Hochschild Complex as an Differential Graduated Lie Algebra: HKR Theorem and Deformation Theory

Hochschild Complex as an Differential Graduated Lie Algebra: HKR Theorem and Deformation Theory.

Published by Gustav on July 10, 2026 Philosophy

§1.Introduction

The cohomology of algebras, which was introduced by Gerhard Hochschild in 1945 as purely algebraic construction, has evolved into a fundamental tool bridging several areas of mathematics and mathematical physics. Hochschild cohomology, in particular, measures the failure of certain algebraic structures to be rigid, and in doing so, it provides a natural language for deformation theory, noncommutative geometry, and, as we shall soon see, quantum mechanics. The central object of this post is the Hochschild cohomology of commutative algebras, with particular emphasis on the Hochschild–Kostant–Rosenberg (HKR) theorem and its applications to deformation quantisation. The HKR theorem for cohomologies establishes that for a smooth commutative algebra $A$ over a field of characteristic zero, the Hochschild cohomology $HH^\bullet(A)$ is isomorphic to the space of polyvector fields $\bigwedge^\bullet_A \operatorname{Der}_k(A)$ . This deceptively simple isomorphism works as mediator that translates the purely algebraic objects of deformations of $A$ into geometric objects, like Poisson structures and multivector fields. The story we tell unfolds in three acts.

Act I: Foundations. We begin by establishing the definition of the Hochschild cohomology $HH^\bullet(A)$ via the bar resolution and explore its interpretation in low degrees. We also present some explicit examples where we develop computational familiarity with these invariants.

Act II: The HKR Theorem. The heart of this work is the proof of the HKR theorem for both homology and cohomology. For homology, the theorem identifies $HH_n(A)$ with the module of Kähler differentials $\Omega^n_{A/k}$ ; for cohomology, it identifies $HH^n(A)$ with polyvector fields. We present a detailed proof using Koszul complexes for polynomial algebras, followed by extension of the result to arbitrary smooth commutative algebras. Along the way, we introduce the Gerstenhaber bracket, which endows $HH^\bullet(A)$ with the structure of a Gerstenhaber algebra. This bracket, when combined with the cup product, makes Hochschild cohomology a graded Lie algebra that governs deformation theory. Crucially, the HKR isomorphism is identified as an isomorphism of Gerstenhaber algebras, identifying the Gerstenhaber bracket on $HH^\bullet(A)$ with the Schouten–Nijenhuis bracket on polyvector fields.

Act III: Deformation Quantisation. The final act connects the algebraic machinery to mathematical physics via deformation quantisation. A Poisson manifold $(M, \pi)$ gives rise to a Poisson algebra structure on $C^\infty(M)$ . The HKR theorem identifies $HH^2(A)$ with bivector fields, so a Poisson structure $\pi$ corresponds to an infinitesimal deformation of the algebra $A$ . The condition that $\pi$ defines a Poisson structure — namely, $[\pi, \pi]_{SN} = 0$ — is precisely the condition that the primary obstruction to extending the deformation vanishes. Thus, every Poisson structure on a smooth commutative algebra gives rise to a formal deformation, i.e., a star product. This is the program of deformation quantization, which interprets quantum mechanics as a deformation of classical mechanics.

This post offers an extended, self-contained introduction to deformation quantization, contextualizing it within the history of physics from Newtonian mechanics through Hamiltonian mechanics to quantum mechanics, although it does not go into more technical detail regarding issues of obstructions and cohomology, one may regard this post as an extension of it, incorporating (co)homological considerations and serving as a foundation for another post in which I will formulate a theorem that unifies Morita equivalence of formal Poisson algebras with gauge transformations of Poisson structures, ultimately connecting everything to deformation quantization. Moreover, the linked post traces the conceptual evolution from forces to Lagrangians to Hamiltonians, culminating in the algebraic reformulation of classical mechanics as a Poisson algebra and quantum mechanics as its noncommutative deformation. Here, however, our aim is to show that Hochschild cohomology applications to deformation quantisation.

§2. Commutative Algebra Background

The material presented in Subsections 1.1 to 1.4 is written based primarily on the classical text of Atiyah and Macdonald, which serves as my standard reference for commutative algebra. Beginning with Subsection 1.5, the exposition follows mainly the treatment of L. R. Vermani. Readers already familiar with the fundamentals of abstract algebra may safely skip this section without any loss of content.

Rings and Ideals

Definition 1 (Ring). A ring $R$ is a set equipped with two binary operations, namely $\cdot: R \times R \rightarrow R$ and $+: R \times R \rightarrow R$ , called multiplication and addition respectively. These operations satisfy the following conditions:

$R$ is an abelian group under addition. That is, there exist elements $0_R, -r \in R$ such that $r + 0_R = r$ and $r + (-r) = 0_R$ for all $r \in A$ ;
Multiplication is associative, i.e. $(xy)z = x(yz)$ , and distributive over addition, i.e. $x(y + z) = xy + xz$ and $(y + z)x = yx + zx$ for all $x,y,z\in R$ .

For the purposes of this dissertation, we shall restrict our attention to commutative rings with unit; that is, rings for which $xy = yx$ holds for all $x,y\in R$ and which contain a unique element $1_R \in R$ such that $x1_R = 1_Rx = x$ for all $x\in R$ .

Example 1. The ring of integers $\mathbb{Z}$ is the archetypal commutative ring with identity. Its units are $\pm 1$ , and it is an integral domain but not a field.

Example 2. For any integer $n > 0$ , the quotient ring $\mathbb{Z}/n\mathbb{Z}$ is a finite commutative ring with identity. It is a field if and only if $n$ is prime.

Example 3. Let $k$ be a field. The polynomial ring $k[x_1,\dots,x_n]$ in $n$ indeterminates is a commutative ring with identity. It is an integral domain and a finitely generated $k$ -algebra. Its units are precisely the non-zero constants.

Example 4. The formal power series ring $k[[x_1,\dots,x_n]]$ over a field $k$ is a commutative local ring whose maximal ideal is $(x_1,\dots,x_n)$ . It plays a central role in the study of completions.

Example 5. The product ring $R \times R'$ of two commutative rings $R$ and $R'$ (with componentwise addition and multiplication) is again a commutative ring with identity $(1_R,1_{R'})$ . Its ideals are not simply products of ideals; for instance, the diagonal $\{(r,r): r\in R\}$ is an ideal in $R\times R$ only when $R$ has additional structure.

Definition 2 (Ring Homomorphism). A ring homomorphism is a map $f: R \rightarrow R'$ , where $R$ and $R'$ are rings, that respects the algebraic structures of $R$ and $R'$ ; i.e., it satisfies:

$f(x + y) = f(x) + f(y)$ ,
$f(xy) = f(x)f(y)$ ,
$f(1_R) = 1_{R'}$ .

The composition of two ring homomorphisms is again a ring homomorphism.

Example 6. The inclusion map $\iota: \mathbb{Z} \hookrightarrow \mathbb{Q}$ is an injective ring homomorphism.

Example 7. The projection $\pi:R \to R/I$ onto a quotient ring is a surjective ring homomorphism with kernel $I$ .

Example 8. The evaluation map $k[x] \to k$ given by $f(x) \mapsto f(\alpha)$ for a fixed $\alpha \in k$ is a surjective ring homomorphism whose kernel is the maximal ideal $(x-\alpha)$ .

Example 9. Localisation: for a multiplicatively closed subset $S$ of $A$ , the canonical map $A \to S^{-1}A$ , $a \mapsto a/1$ , is a ring homomorphism.

Example 10. The map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ sending an integer to its residue class mod $n$ is a surjective ring homomorphism.

Definition 3 (Subring). Let $A$ be a ring. A subset $S \subseteq R$ is called a subring of $A$ if it satisfies the following conditions:

$S$ is closed under addition: for all $x, y \in S$ , we have $x + y \in S$ .
$S$ contains the additive identity $0_R$ of $R$ .
$S$ is closed under additive inverses: for every $x \in S$ , we have $-x \in S$ .
$S$ is closed under multiplication: for all $x, y \in S$ , we have $xy \in S$ .
$S$ contains the multiplicative identity $1_R$ of $R$ .

Equivalently, $S$ is an additive subgroup of $R$ that is closed under multiplication and contains $1_A$ . With the operations inherited from $R$ , $S$ itself becomes a commutative ring with identity (the identity being $1_R$ ).

Definition 4 (Ideal). An ideal $I$ of a ring $R$ is a subset of $R$ that is an additive subgroup and satisfies $RI \subseteq I$ ; i.e., for all $x \in R$ and $y \in I$ , we have $xy \in I$ .

The quotient group $R/I$ inherits a uniquely defined multiplication from $A$ , thereby becoming a ring, which we call the quotient ring $A/I$ . Its elements are the cosets of $I$ in $R$ , i.e. sets of the form $r + I = \{r + x \mid x \in I\}$ . The mapping $\phi: R \rightarrow R/I$ , given by $x \mapsto x + r$ is a surjective ring homomorphism.

Proposition 1 (Correspondence Theorem). There exists a one-to-one, order-preserving correspondence between the ideals $J$ of $R$ that contain $I$ and the ideals $\bar{J}$ of the quotient ring $R/I$ , given by $J = \phi^{-1}(\bar{J})$ , where $\phi: R \rightarrow R/I$ denotes the canonical projection.

