. The bar resolution is a resolution: the augmented complex $B_\bullet(A)\to A\to0$ is exact as a complex of $A$-bimodules, and each $B_n(A)$ is a free $A^{\mathrm{e}}$-module.

Proof. The module $B_n(A)=A\otimes_kA^{\otimes n}\otimes_kA\cong A^{\mathrm{e}}\otimes_kA^{\otimes n}$ is free over $A^{\mathrm{e}}=A\otimes_kA^{\mathrm{op}}$; the isomorphism sends $a_0\otimes\cdots\otimes a_{n+1}$ to $(a_0\otimes a_{n+1})\otimes(a_1\otimes\cdots\otimes a_n)$. For exactness, the maps $s:B_n(A)\to B_{n+1}(A)$, $s(a_0\otimes\cdots\otimes a_{n+1})=1\otimes a_0\otimes\cdots\otimes a_{n+1}$, satisfy $bs+sb=\operatorname{id}$ on positive degrees and $bs=1-\eta\varepsilon$ with $\eta$ the unit and $\varepsilon$ the augmentation; hence the augmented complex is contractible as a complex of $k$-modules and therefore exact.

Homology and Cohomology

Definition. Let $M$ be an $A$-bimodule. The Hochschild complex of $A$ with coefficients in $M$ is $C_\bullet(A,M)=M\otimes_{A^{\mathrm{e}}}B_\bullet(A)$, with terms $M\otimes_kA^{\otimes n}$ and boundary

$$ b(m\otimes a_1\otimes\cdots\otimes a_n)=ma_1\otimes a_2\otimes\cdots\otimes a_n+\sum_{i=1}^{n-1}(-1)^im\otimes a_1\otimes\cdots\otimes a_ia_{i+1}\otimes\cdots\otimes a_n+(-1)^na_nm\otimes a_1\otimes\cdots\otimes a_{n-1}. $$

The Hochschild homology is $HH_n(A,M)=H_n(C_\bullet(A,M))$. The Hochschild cohomology is $HH^n(A,M)=H^n(\operatorname{Hom}_{A^{\mathrm{e}}}(B_\bullet(A),M))$, computed from the complex with terms $\operatorname{Hom}_k(A^{\otimes n},M)$ and the dual boundary.

Theorem. There are natural isomorphisms

$$ HH_n(A,M)\cong\operatorname{Tor}_n^{A^{\mathrm{e}}}(A,M), \qquad HH^n(A,M)\cong\operatorname{Ext}_{A^{\mathrm{e}}}^n(A,M), $$

where $A$ is regarded as an $A$-bimodule via the multiplication. In particular Hochschild homology is the left derived functor of the coinvariants $M\mapsto M_A=M\otimes_{A^{\mathrm{e}}}A$ and Hochschild cohomology is the right derived functor of the invariants $M\mapsto M^A=\operatorname{Hom}_{A^{\mathrm{e}}}(A,M)$.

Proof. The bar resolution is a free resolution of $A$ over $A^{\mathrm{e}}$ by the proposition, so applying $M\otimes_{A^{\mathrm{e}}}(-)$ computes the Tor groups and applying $\operatorname{Hom}_{A^{\mathrm{e}}}(-,M)$ computes the Ext groups; the identification of the complexes with the displayed Hochschild complexes is the isomorphism $M\otimes_{A^{\mathrm{e}}}B_n(A)\cong M\otimes_kA^{\otimes n}$ and its dual. The functorial description follows because $M\otimes_{A^{\mathrm{e}}}A$ is the coinvariant quotient of $M$ by the relations $am=ma$, and $\operatorname{Hom}_{A^{\mathrm{e}}}(A,M)$ is the invariant submodule.

Corollary. The theory is functorial: a $k$-algebra homomorphism $A\to A'$ and a compatible bimodule map give maps on Hochschild homology and cohomology, and a Morita equivalence of algebras gives isomorphisms.

Example. For $A=k$ one has $A^{\mathrm{e}}=k$ and $HH_n(k,M)=M$ for $n=0$ and $0$ for $n\ge1$, since $k$ is free of rank one as a $k$-module and the bar resolution is the resolution $0\to k\to0$ in positive degrees. The Hochschild complex of a free algebra is computed .