Moreover, given a ring homomorphism $f: R \rightarrow R'$ , its kernel $\ker f = f^{-1}(0)$ is an ideal $I$ of $R$ , and its image $\text{Im} f$ is a subring $S$ of $R'$ ; furthermore, $f$ induces a ring isomorphism $R/I \cong S$ given by $\overline{a} \mapsto f(r)$ for each $r \in R$ .

Example 11. In any ring $R$ , the zero ideal $\{0\}$ and the unit ideal $R$ are trivial ideals. An ideal is proper if it is not equal to $R$ .

Example 12. In the polynomial ring $k[x]$ , the ideal $(x)$ consisting of all polynomials with zero constant term is maximal; more generally, for any irreducible polynomial $f$ , the ideal $(f)$ is prime (in fact maximal when $k$ is a field).

Example 13. In $\mathbb{Z}$ , every ideal is of the form $n\mathbb{Z}$ for a unique $n \geqslant 0$ . The ideal $p\mathbb{Z}$ with $p$ prime is maximal, and also prime. The zero ideal is prime but not maximal.

§2. Nilpotent Elements, Zero-Divisors and Units

We now introduce several classes of elements that play a fundamental role in the structure of a commutative ring.

Definition 5. Let $A$ be a commutative ring with identity.

An element $x \in R$ is called a zero-divisor if there exists a non-zero element $y \in R$ such that $xy = 0$ .
An element $x \in R$ is called nilpotent if there exists a positive integer $n$ such that $x^{n} = 0$ .
An element $x \in R$ is called a unit if there exists $y \in R$ such that $xy = 1$ ; the element $y$ is then uniquely determined and denoted by $x^{-1}$ .

A ring in which $1 \neq 0$ and which has no zero-divisors other than $0$ is called an integral domain. A ring in which $1 \neq 0$ and every non-zero element is a unit is called a field.

Clearly every field is an integral domain, but the converse is false (e.g., $\mathbb{Z}$ is an integral domain but not a field). Moreover, every nilpotent element is a zero-divisor (unless $R = 0$ ), but the converse need not hold; for instance, in the ring $\mathbb{Z}/6\mathbb{Z}$ the element $2$ is a zero-divisor but not nilpotent.

Example 14.

In the ring $\mathbb{Z}$ , the only nilpotent element is $0$ , and the only units are $\pm 1$ . The zero-divisors are precisely $0$ .
In the ring $\mathbb{Z}/8\mathbb{Z} = \{0,1,\dots,7\}$ , the element $2$ is nilpotent because $2^{3}=8\equiv 0$ . The units are the odd numbers, and the zero-divisors are the even numbers.
Let $k$ be a field and consider the polynomial ring $k[x]$ . An element $f \in k[x]$ is a unit iff it is a non-zero constant. It is a zero-divisor iff it is $0$ . It is nilpotent only when $f = 0$ .
More generally, in the quotient ring $k[x]/(x^{n})$ (with $n \ge 2$ ), the image of $x$ is nilpotent (since $x^{n}=0$ ), and every non-zero element of the maximal ideal $(x)/(x^{n})$ is a zero-divisor.

The following proposition gives several equivalent characterisations of a field; it is a standard result whose proof can be found in any introductory text on commutative algebra (see e.g. Atiyah–Macdonald, Proposition 1.2).

Proposition 2. Let $R$ be a non-zero commutative ring. Then the following are equivalent:

$R$ is a field;
the only ideals of $A$ are $0$ and $R$ ;
every ring homomorphism from $R$ to a non-zero ring is injective.

The Nilradical and the Jacobson Radical

The set of all nilpotent elements of a ring $A$ has a particularly simple structure.

Definition 6. The nilradical of $A$ , denoted by $\mathfrak{N}(R)$ or $\text{nil}(R)$ , is the set of all nilpotent elements of $R$ .

It is an ideal of $R$ , and the quotient ring $R/\mathfrak{N}(R)$ has no non-zero nilpotent elements. Moreover, the nilradical equals the intersection of all prime ideals of $R$ : $\mathfrak{N}(R) = \bigcap_{\mathfrak{p} \in \text{Spec} R} \mathfrak{p}.$ This is a fundamental result (Atiyah–Macdonald, Proposition 1.8) and explains why nilpotent elements are sometimes thought of as “infinitesimal” objects.

In parallel, the Jacobson radical is defined as the intersection of all maximal ideals.

Definition 7. The Jacobson radical of $R$ , denoted by $\mathfrak{R}(R)$ , is the intersection of all maximal ideals of $R$ : $\mathfrak{R}(R) = \bigcap_{\mathfrak{m} \in \text{Max} R} \mathfrak{m}.$

It admits the following useful characterisation (Atiyah–Macdonald, Proposition 1.9): $x \in \mathfrak{R}(R) \quad\Longleftrightarrow\quad 1 - xy \text{ is a unit in } R \text{ for every } y \in R.$

Local Rings

A particularly important class of rings is that of local rings, which arise naturally in algebraic geometry and number theory when one concentrates attention near a point or a prime.

Definition 8. A ring $R$ is called a local ring if it has exactly one maximal ideal. That maximal ideal is usually denoted by $\mathfrak{m}$ , and the field $R/\mathfrak{m}$ is called the residue field of $R$ .

Example 15. Any field is a local ring (its unique maximal ideal is $\{0\}$ ).

Example 16. The ring $\mathbb{Z}_{(p)} = \{ a/b \in \mathbb{Q} : p \nmid b \}$ , for a prime $p$ , is a local ring with maximal ideal $p\mathbb{Z}_{(p)}$ . Its residue field is $\mathbb{F}_{p}$ .

Example 17. Let $k$ be a field and consider the polynomial ring $k[x]$ localized at the prime ideal $(x)$ ; we obtain the local ring $k[x]_{(x)} = \{ f/g \in k(x) : g(0) \neq 0 \}$ , with maximal ideal $x k[x]_{(x)}$ . Its residue field is $k$ .

Localisation at a prime ideal allows one to study an algebra locally on its prime spectrum, a perspective essential for local properties such as smoothness and regularity; in the smooth commutative case, the Hochschild–Kostant–Rosenberg theorem then identifies Hochschild (co)homology with differential forms and polyvector fields. Nilpotent elements are equally central to deformation theory: infinitesimal and formal deformations are encoded by square-zero extensions and formal power series rings, with Hochschild cohomology governing first-order deformations (via $HH^2$ ) and obstructions (via $HH^3$ ). These ideas underpin deformation quantisation, which will serve as our case study in the final section.

Modules and Module Homomorphisms

Definition 9 (Left and Right $R$ -Modules). Let $R$ be a ring with unity. An additive Abelian group $M$ is called a left $R$ -Module over $R$ if there exists, for every $r \in R$ and $a \in M$ , a uniquely determined element $ra \in M$ such that the following hold:

$(r + s)a = ra + sa$ for every $r,s \in R$ and $a \in M$ ;
$(rs)a = r(sa)$ for every $r,s \in R$ and $a \in M$ ;
$r(a + b) = ra + rb$ for every $r\in R$ and $a, b \in M$ ;
$1_R a= a$ for every $a \in M$ .

The definition of a right $R$ -module is entirely analogous, with the order of scalar multiplication reversed so that, in particular, $a \cdot 1_R = a$ for all $a$ in the module.

Now let $R$ be a commutative ring and let $M$ be a left $R$ -module. For each $a \in M$ and $r \in R$ , we may define a right scalar multiplication by setting $a \cdot r := r a$ . Because $R$ is commutative, this definition is well‑behaved. Moreover, one gets for all $a \in M$ and $r, s \in R$ , $a \cdot (rs) = (rs)a = (sr)a = s(ra) = (a \cdot r) \cdot s$ , $(r + s)a = ra + sa = a \cdot r + a \cdot s$ , $r(a + b) = ra + rb = a \cdot r + b \cdot r$ , and $1_R \cdot a = a = a \cdot 1_R$ . Thus every left module over a commutative ring is simultaneously a right module, and similarly every right module can be regarded as a left module. In other words, modules over commutative rings are naturally symmetric; there is no distinction between left and right modules.

Example 18. Let $M$ and $N$ be $R$ -modules. Their direct sum $M \oplus N = \{(m,n) : m \in M,\; n \in N\}$ is an $R$ -module with componentwise addition and scalar multiplication: $r \cdot (m,n) = (r m, r n)$ for all $r \in R$ . More generally, the direct sum of any family of $R$ -modules is again an $R$ -module.

Example 19. Let $M$ be an $R$ -module and let $N \subseteq M$ be a submodule. The quotient module $M/N = \{m + N : m \in M\}$ is an $R$ -module with addition defined by $(m_1 + N) + (m_2 + N) = (m_1 + m_2) + N$ and scalar multiplication by $r \cdot (m + N) = r m + N$ for all $r \in R$ . The natural projection $\pi: M \to M/N$ is a surjective $R$ -module homomorphism with kernel $N$ .

Example 20. The ring $R$ itself, considered with its own addition and multiplication, forms an $R$ -module, called the regular module. For $r \in R$ and $a \in R$ , the scalar multiplication is given by $r \cdot a = r a$ (the ring product). This module is free of rank one.

Definition 10 (Submodule). Consider $M$ a left $R$ -Module and an additive group. A subgorup $N$ of $M$ is called a submodule of $M$ if, for every $a \in N$ and $r \in R$ , we have $ra \in N$ .