Low Dimensions

Degree Zero and One

Definition. For an $A$-bimodule $M$, the module of coinvariants is $M_A=M/\langle am-ma\rangle$ and the module of invariants is $M^A=\{m\in M: am=ma\text{ for all }a\in A\}$; these are the degree-zero Hochschild homology and cohomology.

Proposition. $HH_0(A,M)=M_A$ and $HH^0(A,M)=M^A$. In particular $HH_0(A,A)=A/[A,A]$, the vector space of traces, where $[A,A]$ is the $k$-submodule generated by the commutators $ab-ba$, and $HH^0(A,A)=Z(A)$, the centre.

Proof. The degree-zero complex is $M\to0$, giving the cokernel $M_A$. For cohomology the degree-zero term is $\operatorname{Hom}_{A^{\mathrm{e}}}(A,M)=M^A$, the invariants. With $M=A$ the coinvariants are $A/[A,A]$ and the invariants are the centre by definition.

Definition. A derivation $D:A\to M$ of $A$ with values in the $A$-bimodule $M$ is a $k$-linear map with

$$ D(ab)=D(a)b+aD(b). $$

The $k$-module of derivations is $\operatorname{Der}_k(A,M)$, and the inner derivations are the maps $D_m(a)=am-ma$ for $m \in M$, forming the submodule $\operatorname{Inn}_k(A,M)$.

Proposition. $HH^1(A,M)\cong\operatorname{Der}_k(A,M)/\operatorname{Inn}_k(A,M)$. In particular $HH^1(A,A)$ is the quotient of the derivations of $A$ by the inner derivations, that is, the Lie algebra of the outer derivations when $A$ is commutative.

Proof. A Hochschild $1$-cocycle is a $k$-linear $D:A\to M$ with $aD(b)-D(ab)+D(a)b=0$, which is the derivation condition after the sign change; a $1$-coboundary is a map $D(a)=am-ma$ with $m\in M$, that is an inner derivation. Hence the first cohomology is the quotient. For commutative $A$ the inner derivations vanish, and $HH^1(A,A)=\operatorname{Der}_k(A,A)$.

Proposition. Let $I=\ker(\mu:A\otimes_kA\to A)$ be the augmentation ideal of the multiplication and let $\Omega^1_{A/k}=I/I^2$ be the $A$-bimodule of Kähler differentials, with $d:A\to\Omega^1_{A/k}$, $da=a\otimes1-1\otimes a$. It represents the derivations:

$$ \operatorname{Hom}_{A^{\mathrm{e}}}\bigl(\Omega^1_{A/k},M\bigr)\cong\operatorname{Der}_k(A,M), $$

naturally in the $A$-bimodule $M$. For a commutative $k$-algebra $A$ the degree-one Hochschild homology is

$$ HH_1(A,A)\cong\Omega^1_{A/k}, $$

and for a symmetric bimodule $M$, that is an $A$-module, $HH_1(A,M)\cong M\otimes_A\Omega^1_{A/k}$. In particular $HH_1(k[x_1,\dots,x_m])\cong A^m$ with basis $dx_1,\dots,dx_m$.

Proof. The universal property is the standard one: a $k$-derivation $D:A\to M$ is the same thing as an $A$-bimodule map $\varphi:\Omega^1_{A/k}\to M$, the correspondence being $D(a)=\varphi(da)$, and the Leibniz rule is exactly what makes the prescription well defined on the quotient of $I$ by $I^2$. For commutative $A$ the degree-one part of the Hochschild complex with coefficients in $A$ is $A\otimes_kA\xrightarrow{\mu}A$ with $\mu$ the multiplication, and the image of the degree-two boundary in $A\otimes_kA$ is the submodule generated by the Leibniz relations of the multiplication; the standard computation gives $HH_1(A,A)=I/I^2=\Omega^1_{A/k}$, and tensoring the identification with the symmetric bimodule $M$ gives $HH_1(A,M)=M\otimes_A\Omega^1_{A/k}$.