Equivalently, a non-empty subset $N$ of a left $R$ -module $M$ is a submodule of $M$ if and only if, for every $a, b \in N$ and every $r \in R$ , we have $a - b \in N$ and $r a \in N$ . It should be obvious at this point, but an analogous construction holds for right submodules.

Now let $M$ be a left $R$ -module and let $N$ be a submodule of $M$ . We define the quotient module of $M$ by $N$ , denoted $M/N$ , as the set of cosets $\{m + N : m \in M\}$ with addition and scalar multiplication given by $(m_1 + N) + (m_2 + N) = (m_1 + m_2) + N, \qquad r \cdot (m + N) = r m + N,$ for all $m, m_1, m_2 \in M$ and $r \in R$ . These operations are well-defined because $N$ is a submodule, and they endow $M/N$ with the structure of a left $R$ -module. The natural projection $\pi: M \to M/N$ , defined by $\pi(m) = m + N$ , is a surjective $R$ -module homomorphism whose kernel is precisely $N$ .

Definition 11. Let $M$ , $N$ be $R$ -Modules. A $R$ -homomorphism or module homomorphism is a map $f: M \rightarrow N$ that satisfies

$f(a + b) = f(a) + f(b)$ for every $a, b \in M$ ;
$f(ra) = r f(a)$ for every $r \in R$ and $a \in M$ .

Example 21. Let $A$ be a ring and let $M$ be a left $A$ -module. For each $a \in A$ , the map $\varphi_a : M \to M, \qquad \varphi_a(m)=am$ is an $A$ -module homomorphism.

Example 22. Let $A$ be a commutative ring and let $M,N$ be $A$ -modules. The projection maps $\pi_M : M \oplus N \to M, \qquad \pi_N : M \oplus N \to N$ defined by $\pi_M(m,n)=m, \qquad \pi_N(m,n)=n$ are $A$ -module homomorphisms.

Example 23. Let $A$ be a ring and let $I\subseteq A$ be a left ideal. The quotient map $\pi : A \to A/I, \qquad \pi(a)=a+I$ is an $A$ -module homomorphism.

Example 24. Let $A$ be a ring and let $M$ be an $A$ -module. If $N\subseteq M$ is a submodule, then the inclusion map $\iota : N \hookrightarrow M, \qquad \iota(n)=n$ is an $A$ -module homomorphism.

Example 25. Let $A$ be a $k$ -algebra. A derivation $d:A\to A$ satisfying $d(ab)=ad(b)+d(a)b$ for all $a,b\in A$ is generally not an $A$ -module homomorphism, but it is a $k$ -linear map. Derivations play a central role in Hochschild cohomology, where $HH^1(A)\cong \mathrm{Der}(A)/\mathrm{InnDer}(A).$

Module Isomorphism Theorems

We now recall the fundamental isomorphism theorems for modules over a ring $R$ . These results are essential for working with quotient modules and homomorphisms.

Theorem 12 (First Isomorphism Theorem). Let $M$ and $N$ be $R$ -modules and let $f: M \to N$ be an $R$ -module homomorphism. Then $M / \ker f \cong \text{im} f.$ In particular, if $f$ is surjective then $M / \ker f \cong N$ .

Theorem 13 (Second Isomorphism Theorem). Let $M$ be an $R$ -module and let $A, B$ be submodules of $M$ . Then $(A + B) / B \cong A / (A \cap B).$ Moreover, the natural map $A \to (A + B)/B$ induces this isomorphism.

Theorem 14 (Third Isomorphism Theorem). Let $M$ be an $R$ -module and let $N \subseteq P$ be submodules of $M$ . Then $(M / N) / (P / N) \cong M / P.$ The isomorphism is given by $(m + N) + (P / N) \mapsto m + P$ .

Theorem 15 (Correspondence Theorem (Fourth Isomorphism Theorem)). Let $M$ be an $R$ -module and let $N$ be a submodule of $M$ . Then there exists a one‑to‑one, order‑preserving correspondence between submodules of the quotient module $M/N$ and submodules of $M$ that contain $N$ . Explicitly, if $\pi: M \to M/N$ denotes the canonical projection, then for any submodule $\overline{S}$ of $M/N$ the preimage $\pi^{-1}(\overline{S})$ is a submodule of $M$ containing $N$ ; conversely, for any submodule $S$ of $M$ with $N \subseteq S$ , the image $\pi(S) = S/N$ is a submodule of $M/N$ . These correspondences are inverses of each other.

These theorems are used throughout homological algebra, particularly in the construction and analysis of exact sequences and chain complexes, both central concepts for this work. Their proofs were given during the lectures and are straightforward and can be found in any standard text on algebra or module theory.

Exact Sequences

We now turn to one of the central ideas in homological algebra: the notion of an exact sequence.

Definition 16 (Exact sequence). A sequence of $A$ -modules and homomorphisms $\cdots \longrightarrow M_{n+1} \xrightarrow{f_{n+1}} M_n \xrightarrow{f_n} M_{n-1} \longrightarrow \cdots$ is said to be exact at $M_n$ if $\text{Im} f_{n+1} = \text{Ker} f_n$ . The sequence is exact if it is exact at every module in the chain.

A particularly important case is the short exact sequence $0 \longrightarrow M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \longrightarrow 0,$ which encodes the fact that $\alpha$ is injective, $\beta$ is surjective, and $\text{Im} \alpha = \text{Ker} \beta$ . In this situation $M'' \cong M / \text{Im} \alpha$ , so a short exact sequence can be thought of as an extension of $M''$ by $M'$ .

Definition 17 (Split exact sequence). A short exact sequence $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ is said to split if there exists a homomorphism $\gamma: M'' \to M$ such that $\beta \gamma = 1_{M''}$ (a right splitting), or equivalently a homomorphism $\delta: M \to M'$ such that $\delta \alpha = 1_{M'}$ (a left splitting). In that case $M \cong M' \oplus M''$ .

Split exact sequences are the “trivial” extensions; they occur, for example, when $M''$ is projective (see the next subsection) or when $M'$ is injective. For the study of Hochschild cohomology, split exact sequences appear naturally when one considers the bar resolution: the normalisation map provides a splitting of certain complexes, which is why one can work with the reduced bar resolution.

Example 26 (Splitting in $\mathbb{Z}$ -modules). Consider the short exact sequence of abelian groups $0 \longrightarrow \mathbb{Z} \xrightarrow{2} \mathbb{Z} \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 0.$ This sequence does not split, because $\mathbb{Z}$ has no element of order $2$ and consequently there is no homomorphism $\mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}$ that splits the projection. In contrast, the sequence $0 \longrightarrow \mathbb{Z} \xrightarrow{\iota} \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/2\mathbb{Z} \longrightarrow 0$ (with $\iota(n)=(n,0)$ and $\pi$ the projection onto the second factor) splits – indeed the obvious section $\gamma(a)=(0,a)$ works. This example will reappear when we discuss projective modules.

A fundamental tool for working with exact sequences is the Five Lemma, that appeared on the second problem set of the course, if I’m not mistaken.

Lemma 18 (Five Lemma). Consider a commutative diagram of $A$ -modules $\begin{array}{ccccccccc} M_1 & \xrightarrow{} & M_2 & \xrightarrow{} & M_3 & \xrightarrow{} & M_4 & \xrightarrow{} & M_5 \\ \downarrow f_1 & & \downarrow f_2 & & \downarrow f_3 & & \downarrow f_4 & & \downarrow f_5 \\ N_1 & \xrightarrow{} & N_2 & \xrightarrow{} & N_3 & \xrightarrow{} & N_4 & \xrightarrow{} & N_5 \end{array}$ with exact rows. If $f_1$ is an epimorphism, $f_2$ and $f_4$ are monomorphisms, and $f_5$ is a monomorphism, then $f_3$ is a monomorphism. If $f_1$ is an epimorphism, $f_2$ and $f_4$ are epimorphisms, and $f_5$ is a monomorphism, then $f_3$ is an epimorphism. Consequently, if $f_1, f_2, f_4, f_5$ are isomorphisms, then so is $f_3$ .

The Five Lemma is of immense practical value. In Hochschild cohomology, one frequently encounters long exact sequences coming from short exact sequences of algebras or modules, and the Five Lemma is the standard device for transferring isomorphisms from one degree to another.

Finitely Generated and Free Modules

In homological algebra we rarely work with arbitrary modules; we prefer those that are “not too large”. The most convenient finiteness condition is that of being finitely generated.

Definition 19. An $A$ -module $M$ is finitely generated if there exist elements $x_1,\dots,x_n \in M$ such that every element of $M$ can be expressed as $\sum_{i=1}^n a_i x_i$ with $a_i \in A$ . Equivalently, there is a surjective homomorphism $A^{\oplus n} \to M$ .

The simplest finitely generated modules are the free modules.

Definition 20. An $A$ -module $F$ is free on a set $X$ if every element of $F$ can be written uniquely as a finite linear combination $\sum_{i} a_i x_i$ with $a_i \in A$ and $x_i \in X$ . The cardinality of $X$ is called the rank of $F$ . When $X = \{x_1,\dots,x_n\}$ we write $F = A^{\oplus n}$ (or simply $A^n$ ).

Free modules are the algebraic analogues of vector spaces. Their most fundamental property – the invariance of rank – is as follows.

Proposition 3 (Invariance of rank). If $A$ is a non‑zero commutative ring, then $A^m \cong A^n$ as $A$ -modules implies $m=n$ . In other words, the rank of a free module is well defined.