Example. For $A=k[x]$ the module of Kähler differentials is $\Omega^1_{A/k}=A\,dx$, free of rank one; hence $HH_1(k[x])\cong k[x]$ and, more generally, $HH_1(k[x_1,\dots,x_n])\cong A^n$ with basis $dx_1,\dots,dx_n$.

Degree Two and Deformations

Definition. A square-zero extension of $A$ by an $A$-bimodule $M$ is a $k$-algebra $E$ with an ideal $I$ with $I^2=0$ and an isomorphism $E/I\cong A$ such that $I\cong M$ as $A$-bimodules. Two such extensions are equivalent when there is an isomorphism of $k$-algebras over $A$ and compatible with the identifications.

Theorem. $HH^2(A,M)$ is in natural bijection with the equivalence classes of square-zero extensions of $A$ by $M$, and the element $0$ corresponds to the split extension $A\oplus M$ with the multiplication of the trivial bimodule.

Proof. Choose a $k$-linear section $s:A\to E$ of the projection; the deviation of $s$ from multiplicative is the function $f(a,b)=s(a)s(b)-s(ab) \in I\cong M$, which is a Hochschild $2$-cocycle, and the cocycle condition is exactly the associativity of the multiplication on $E=A\oplus M$. Changing the section by a $k$-linear map $A\to M$ changes $f$ by a coboundary. Two extensions are equivalent exactly when the corresponding cocycles differ by a coboundary, so the classes form $HH^2(A,M)$.

Remark. The theorem is the infinitesimal case of the deformation theory; the whole deformation functor is controlled by $HH^2$ with the Gerstenhaber bracket supplying the higher obstructions, which is the subject of that article .

Proposition. $HH_2(A,A)$ carries the shuffle product with $HH_1(A,A)$ and, in the commutative case, is the degree-two part of the exterior algebra $\Lambda_A\Omega^1_{A/k}$ when $A$ is smooth; this is part of the Hochschild–Kostant–Rosenberg theorem below.

The Hochschild–Kostant–Rosenberg Theorem

Statement

Definition. A commutative $k$-algebra $A$ is smooth over $k$ if it is finitely presented and formally smooth: for every $k$-algebra surjection $A_1\to A_0$ with square-zero kernel, every $k$-algebra map $A\to A_0$ lifts to a $k$-algebra map $A\to A_1$. Equivalently, $A$ is smooth if it is a localisation of a polynomial algebra over $k$ followed by an étale extension; the equivalent formulations and the geometric content belong to Part II, and the algebraic definition just given suffices here. Write $\Omega^p_{A/k}=\Lambda^p_A\Omega^1_{A/k}$ for the $p$-th exterior power of the module of Kähler differentials.

Theorem (Hochschild–Kostant–Rosenberg). Let $A$ be a commutative $k$-algebra that is smooth over $k$. Then there are natural isomorphisms

$$ HH_n(A,A)\cong\Omega^n_{A/k}, \qquad HH^n(A,A)\cong\bigoplus_{p+q=n}\Lambda^p\operatorname{Der}_k(A,A)\otimes_A\Omega^q_{A/k} $$

under which the Hochschild boundary corresponds to the de Rham differential and the shuffle products correspond to the exterior products.

Example. For the polynomial algebra $A=k[x_1,\dots,x_m]$ one has $\Omega^p_{A/k}$ free over $A$ with basis $dx_{i_1}\wedge\cdots\wedge dx_{i_p}$ for $i_1<\cdotsm$. In particular $HH_0=A$, $HH_1=A^m$ and $HH_m(A,A)\cong A$.

Proof of the example. The polynomial algebra is smooth over $k$, so the theorem applies; the exterior powers of a free module of rank $m$ are free of the stated ranks.

The Differential-Form Interpretation

Proposition. Under the theorem, the Hochschild $n$-cocycles for a smooth commutative algebra are represented by the alternating multilinear maps, and the Hochschild cochain complex contains the subcomplex of the alternating cochains whose cohomology is $\Omega^n_{A/k}$ with the zero differential; the remaining cohomology comes from the symmetric part, the derivations. This is the split of the cohomology displayed in the theorem.

Example. For $A=k[x]$ the Hochschild complex computes $HH_n(k[x])\cong k[x]$ for $n=0,1$ and $0$ for $n\ge2$; the identification is explicit: a Hochschild $n$-cocycle is represented by a polynomial, with $f(x)$ in degree one corresponding to the Kähler differential $f\,dx$, and the boundary is the de Rham differential $d$, whose kernel in degree one is the constant polynomials, matching $HH_0=k[x]$.

Example (the dual numbers). Let $A=k[x]/(x^2)$ over a field $k$ of characteristic not $2$. The module of Kähler differentials is $\Omega^1_{A/k}=A\,dx/(2x\,dx)\cong A/(x)\cong k$, so $HH_1(A,A)\cong k$. The whole Hochschild homology is $HH_0(A,A)\cong A$, of dimension $2$ over $k$, and $HH_n(A,A)\cong k$, of dimension $1$, for every $n\ge1$. The computation is explicit. Writing the basis $\{1,x\}$ of $A$ as the index set $\{0,1\}$ and a Hochschild $n$-chain as a tuple of $n+1$ indices, the Hochschild complex is $C_n=A^{\otimes(n+1)}$ of dimension $2^{n+1}$ over $k$ with the boundary at degree $n$ given by the $n+1$ contractions $$ (a_0,\dots,a_n)\mapsto\sum_{i=0}^{n-1}(-1)^i(a_0,\dots,a_ia_{i+1},\dots,a_n)+(-1)^n(a_na_0,a_1,\dots,a_{n-1}), $$ where a contraction with the product $x\cdot x=0$ is omitted. The ranks of the boundary maps $b_n$ for $n=1,\dots,6$, computed by exact row reduction over $\mathbb{Q}$ on matrices of size $2^n\times2^{n+1}$, are $0,3,4,11,20,43$; hence $\dim HH_0=\dim\ker b_0-\operatorname{rank}b_1=2-0=2$ and $\dim HH_n=\dim\ker b_n-\operatorname{rank}b_{n+1}=1$ for $n=1,2,3,4,5$, confirming the stated answer in low degrees. The result depends on $2$ being invertible in $k$: when $2=0$ the differential form $2x\,dx$ vanishes and $\Omega^1_{A/k}\cong A$, of dimension $2$, so $HH_1$ jumps.

Products and the Gerstenhaber Structure

The Cup Product

Definition. For $A$-bimodules the cup product is the map

$$ \smile:HH^p(A,M)\otimes_kHH^q(A,N)\to HH^{p+q}(A,M\otimes_AN) $$

induced by the composition of cochains $\varphi\smile\psi(a_1,\dots,a_{p+q})=\varphi(a_1,\dots,a_p)\otimes\psi(a_{p+1},\dots,a_{p+q})$ composed with the bimodule map. Applied with $M=N=A$, it makes $HH^{\bullet}(A,A)$ a graded commutative $k$-algebra when $A$ is commutative, and in general a graded ring with a skew action of the centre.

Proposition. The cup product is associative and, for $A$ commutative, graded commutative on the cohomology: $x\smile y=(-1)^{pq}y\smile x$ for $x \in HH^p$, $y \in HH^q$. Under the Hochschild–Kostant–Rosenberg theorem it corresponds to the exterior product of differential forms.

Proof. Associativity is the associativity of composition of cochains at the level of the complex and survives in cohomology. Graded commutativity is a computation with the shuffle permutation of the $p+q$ variables, whose sign is $(-1)^{pq}$, and it uses the commutativity of $A$ to move the factors past one another. The identification with the exterior product is part of the Hochschild–Kostant–Rosenberg theorem.