Sketch. Let $\mathfrak{m}$ be a maximal ideal of $A$ and let $k = A/\mathfrak{m}$ be the residue field. Tensor the isomorphism $A^m \cong A^n$ with $k$ over $A$ ; we obtain an isomorphism of $k$ -vector spaces $k^m \cong k^n$ , whence $m=n$ . The details are in Vermani, Proposition 1.2.11 (though there the proof uses a maximal ideal and the fact that tensor product is right exact). ◻

This simple observation is surprisingly powerful. For instance, it guarantees that the dimension of a projective module (when defined) is unique. In Hochschild cohomology, when we compute $\text{HH}_0(A) = A / [A,A]$ for a free algebra, the rank of the free module that appears in the bar resolution is finite and well defined, so we can speak of its dimension as a vector space over a field.

Example 27 (Free modules in Hochschild theory). Let $k$ be a field and let $A = k[x]$ . Then the bar resolution of $A$ as an $A$ -bimodule is a complex of free $A$ -modules; the term in degree $n$ is $A \otimes_k A^{\otimes_k n} \otimes_k A$ , which is free of infinite rank over $A$ (since $k$ is a field, $A^{\otimes_k n}$ is an infinite‑dimensional vector space). Nevertheless, each component is free, and the invariance of rank tells us that the length of any free basis is uniquely determined.

The Hom Functor and Its Exactness

For any two $A$ -modules $M,N$ , the set of all $A$ -linear maps from $M$ to $N$ is denoted $\text{Hom}_A(M,N)$ . It is an abelian group under pointwise addition, and when $A$ is commutative it becomes an $A$ -module via $(a f)(m) = a f(m)$ .

The functor $\text{Hom}_A(-,-)$ is a bifunctor: contravariant in the first argument and covariant in the second. Explicitly, a homomorphism $f: M \to M'$ induces $f^* : \text{Hom}_A(M',N) \to \text{Hom}_A(M,N),\quad \varphi \mapsto \varphi \circ f,$ while $g: N \to N'$ induces $g_* : \text{Hom}_A(M,N) \to \text{Hom}_A(M,N'),\quad \psi \mapsto g \circ \psi.$

A cornerstone of homological algebra is that $\text{Hom}_A(M,-)$ and $\text{Hom}_A(-,N)$ are left exact functors. This means that they preserve kernels but not necessarily cokernels – the failure of right exactness is precisely what the $Ext$ functors measure.

Proposition 4 (Left exactness of Hom). For any fixed $A$ -module $M$ , the functor $\text{Hom}_A(M,-)$ is left exact. That is, given a short exact sequence $0 \to N' \xrightarrow{\alpha} N \xrightarrow{\beta} N'' \to 0,$ the induced sequence $0 \to \text{Hom}_A(M,N') \xrightarrow{\alpha_*} \text{Hom}_A(M,N) \xrightarrow{\beta_*} \text{Hom}_A(M,N'')$ is exact. Similarly, for fixed $N$ , the contravariant functor $\text{Hom}_A(-,N)$ is left exact: from $0 \to M' \xrightarrow{f} M \xrightarrow{g} M'' \to 0$ we obtain $0 \to \text{Hom}_A(M'',N) \xrightarrow{g^*} \text{Hom}_A(M,N) \xrightarrow{f^*} \text{Hom}_A(M',N).$

The proof is a straightforward verification (see Vermani, Theorem 2.3.1). Why should we care? In Hochschild cohomology, the cochain complex is obtained by applying $\text{Hom}$ to a projective resolution. The left exactness of $\text{Hom}$ guarantees that the zeroth cohomology group $\text{Hom}_A(M,N)$ is exactly the set of maps that commute with the differentials – i.e., the module of “cycles” in degree zero. The higher cohomology groups then arise from the failure of $\text{Hom}(-,-)$ to be exact in higher degrees.

Example 28 (A non‑exact Hom). Consider the short exact sequence of $\mathbb{Z}$ -modules $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/2\mathbb{Z} \to 0.$ Applying $\text{Hom}_{\mathbb{Z}}(-,\mathbb{Z})$ we obtain $0 \to \text{Hom}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}) \to \text{Hom}(\mathbb{Z},\mathbb{Z}) \xrightarrow{2^*} \text{Hom}(\mathbb{Z},\mathbb{Z}) \to \cdots$ Now $\text{Hom}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}) = 0$ and $\text{Hom}(\mathbb{Z},\mathbb{Z}) \cong \mathbb{Z}$ . The map $2^*$ is multiplication by $2$ , which is injective but not surjective. Hence the sequence $0 \to \text{Hom}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}) \to \text{Hom}(\mathbb{Z},\mathbb{Z}) \xrightarrow{2^*} \text{Hom}(\mathbb{Z},\mathbb{Z})$ is exact at the first two terms, but the last map is not an epimorphism – it fails to be exact at the third term because $\text{Hom}(\mathbb{Z},\mathbb{Z}) / \text{Im}(2^*) \cong \mathbb{Z}/2\mathbb{Z}$ is non‑zero. This $\mathbb{Z}/2\mathbb{Z}$ is precisely $\text{Ext}_{\mathbb{Z}}^1(\mathbb{Z}/2\mathbb{Z},\mathbb{Z})$ , a first taste of derived functors.

Projective Modules

Projective modules are the “free modules without a basis”. They are the objects that behave like free modules with respect to lifting homomorphisms.

Definition 21. An $A$ -module $P$ is called projective if for every diagram of $A$ -modules $\begin{array}{c} P \\ \downarrow \exists ? \\ M \xrightarrow{\alpha} N \to 0 \end{array}$ with the bottom row exact (i.e., $\alpha$ surjective), and for every homomorphism $f: P \to N$ , there exists a homomorphism $g: P \to M$ such that $\alpha \circ g = f$ . In words: every homomorphism from $P$ onto a quotient of $M$ lifts to a homomorphism into $M$ .

This lifting property is often called the projective lifting property. The following equivalent characterisations are extremely useful.

Proposition 5. For an $A$ -module $P$ , the following are equivalent:

$P$ is projective.
Every short exact sequence $0 \to M \to N \to P \to 0$ splits.
$P$ is a direct summand of a free module: there exists a free module $F$ such that $F \cong P \oplus Q$ for some module $Q$ .
The functor $\text{Hom}_A(P,-)$ is exact (i.e., preserves epimorphisms).

The equivalence of (i) and (iv) is particularly illuminating: projective modules are precisely those for which $\text{Hom}$ is exact, so they behave like “free modules” in cohomological computations. The proof of these equivalences can be found in Vermani, Section 3.1.

Example 29 (Free modules are projective). Any free module is projective. Indeed, let $F = A^{\oplus I}$ be free with basis $\{e_i\}$ . Given a surjection $\alpha: M \to N$ and a map $f: F \to N$ , choose for each basis element $e_i$ a preimage $m_i \in M$ of $f(e_i)$ (possible because $\alpha$ is surjective). Extend linearly to a homomorphism $g: F \to M$ ; then $\alpha g = f$ . This is the standard argument, and it is the reason why projective resolutions exist: every module has a projective resolution because we can always resolve by free modules.

Example 30 (A projective module that is not free). Let $A = \mathbb{Z}/6\mathbb{Z}$ . The module $2\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/3\mathbb{Z}$ is projective because it is a direct summand of the free module $A = \mathbb{Z}/6\mathbb{Z} \cong 2\mathbb{Z}/6\mathbb{Z} \oplus 3\mathbb{Z}/6\mathbb{Z}$ (Chinese remainder theorem). Yet it is not free, because a free $\mathbb{Z}/6\mathbb{Z}$ -module would have order a power of $6$ , whereas the order of $2\mathbb{Z}/6\mathbb{Z}$ is $3$ . This example, although elementary, already shows that projectivity is a genuinely weaker condition than freeness.

In the context of Hochschild cohomology, we typically work over a field $k$ and consider algebras that are projective as $k$ -modules (e.g., all algebras are projective because every module over a field is free). So the subtlety of non‑free projectives does not appear there. However, when one considers algebras over more general rings (e.g., $\mathbb{Z}$ ), the distinction becomes important. I find it fascinating that the notion of projectivity, born from the desire to solve lifting problems, is exactly the right setting for constructing resolutions and that resolutions are precisely the engine that drives Hochschild (co)homology.

Remark 1 (Connection to Hochschild cohomology). The bar resolution of an algebra $A$ over a commutative ring $k$ is a projective resolution of $A$ as an $A$ -bimodule. This resolution is the starting point for defining Hochschild cohomology: $\text{HH}^n(A,M) = H^n\bigl( \text{Hom}_{A\otimes A^{\mathrm{op}}}( \mathrm{Bar}_\bullet(A), M ) \bigr).$ Because the bar modules $\mathrm{Bar}_n(A) = A^{\otimes (n+2)}$ are free over $A\otimes A^{\mathrm{op}}$ , they are in particular projective. Hence the existence of such a resolution is guaranteed, and the theory of derived functors applies directly. The projective lifting property ensures that any two projective resolutions are chain‑homotopy equivalent, so the cohomology groups do not depend on the choice of resolution.

§3.Homological Algebra

Category Theory Essentials

Category theory provides a unifying language for homological algebra. While not strictly necessary for the computational aspects of Hochschild cohomology, it clarifies the relationships between functors and explains why certain constructions are “natural”. In this subsection, we give only the basic and strictly necessary definitions, but one can safely skip to the next section without loss.

Definition 22. A category $\mathcal{C}$ consists of a class of objects, for each pair of objects $X,Y$ a set $\text{Hom}_{\mathcal{C}}(X,Y)$ of morphisms, a composition law that is associative, and an identity morphism $1_X \in \text{Hom}_{\mathcal{C}}(X,X)$ for each object $X$ . Morphisms are often drawn as arrows $f: X \to Y$ .

Examples abound. The category $\mathbf{Mod}_A$ of modules over a commutative ring $A$ has $A$ -modules as objects and $A$ -linear maps as morphisms. The category $\mathbf{Set}$ of sets, the category $\mathbf{Ab}$ of abelian groups, and the category $\mathbf{Top}$ of topological spaces are other familiar examples.

A functor is a map between categories that preserves structure. A covariant functor $F: \mathcal{C} \to \mathcal{D}$ sends objects to objects and morphisms $f: X \to Y$ to morphisms $F(f): F(X) \to F(Y)$ , respecting composition and identities. A contravariant functor reverses arrows: $F(f): F(Y) \to F(X)$ . For example, $\text{Hom}_A(M,-): \mathbf{Mod}_A \to \mathbf{Ab}$ is covariant, while $\text{Hom}_A(-,N)$ is contravariant.

Definition 23. Given two functors $F,G: \mathcal{C} \to \mathcal{D}$ , a natural transformation $\eta: F \Rightarrow G$ is a family of morphisms $\eta_X: F(X) \to G(X)$ such that for every $f: X \to Y$ the diagram $\begin{array}{ccc} F(X) & \xrightarrow{F(f)} & F(Y) \\ \downarrow \eta_X & & \downarrow \eta_Y \\ G(X) & \xrightarrow{G(f)} & G(Y) \end{array}$ commutes. If each $\eta_X$ is an isomorphism, $\eta$ is a natural isomorphism and we write $F \cong G$ .

Naturality is the precise way of saying that a construction is “independent of choices”. For instance, the isomorphism $A \otimes_A M \cong M$ is natural in $M$ . This will be important when we claim that different projective resolutions give the same derived functors up to natural isomorphism.

The Yoneda lemma (stated without proof) is a remarkable result of category theory. It says that for any object $X$ in a category $\mathcal{C}$ , the functor $\text{Hom}_{\mathcal{C}}(X,-)$ determines $X$ up to isomorphism. More concretely, natural transformations from $\text{Hom}_{\mathcal{C}}(X,-)$ to a functor $F$ correspond bijectively to elements of $F(X)$ . In this work we will not need the Yoneda lemma explicitly, but it underlies the philosophy that objects are determined by their morphisms.

Definition 24. Two functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ are adjoint (with $F$ left adjoint to $G$ ) if there exists a natural isomorphism $\text{Hom}_{\mathcal{D}}(F(X), Y) \cong \text{Hom}_{\mathcal{C}}(X, G(Y))$ for all objects $X \in \mathcal{C}$ , $Y \in \mathcal{D}$ .

The most important adjunction for us is the tensor–Hom adjunction: for modules over a commutative ring $A$ , $\text{Hom}_A(M \otimes_A N, L) \cong \text{Hom}_A(M, \text{Hom}_A(N, L)).$ This adjunction is used repeatedly when we discuss the bar resolution and when we identify Hochschild cohomology with $\text{Ext}$ over the enveloping algebra. The free–forgetful adjunction between sets and modules also appears when we construct free modules, but we will not need its categorical formulation.

Finally, an abelian category is a category in which one can do homological algebra: it has a zero object, all finite products and coproducts, every morphism has a kernel and a cokernel, and every monomorphism is a kernel of its cokernel while every epimorphism is a cokernel of its kernel. The category of modules over a ring is the prototypical example. The language of abelian categories allows one to state theorems about kernels, cokernels, exact sequences, and derived functors without ever referring to elements. For our purposes, working with modules over a commutative ring is sufficient, and we will rarely need the full machinery of abelian categories. However, knowing that $\mathbf{Mod}_A$ is abelian assures us that all the usual diagram lemmas (the five lemma, the snake lemma, etc.) hold.

Remark 2. This short excursion into category theory is not essential for the computations that follow. The reader who prefers a more concrete approach may safely ignore categories and think of “natural isomorphisms” as explicit formulas, of “adjunctions” as the fact that homomorphisms out of a tensor product correspond to bilinear maps, and of “abelian categories” as the statement that modules behave like abelian groups with extra structure. Nevertheless, the categorical viewpoint has permeated homological algebra, and a passing familiarity with its language will help in reading advanced texts. For deformation quantisation, the most categorical notion we use is the tensor–Hom adjunction, which we will invoke without fanfare.

Chain Complexes and Homology

The central objects of homological algebra are chain complexes. They encode linearised versions of algebraic structures, and their homology measures the obstruction to exactness.

Definition 25. A chain complex $(C_\bullet,\partial_\bullet)$ of modules over a commutative ring $A$ consists of a family $\{C_n\}_{n\in\mathbb{Z}}$ of $A$ -modules together with $A$ -linear maps $\partial_n: C_n \to C_{n-1}$ called differentials such that $\partial_{n-1} \circ \partial_n = 0$ for every $n$ . A cochain complex $(C^\bullet, d^\bullet)$ is defined similarly with differentials $d^n: C^n \to C^{n+1}$ satisfying $d^{n+1}\circ d^n = 0$ .

The condition $\partial^2 = 0$ implies that $\text{Im}\partial_{n+1} \subseteq \text{Ker}\partial_n$ . We define the module of $n$ -cycles as $Z_n(C) = \text{Ker}\partial_n$ and the module of $n$ -boundaries as $B_n(C) = \text{Im}\partial_{n+1}$ . The quotient $H_n(C) = Z_n(C) / B_n(C)$ is the $n$ th homology module of the complex. For a cochain complex we define cohomology $H^n(C) = \text{Ker}d^n / \text{Im}d^{n-1}$ .

A chain map $f_\bullet: C_\bullet \to D_\bullet$ is a family of homomorphisms $f_n: C_n \to D_n$ such that $f_{n-1}\partial_n^C = \partial_n^D f_n$ for all $n$ . Such a map induces homomorphisms $H_n(f): H_n(C) \to H_n(D)$ . Two chain maps are homotopic if there exist homomorphisms $h_n: C_n \to D_{n+1}$ satisfying $f_n - g_n = \partial_{n+1}^D h_n + h_{n-1} \partial_n^C$ . Homotopic maps induce the same homomorphism on homology.

Given a short exact sequence of complexes $0 \longrightarrow C_\bullet' \xrightarrow{f_\bullet} C_\bullet \xrightarrow{g_\bullet} C_\bullet'' \longrightarrow 0,$ there exists a canonical long exact homology sequence $\cdots \to H_n(C') \xrightarrow{H_n(f)} H_n(C) \xrightarrow{H_n(g)} H_n(C'') \xrightarrow{\delta_n} H_{n-1}(C') \to \cdots .$ The connecting homomorphism $\delta_n$ is defined by a diagram chase; its construction is the heart of many proofs in homological algebra. We will use this long exact sequence repeatedly, especially when we study derived functors.

Example 31 (The bar complex for Hochschild homology). Let $A$ be an algebra over a commutative ring $k$ . The bar complex $B_\bullet(A)$ is defined by $B_n(A) = A^{\otimes (n+2)}$ with differentials that sum over insertions of the unit. Its homology is the Hochschild homology $HH_\bullet(A)$ . The fact that $B_\bullet(A)$ is a chain complex (i.e., $\partial^2 = 0$ ) follows from a telescoping cancellation, and the long exact sequence associated to a short exact sequence of algebras will later give us powerful computational tools. This example will be central in the second part of this dissertation.

Projective and Injective Resolutions

To extract derived functors we need to replace a module by a complex of particularly nice modules – either projectives or injectives. These resolutions are the algebraic analogue of a “free cover” or an “injective hull”.

Definition 26. A projective resolution of an $A$ -module $M$ is an exact sequence $\cdots \to P_2 \to P_1 \to P_0 \to M \to 0$ where each $P_n$ is a projective $A$ -module. Dually, an injective resolution is an exact sequence $0 \to M \to I^0 \to I^1 \to I^2 \to \cdots$ with each $I^n$ injective.

Every module admits a projective resolution: one can take a free module on a set of generators, then a free module on the kernel, and so on. Similarly, every module can be embedded into an injective module (over a PID divisible modules are injective, and over a field every module is injective), so injective resolutions exist. For the category of modules over a commutative ring, these facts are proved using Zorn’s lemma or by constructing injective hulls (see Vermani, Sections 5.1 and 5.2).

Proposition 6 (Comparison theorem). Let $M$ and $N$ be modules, and let $f: M \to N$ be a homomorphism. Given a projective resolution $P_\bullet \to M$ and a resolution $Q_\bullet \to N$ (not necessarily projective), there exists a chain map $f_\bullet: P_\bullet \to Q_\bullet$ lifting $f$ . Any two such lifts are homotopic. An analogous statement holds for injective resolutions.

This theorem guarantees that, up to homotopy equivalence, the choice of resolution does not matter. Consequently, any functor that we apply to a resolution will yield well‑defined homology groups independent of the resolution.