The Gerstenhaber Bracket

Definition. The Gerstenhaber bracket on $HH^{\bullet}(A,A)$ is induced by the graded commutator of the composition product $\circ$ of cochains,

$$ [\varphi,\psi]=\varphi\circ\psi-(-1)^{(p-1)(q-1)}\psi\circ\varphi, $$

where for $\varphi$ of degree $p$ and $\psi$ of degree $q$ the composition is $$ (\varphi\circ\psi)(a_1,\dots,a_{p+q-1})=\sum_{i=0}^{p-1}(-1)^{(q-1)i}\varphi(a_1,\dots,a_i,\psi(a_{i+1},\dots,a_{i+q}),a_{i+q+1},\dots). $$

Theorem (Gerstenhaber). The bracket makes $HH^{\bullet}(A,A)$ a graded Lie algebra of degree $-1$: it satisfies $[\varphi,\psi]=-(-1)^{(p+1)(q+1)}[\psi,\varphi]$ and the graded Jacobi identity $$ (-1)^{(p+1)(r+1)}[[\varphi,\psi],\theta]+(-1)^{(q+1)(p+1)}[[\psi,\theta],\varphi]+(-1)^{(r+1)(q+1)}[[\theta,\varphi],\psi]=0 $$ on classes of degrees $p,q,r$, and the bracket is compatible with the cup product in the sense of a Poisson-type identity $$ [\varphi,\psi\smile\theta]=[\varphi,\psi]\smile\theta+(-1)^{(p-1)q}\psi\smile[\varphi,\theta]. $$ The bracket is defined on the whole Hochschild cochain complex, not only on the cohomology, and it descends to cohomology because the bracket of a cochain with a cocycle is again a cocycle and the bracket with a coboundary is a coboundary.

Proof. The composition product $\circ$ is homogeneous of degree $-1$ and is associative up to explicit signs on the cochain complex, making it a graded pre-Lie algebra; the graded commutator of a pre-Lie product is a graded Lie bracket, which gives the antisymmetry and the Jacobi identity. The compatibility with the cup product is a computation in the operad of cochains. The descent to cohomology uses that the bracket of a cochain with a cocycle is a cocycle and the bracket with a coboundary is a coboundary.

Example. For a commutative algebra $A$ the bracket of two degree-one cocycles $D,D'$, that is two derivations, is the commutator of the derivations, $[D,D']=D\circ D'-D'\circ D$; the bracket of a degree-one cocycle $D$ and a degree-zero cocycle $a \in A$ is the derivation $D(a)$. Thus on $HH^0\oplus HH^1=Z(A)\oplus\operatorname{Der}_k(A,A)$ the bracket restricts to the semidirect product of the Lie algebra of derivations acting on the centre. This is the sense in which the Hochschild complex linearises the deformation theory of the algebra.

Corollary. The deformation theory is controlled by the differential graded Lie algebra $(HH^{\bullet}(A,A),[,],\smile)$: first-order deformations are the cycles in $HH^2$, the gauge group acts by the degree-one part, and the obstructions lie in $HH^3$.

Morita Invariance and Change of Ground Ring

Theorem. Let $A$ and $A'$ be Morita equivalent $k$-algebras, so that $A'\text{-}\mathbf{Mod}\simeq A\text{-}\mathbf{Mod}$ as $k$-linear categories; then $HH_{\bullet}(A,A)\cong HH_{\bullet}(A',A')$ and $HH^{\bullet}(A,A)\cong HH^{\bullet}(A',A')$, the isomorphisms respecting the products.

Proof. Morita equivalence is realised by a bimodule ${}_AP_{A'}$ which is finitely generated projective and a generator, with $\operatorname{End}_{A'}(P)\cong A$ and $\operatorname{End}_A(P)\cong A'$. The functor $-\otimes_A^{\mathbb{L}}P$ and its inverse identify the categories of bimodules and the enveloping algebras, and the bar resolution is carried to a projective resolution computing the same derived functors, giving the isomorphisms. The products are carried to the products because they are induced by the composition of cochains, which the equivalence preserves.

Example. For the matrix algebra $M_n(k)$ the Hochschild homology and cohomology agree with those of $k$: $HH_0(M_n(k))\cong k$, $HH_n(M_n(k))=0$ for $n\ge1$, and $HH^{\bullet}(M_n(k))=k$ in degree zero. The ring $k[x]$ and the algebra of $k$-linear endomorphisms of a finitely generated projective $k[x]$-module have the same Hochschild groups, so the theory is an invariant of the module category, not of the presentation.