Example 32 (The bar resolution as a projective resolution). For an algebra $A$ over a commutative ring $k$ , the augmented bar complex $\cdots \to A^{\otimes (n+2)} \to \cdots \to A \otimes A \to A \to 0$ (with the appropriate differentials) is a projective resolution of $A$ as an $A$ -bimodule. Here each $A^{\otimes (n+2)}$ is a free, hence projective, $A\otimes A^{\mathrm{op}}$ -module. This resolution will be our main tool for defining Hochschild (co)homology. Note that the injective resolution of the coefficient module is rarely needed in this construction because we always resolve the algebra itself, not the coefficient module.

§4. Direct Sums, Tensor Products and Flat Modules

Before we introduce derived functors, we need to recall two fundamental constructions: direct sums and tensor products. Direct sums are the categorical coproduct in module categories; tensor products allow us to “multiply” modules and are essential for the bar resolution. Flat modules are those for which tensoring preserves exactness, and they appear naturally when studying Tor.

Definition 27. Let $\{M_i\}_{i\in I}$ be a family of $A$ -modules. Their direct sum $\bigoplus_{i\in I} M_i$ is the set of all tuples $(x_i)$ with $x_i\in M_i$ and only finitely many non‑zero components, endowed with componentwise addition and scalar multiplication. The direct product $\prod_{i\in I} M_i$ allows arbitrarily many non‑zero components.

For finite index sets the direct sum and direct product coincide. Direct sums are characterised by a universal property: given a family of homomorphisms $f_i: M_i \to N$ , there exists a unique homomorphism $\bigoplus M_i \to N$ extending them. This property will be used implicitly when we construct projective resolutions from free modules.

The tensor product is a more subtle construction. Given a right $A$ -module $M$ and a left $A$ -module $N$ , their tensor product $M \otimes_A N$ is an abelian group generated by symbols $m \otimes n$ subject to the bilinearity relations $(m_1+m_2)\otimes n = m_1\otimes n + m_2\otimes n,\quad m\otimes (n_1+n_2) = m\otimes n_1 + m\otimes n_2,\quad (ma)\otimes n = m\otimes (an).$ When $A$ is commutative, $M\otimes_A N$ becomes an $A$ -module via $a(m\otimes n)= (am)\otimes n = m\otimes (an)$ . We will always work over a commutative ground ring $k$ , so tensor products are taken over $k$ unless otherwise specified. For an algebra $A$ over $k$ , the enveloping algebra $A^e = A\otimes_k A^{\mathrm{op}}$ appears naturally, and the tensor product over $k$ is used to build the bar resolution.

The functor $-\otimes_A N$ is right exact but not exact in general. Modules for which it is exact deserve a special name.

Definition 28. An $A$ -module $F$ is called flat if the functor $-\otimes_A F$ is exact. Equivalently, for every injective homomorphism $M\to N$ , the induced map $M\otimes_A F \to N\otimes_A F$ is injective.

Over a field every module is flat because tensor products are exact over a field (indeed, $k$ -vector spaces are free, hence flat). Over a principal ideal domain, flat modules are precisely the torsion‑free modules. For Hochschild cohomology over a field $k$ , flatness is automatic – a comforting simplification. However, when one considers algebras over more general rings (e.g., $\mathbb{Z}$ ), flatness becomes a genuine restriction.

Example 33 (Flat modules and the bar resolution). Let $k$ be a field. The bar resolution $B_\bullet(A)$ uses tensor powers $A^{\otimes_k n}$ , each of which is a flat $k$ -module because every $k$ -module is flat. Moreover, when we later consider the Hochschild homology $HH_\bullet(A,M) = \text{Tor}_\bullet^{A^e}(A,M)$ , the flatness of the modules in the resolution is not required – we need projectivity. But the fact that tensoring over $k$ is exact simplifies many computations: for example, the Hochschild–Kostant–Rosenberg theorem for smooth commutative algebras relies on the fact that $k$ is a field.

Remark 3. The direct sum and direct product will appear when we construct free modules and when we discuss the Eilenberg–Zilber theorem for the bar complex (not needed here). Tensor products are everywhere: the enveloping algebra is a tensor product, the bar modules are tensor powers, and the differentials involve alternating sums of insertions. Flatness, although not strictly necessary over a field, is mentioned here for completeness. If you ever work over $\mathbb{Z}$ (e.g., integral group cohomology), flat modules become important. For our deformation quantisation over $\mathbb{C}[[\hbar]]$ , the ring is a field when reduced modulo $\hbar$ , but as a formal power series ring it is a local principal ideal domain, so flatness still holds for torsion‑free modules – but we will not need to dwell on it.

With these constructions in hand, we are ready to build projective resolutions and define derived functors. The tensor–Hom adjunction $\text{Hom}_A(M\otimes_A N, L) \cong \text{Hom}_A(M, \text{Hom}_A(N,L))$ will be used implicitly when we discuss the bar resolution’s contracting homotopy, but the reader need not memorise the categorical language; the algebraic formulas will guide us.

§5. Derived Functors: Ext and Tor

Derived functors measure the failure of an additive functor to be exact. They are defined by applying the functor to a resolution and then taking homology.

Definition 29. Let $F: A\text{-}\mathbf{Mod} \to \mathbf{Ab}$ be an additive covariant functor.

If $F$ is right exact, its $n$ th left derived functor $L_nF$ is defined as follows. Take a projective resolution $P_\bullet \to M$ . Apply $F$ to obtain a complex $F(P_\bullet)$ and set $L_nF(M) = H_n(F(P_\bullet))$ .
If $F$ is left exact, its $n$ th right derived functor $R^nF$ is defined using an injective resolution $M \to I^\bullet$ and setting $R^nF(M) = H^n(F(I^\bullet))$ .

The comparison theorem guarantees that these definitions are independent of the chosen resolution, and the resulting functors are additive.

The most important derived functors in this work are $\text{Ext}$ and $\text{Tor}$ . For modules $M,N$ over a commutative ring $A$ we set $\text{Ext}_A^n(M,N) = R^n\bigl(\text{Hom}_A(M,-)\bigr)(N) = R^n\bigl(\text{Hom}_A(-,N)\bigr)(M),$ $\text{Tor}_n^A(M,N) = L_n\bigl(M\otimes_A -\bigr)(N) = L_n\bigl(-\otimes_A N\bigr)(M).$ In words, $\text{Ext}^n$ is the $n$ th right derived functor of $\text{Hom}$ in either variable, and $\text{Tor}_n$ is the $n$ th left derived functor of the tensor product. These functors satisfy $\text{Ext}^0 \cong \text{Hom}$ and $\text{Tor}_0 \cong \otimes$ .

Given a short exact sequence $0 \to M' \to M \to M'' \to 0$ , we obtain long exact sequences $\cdots \to \text{Ext}^n_A(M'',N) \to \text{Ext}^n_A(M,N) \to \text{Ext}^n_A(M',N) \to \text{Ext}^{n+1}_A(M'',N) \to \cdots$ and similarly for $\text{Tor}$ . These long exact sequences are the workhorses of homological algebra. For Hochschild (co)homology, we will apply these ideas to the enveloping algebra $A^e = A \otimes A^{\mathrm{op}}$ . Indeed, $\text{HH}^n(A,M) = \text{Ext}^n_{A^e}(A,M), \qquad \text{HH}_n(A,M) = \text{Tor}_n^{A^e}(A,M).$ Thus all the machinery of derived functors is directly available.

Example 34 (First Hochschild cohomology group). For an algebra $A$ over a field $k$ , $\text{HH}^1(A,A) = \text{Der}(A,A) / \text{InnDer}(A,A)$ , the space of derivations modulo inner derivations. This is a special case of $\text{Ext}^1_{A^e}(A,A)$ . The long exact sequence of $\text{Ext}$ for a short exact sequence of algebras can sometimes be used to compute this space, for instance when deforming a polynomial algebra to a non‑commutative algebra.

Remark 4. The theory of derived functors can be developed in any abelian category with enough projectives or injectives. However, for our purposes the category of modules over a commutative ring suffices. The categorical language (functors, natural transformations, adjunctions) is not strictly necessary, but the tensor‑Hom adjunction $\text{Hom}_A(M\otimes_A N, L) \cong \text{Hom}_A(M, \text{Hom}_A(N,L))$ will be used implicitly when we discuss the bar resolution. The reader comfortable with basic category theory may recognise this adjunction; those who are not may simply accept the algebraic definitions, which is perfectly adequate for this dissertation.

This concludes the necessary background in homological algebra. In the next section we will specialise to the bar resolution and define Hochschild homology and cohomology, then apply these concepts to deformation quantisation.

§4.Hochschild (Co)Homology

The Bar Complex and Hochschild Homology

In this subsection we work over a field $k$ (or more generally a commutative ring, but a field suffices for our purposes). Let $A$ be a $k$ -algebra. We denote by $A^{\otimes n}$ the $n$ -fold tensor product over $k$ , which carries a natural $A$ -bimodule structure. Define for each $n\ge 0$ the $A$ -bimodule $B_n = A^{\otimes (n+2)}$ . The differential $d_n : B_n \to B_{n-1}$ (for $n\ge 1$ ) is given on elementary tensors by

$d_n(a_0\otimes a_1\otimes\cdots\otimes a_{n+1}) = \sum_{i=0}^{n} (-1)^i\, a_0\otimes\cdots\otimes a_i a_{i+1}\otimes\cdots\otimes a_{n+1}.$

For $n=0$ we set $d_0 = 0$ . The multiplication map $\pi : A\otimes A \to A,\; \pi(a\otimes b)=ab$ gives an augmentation. The sequence

$\cdots \xrightarrow{d_3} A^{\otimes 4} \xrightarrow{d_2} A^{\otimes 3} \xrightarrow{d_1} A^{\otimes 2} \xrightarrow{\pi} A \to 0 \tag{1}$

is called the bar complex (or bar resolution) of $A$ as an $A$ -bimodule.

Proposition 7. The maps $d_n$ satisfy $d_{n-1}\circ d_n = 0$ for all $n\ge 2$ ; hence (1) is a chain complex.

Proof. A direct computation shows that applying two consecutive differentials produces each term twice with opposite signs. Indeed, $d_{n-1}d_n(a_0\otimes\cdots\otimes a_{n+1}) = \sum_{i=0}^{n}\sum_{j=0}^{n-1} (-1)^{i+j} \bigl[\text{multiply at }i\text{ then at }j\bigr].$ If $j < i$ the two multiplications act on disjoint adjacent pairs; if $j \ge i$ the same final expression appears with sign $(-1)^{i+j+1}$ . Hence all contributions cancel pairwise, so $d_{n-1}d_n = 0$ . (For a detailed sign analysis see e.g. .) ◻

It can be shown that (1) is acyclic. For that, one notes that $\ker\pi = \text{im}d_1$ and the homology in positive degrees vanishes. Moreover, if $A$ is free as a $k$ -module, each $B_n$ is a free $A^e$ -module, where $A^e = A\otimes_k A^{\mathrm{op}}$ is the enveloping algebra. Consequently, (1) is a free (hence projective) resolution of the $A^e$ -module $A$ . Now, let $M$ be an $A$ -bimodule. Tensoring the bar resolution (1) with $M$ over $A^e$ gives a chain complex of $k$ -vector spaces: $C_\bullet(A,M) = M\otimes_{A^e} B_\bullet(A).$ Using the natural isomorphism $M\otimes_{A^e} A^{\otimes (n+2)} \;\cong\; M\otimes_k A^{\otimes n}, \qquad m\otimes a_0\otimes\cdots\otimes a_{n+1} \;\longmapsto\; a_{n+1} m a_0 \otimes a_1\otimes\cdots\otimes a_n,$ we obtain an explicit description: $C_n(A,M) = M\otimes_k A^{\otimes n}$ , $n\ge 0,$ with differential $\partial_n : C_n(A,M) \to C_{n-1}(A,M)$ given by $\begin{aligned} \partial_n(m\otimes a_1\otimes\cdots\otimes a_n) =&\; m a_1\otimes a_2\otimes\cdots\otimes a_n \\ &+ \sum_{i=1}^{n-1} (-1)^i\, m\otimes a_1\otimes\cdots\otimes a_i a_{i+1}\otimes\cdots\otimes a_n \\ &+ (-1)^n a_n m\otimes a_1\otimes\cdots\otimes a_{n-1}. \end{aligned}$ The homology of this complex is the Hochschild homology of $A$ with coefficients in $M$ : $HH_n(A,M) := H_n ( M \otimes A^{\otimes \bullet}) = \text{Tor}^{A^e}_n (M, A),\qquad n\ge 0,$ where we set $\partial_0 \equiv 0$ . In other words, $HH_n(A,M) = \ker \partial_n / \text{Im} \partial_{n + 1}$ . Elements of $\ker \partial_n$ are Hochschild $n$ -cycles and elements of $\text{Im}\partial_{n+1}$ are Hochschild $n$ -boundaries. Next, we present three classical examples. In each case we take the coefficient bimodule to be $A$ itself, and write $\text{HH}_n(A)=\text{HH}_n(A,A)$ .

Example 35 (Polynomial ring). Let $A=k[x]$ , viewed as an associative $k$ -algebra. A free resolution of $A$ as an $A^e$ -module is $0 \longrightarrow A^e \xrightarrow{\,u\,} A^e \xrightarrow{\pi} A \longrightarrow 0,$ where $u(a\otimes b) = ax\otimes b-a\otimes xb.$

Applying the functor $A\otimes_{A^e}(-)$ gives $0 \longrightarrow A\otimes_{A^e}A^e \xrightarrow{\,\bar u\,} A\otimes_{A^e}A^e \longrightarrow 0.$

Using the canonical isomorphism $A\otimes_{A^e}A^e \cong A, \qquad a\otimes (b\otimes c)\mapsto c a b,$ the induced differential becomes $\bar u(a) = xa-ax.$

Hence the Hochschild complex is $0 \longrightarrow A \xrightarrow{\,a\mapsto xa-ax\,} A \longrightarrow 0.$

Therefore $\mathrm{HH}_0(A)\cong A/[x,A], \qquad \mathrm{HH}_1(A)\cong \{a\in A\mid xa=ax\}, \qquad \mathrm{HH}_n(A)=0 \;\;(n\ge2).$

If $A$ is moreover commutative, then $xa-ax=0$ for all $a\in A$ . Thus the differential vanishes and $\mathrm{HH}_0(k[x])\cong k[x], \qquad \mathrm{HH}_1(k[x])\cong k[x], \qquad \mathrm{HH}_n(k[x])=0 \;\;(n\ge2).$

Example 36 (Truncated polynomial ring). Let $A=k[x]/(x^n),$ viewed as an associative $k$ -algebra. A periodic free resolution of $A$ as an $A^e$ -module is $\cdots \xrightarrow{v} A^e \xrightarrow{u} A^e \xrightarrow{v} A^e \xrightarrow{u} A^e \xrightarrow{\pi} A \longrightarrow0,$ where $u(a\otimes b) = ax\otimes b-a\otimes xb,$ and $v(1\otimes1) = \sum_{i=0}^{n-1}x^i\otimes x^{n-1-i}.$

Applying $A\otimes_{A^e}(-)$ and identifying $A\otimes_{A^e}A^e\cong A,$ we obtain the complex $\cdots \xrightarrow{\bar v} A \xrightarrow{\bar u} A \xrightarrow{\bar v} A \xrightarrow{\bar u} A \longrightarrow0.$

The induced maps are $\bar u(a)=xa-ax,$ and $\bar v(a) = \sum_{i=0}^{n-1}x^{n-1-i}ax^i.$

Hence $\mathrm{HH}_0(A)\cong A/\mathrm{Im}(\bar u),$ and for $m\ge0$ , $\mathrm{HH}_{2m+1}(A) \cong \ker(\bar u)/\mathrm{Im}(\bar v),$ $\mathrm{HH}_{2m+2}(A) \cong \ker(\bar v)/\mathrm{Im}(\bar u).$

If $A$ is moreover commutative, then $\bar u=0,$ and $\bar v(a) = \sum_{i=0}^{n-1}x^{n-1}a = n x^{n-1}a.$

Thus the complex becomes $\cdots \xrightarrow{\,n x^{n-1}\,} A \xrightarrow{0} A \xrightarrow{\,n x^{n-1}\,} A \xrightarrow{0} A \longrightarrow0.$

If $\operatorname{char}(k)\nmid n$ , then multiplication by $n x^{n-1}$ has kernel $(x)/(x^{n-1})$ and cokernel $A/(x^{n-1}).$ Therefore $\mathrm{HH}_0(A)\cong A,$ $\mathrm{HH}_{2m}(A)\cong A/(x^{n-1}) \qquad (m\ge1),$ $\mathrm{HH}_{2m+1}(A)\cong (x)/(x^{n-1}) \qquad (m\ge0).$
If $\operatorname{char}(k)\mid n$ , then $n x^{n-1}=0,$ so every differential vanishes. Consequently $\mathrm{HH}_0(A)\cong A, \qquad \mathrm{HH}_m(A)\cong A \qquad (m\ge1).$

Example 37 (Separable algebra). Let $A=k\times k$ , viewed as an associative $k$ -algebra. Since $A$ is separable over $k$ , it is projective as an $A^e$ -module. Hence $\operatorname{Tor}_n^{A^e}(A,A)=0 \qquad (n\ge1),$ and therefore $\mathrm{HH}_n(A)=0 \qquad (n\ge1).$

Moreover, $\mathrm{HH}_0(A)\cong A/[A,A].$

Since $A=k\times k$ is commutative, $[A,A]=0,$ so $\mathrm{HH}_0(k\times k)\cong k\times k, \qquad \mathrm{HH}_n(k\times k)=0 \qquad (n\ge1).$

More generally, for a separable associative algebra $A$ , $\mathrm{HH}_0(A)\cong A/[A,A], \qquad \mathrm{HH}_n(A)=0 \qquad (n\ge1).$