Proposition. If $k\to k'$ is a flat commutative ring homomorphism and $A$ is flat over $k$, there are natural isomorphisms $HH_n(A\otimes_kk',A\otimes_kk')\cong HH_n(A,A)\otimes_kk'$; the corresponding statement for cohomology holds when the relevant modules are flat or finitely generated projective.

Proof. The bar resolution consists of free $A^{\mathrm{e}}$-modules and the flatness hypotheses make the two resolutions compatible after base change, so the derived functors commute with base change by the standard flat-base-change theorem of Derived Functors.

Summary

The bar resolution is a free resolution of a $k$-algebra $A$ over its enveloping algebra $A^{\mathrm{e}}=A\otimes_kA^{\mathrm{op}}$, and applying a bimodule $M$ gives the Hochschild complexes: the Hom complex computes the Hochschild cohomology $HH^{\bullet}(A,M)$ and the tensor complex computes the Hochschild homology $HH_{\bullet}(A,M)$. These are the derived functors of the invariants and the coinvariants, and they agree with $\operatorname{Ext}$ and $\operatorname{Tor}$ over the enveloping algebra. In low degrees the invariant takes its classical form: $HH_0(A,A)=A/[A,A]$, $HH^0(A,A)=Z(A)$, $HH^1(A,A)$ is the derivations modulo the inner derivations, $HH_1(A,A)$ is the module of Kähler differentials and its tensor products, and $HH^2(A,M)$ classifies the square-zero extensions of $A$ by $M$.

For a smooth commutative algebra the Hochschild–Kostant–Rosenberg theorem identifies the homology with the differential forms, $HH_n(A,A)\cong\Omega^n_{A/k}$, and the cohomology with the direct sum of the exterior powers of the differentials tensored with the exterior powers of the derivations; this makes the theory the algebraic substitute for the de Rham complex, with the Hochschild boundary playing the role of the de Rham differential. The Hochschild cohomology carries the cup product and the Gerstenhaber bracket, which make it a Gerstenhaber algebra whose degree-one part acts on the centre, and the resulting differential graded Lie algebra controls the deformation theory of the algebra. The theory is Morita invariant and commutes with flat base change.

Cyclic homology, treated of this category, is built from the same cyclic object obtained by making the Hochschild complex cyclic, with the Connes boundary adjoined; cyclic cohomology, a different invariant despite the name, belongs to Part III. The deformation-theoretic use of the cohomology and its operadic organisation are not covered here.

Summary of Notation

Symbol	Meaning
$k$	commutative ground ring with $1\neq0$
$A$	associative $k$-algebra with unit, not necessarily commutative
$A^{\mathrm{e}}=A\otimes_kA^{\mathrm{op}}$	enveloping algebra, so $A$-bimodules are $A^{\mathrm{e}}$-modules
$M$	an $A$-bimodule of coefficients
$B_n(A)=A^{\otimes(n+2)}$	bar resolution, free over $A^{\mathrm{e}}$
$b$	Hochschild boundary
$C_\bullet(A,M)$	Hochschild complex $M\otimes_kA^{\otimes n}$
$HH_n(A,M)$, $HH^n(A,M)$	Hochschild homology and cohomology
$M_A$, $M^A$	coinvariants and invariants of a bimodule
$\operatorname{Der}_k(A,M)$, $\operatorname{Inn}_k(A,M)$	derivations and inner derivations
$I=\ker\mu$	augmentation ideal of the multiplication
$\Omega^1_{A/k}=I/I^2$, $\Omega^p_{A/k}$	Kähler differentials and their exterior powers
$d:A\to\Omega^1_{A/k}$	the Kähler derivation, $da=a\otimes1-1\otimes a$
$\smile$	cup product on Hochschild cohomology
$[,]$	Gerstenhaber bracket, of degree $-1$
$\circ$	composition product of cochains
$\mu:A\otimes_kA\to A$	multiplication