§2. Hochschild Cohomology

By applying the contravariant functor $\text{Hom}_{A^e}(-,M)$ to the bar resolution (1) yields the cochain complex of Hochschild cochains: $0 \longrightarrow \text{Hom}_{A^e}(A^{\otimes 2}, M) \xrightarrow{d_1^*} \text{Hom}_{A^e}(A^{\otimes 3}, M) \xrightarrow{d_2^*} \text{Hom}_{A^e}(A^{\otimes 4}, M) \xrightarrow{d_3^*} \cdots$ where $d_n^*(f) = f \circ d_{n+1}$ for $f \in \text{Hom}_{A^e}(A^{\otimes (n+2)}, M)$ . It produces a cochain complex $C^*(A,M) \;=\; \bigoplus_{n\ge 0} \text{Hom}_{A^e}\bigl(A^{\otimes (n+2)},\, M\bigr),$ with differential $\delta^n : \text{Hom}_{A^e}(A^{\otimes (n+2)},M) \to \text{Hom}_{A^e}(A^{\otimes (n+3)},M)$ defined by $(\delta^n f)(x) = f(d_{n+1}(x)) \qquad (x\in A^{\otimes (n+3)}).$ In summary, we just took the elements of the chain complex and summed them. There is a canonical $k$ -module isomorphism $\text{Hom}_{A^e}(A^{\otimes (n+2)}, M) \;\cong\; \text{Hom}_k(A^{\otimes n}, M),$ given by $g \;\longmapsto\; \bigl( a_1\otimes\cdots\otimes a_n \;\mapsto\; g(1\otimes a_1\otimes\cdots\otimes a_n\otimes 1) \bigr).$ The inverse sends a $k$ -linear map $h: A^{\otimes n}\to M$ to the $A^e$ -homomorphism $a_0\otimes\cdots\otimes a_{n+1} \;\longmapsto\; a_0\cdot h(a_1\otimes\cdots\otimes a_n)\cdot a_{n+1}.$

Under this identification, the differential $d_n^*$ becomes the explicit Hochschild coboundary operator (still denoted $d_n^*$ by abuse of notation): $\begin{aligned}\label{differentilhoch} (d_n^* h)(a_1\otimes\cdots\otimes a_{n+1}) &= a_1 h(a_2\otimes\cdots\otimes a_{n+1}) \\ &\quad + \sum_{i=1}^{n} (-1)^i h(a_1\otimes\cdots\otimes a_i a_{i+1}\otimes\cdots\otimes a_{n+1}) \\ &\quad + (-1)^{n+1} h(a_1\otimes\cdots\otimes a_n) a_{n+1}, \end{aligned}$ for all $h\in\text{Hom}_k(A^{\otimes n}, M)$ and $a_1,\dots,a_{n+1}\in A$ . (For $n=0$ we interpret $A^{\otimes 0}=k$ and the formula reduces to $(d_0^* m)(a)=am-ma$ .) The cohomology of this cochain complex is the Hochschild cohomology of $A$ with coefficients in $M$ : $HH^n(A,M) := H^n\bigl(\text{Hom}_k(A^{\otimes \bullet}, M), d^*\bigr) = \text{Ext}_{A^e}^n(A,M).$ As in the previous subsection, we now present several examples. These consist in the computation of the Hochschild cohomology groups $HH^n$ for the same three algebras considered in the preceding subsection.

Example 38 (Polynomial ring / Commutative Free Algebra). Let $A$ be a commutative algebra. In the specific case of the polynomial ring $A = k[x]$ , we use the standard resolution. Applying the functor $\mathrm{Hom}_{A^e}(-,A)$ and utilizing the isomorphism (which holds generally for any algebra) $\mathrm{Hom}_{A^e}(A^e,A)\cong A,$ yields the cochain complex $0 \longrightarrow A \xrightarrow{0} A \longrightarrow 0,$ where the differential vanishes precisely because $A$ is commutative (meaning the left and right actions of $A$ on itself coincide, causing the boundary maps $a \cdot x - x \cdot a$ to become zero).

Thus, for the polynomial ring $A = k[x]$ : $\mathrm{HH}^0(k[x])\cong k[x],\quad \mathrm{HH}^1(k[x])\cong k[x],\quad \mathrm{HH}^n(k[x])=0 \quad (n\ge 2).$ More generally, if $A$ is any smooth commutative $k$ -algebra, the Hochschild cohomology decomposes via the Hochschild-Kostant-Rosenberg (HKR) theorem into exterior powers of its derivations: $\mathrm{HH}^n(A) \cong \Omega^n_{A/k} \cong \mathrm{Der}_k(A, A)$ , as we will see later on.

Example 39 (Truncated polynomial ring). Let $A = k[x]/(x^n)$ . Since $A$ is a commutative algebra, the alternated commutator differentials in the resolution simplify. Applying $\mathrm{Hom}_{A^e}(-,A)$ to the periodic resolution yields the cochain complex $0 \longrightarrow A \xrightarrow{0} A \xrightarrow{n x^{n-1}} A \xrightarrow{0} A \xrightarrow{n x^{n-1}} \cdots.$ Because $A$ is commutative, the map induced by the norm element simplifies directly to multiplication by the derivative $n x^{n-1}$ .

Hence, we obtain the following cases based on the characteristic of the base field $k$ :

If $\operatorname{char}(k)\nmid n$ , then $\mathrm{HH}^0(A)\cong A,\quad \mathrm{HH}^{2m}(A)\cong A/(x^{n-1}) \quad (m\ge 1),\quad \mathrm{HH}^{2m+1}(A)\cong (x)/(x^{n-1}) \quad (m\ge 0).$
If $\operatorname{char}(k)\mid n$ , then the scalar $n$ vanishes in $k$ . Consequently, all differentials in the complex vanish, and therefore $\mathrm{HH}^0(A)\cong A,\qquad \mathrm{HH}^m(A)\cong A \quad (m\ge 1).$

Example 40 (Separable algebra $A=k\times k$ ). Let $A = k \times k$ , which is a semisimple, finite-dimensional commutative algebra. Since $A$ is separable, its Hochschild homology and cohomology vanish in higher degrees: $\mathrm{Ext}^{n}_{A^e}(A,A)=0 \qquad (n\ge 1).$

Moreover, because $A$ is a commutative algebra, it is equal to its own center ( $A = Z(A)$ ). Thus, the degree-zero Hochschild cohomology is simply: $\mathrm{HH}^0(k\times k)\cong Z(k\times k)\cong k\times k,$ while $\mathrm{HH}^n(k\times k)=0 \qquad (n\ge 1).$

§3. A geometric construction of Hochschild cohomology

The following geometric viewpoint interprets Hochschild cohomology as the cohomology of the tangent complex to the moduli space of associative algebra structures on a fixed vector space. Let $k$ be a field of characteristic zero (the discussion generalises to characteristic free settings with care). Fix a finite‑dimensional $k$ -vector space $A$ . The space of all $k$ -bilinear products $\mu : A \otimes_k A \to A$ is the vector space $\operatorname{Bilin}(A) = \operatorname{Hom}_k(A^{\otimes 2}, A) \cong (A^* \otimes A^*) \otimes A.$ Inside it, the subset of associative products is $\operatorname{Ass}(A) = \bigl\{ \mu \in \operatorname{Bilin}(A) \mid \mu(\mu(a,b),c) = \mu(a,\mu(b,c))\ \forall a,b,c\in A \bigr\}.$ Define the associativity map $F : \operatorname{Bilin}(A) \longrightarrow \operatorname{Tern}(A),\qquad F(\mu)(a,b,c) = \mu(\mu(a,b),c) - \mu(a,\mu(b,c)),$ where $\operatorname{Tern}(A)=\operatorname{Hom}_k(A^{\otimes 3}, A)$ . Clearly $\operatorname{Ass}(A) = F^{-1}(0)$ . The map $F$ is quadratic (hence smooth), and its derivative at a point $\mu_0\in\operatorname{Ass}(A)$ is $(dF_{\mu_0}\varphi)(a,b,c) = \mu_0(a,\varphi(b,c)) + \varphi(a,\mu_0(b,c)) - \varphi(\mu_0(a,b),c) - \mu_0(\varphi(a,b),c).$ If we identify a bilinear map $\varphi$ with a $2$ -cochain in the Hochschild complex $C^2(A,A)$ , the right‑hand side is precisely the Hochschild coboundary $\delta^2\varphi$ evaluated on $(a,b,c)$ . Hence $dF_{\mu_0} = \delta^2 : C^2(A,A) \longrightarrow C^3(A,A).$

The group $\operatorname{GL}(A)$ acts on $\operatorname{Bilin}(A)$ by $(g\cdot\mu)(a,b)=g\bigl(\mu(g^{-1}a,g^{-1}b)\bigr)$ ; this action preserves $\operatorname{Ass}(A)$ . The tangent space to the orbit of $\mu_0$ at $\mu_0$ consists of the images of derivations: $T_{\mu_0}(\operatorname{GL}(A)\cdot\mu_0) = \bigl\{ \delta^1 D \mid D\in\operatorname{Der}(A) \bigr\} = B^2(A,A) \subseteq C^2(A,A),$ where $\delta^1$ is the Hochschild differential $C^1\to C^2$ . Consequently, the tangent space to the moduli space $\operatorname{Ass}(A)/\operatorname{GL}(A)$ at the point $[\mu_0]$ is $T_{[\mu_0]}\bigl(\operatorname{Ass}(A)/\operatorname{GL}(A)\bigr) \cong \frac{\ker(dF_{\mu_0})}{\operatorname{Im}(\delta^1)} = \frac{Z^2(A,A)}{B^2(A,A)} = \mathrm{HH}^2(A,A).$

Now consider the exact sequence of tangent spaces induced by the orbit‑map sequence: $0 \longrightarrow \operatorname{Der}(A) \xrightarrow{\;\delta^1\;} \operatorname{Bilin}(A) \xrightarrow{\;dF_{\mu_0}\;} \operatorname{Tern}(A) \longrightarrow \operatorname{Coker}(dF_{\mu_0}) \longrightarrow 0.$ A geometric version of the Inverse Function Theorem (or implicit function theorem for varieties) tells us that the cokernel of $dF_{\mu_0}$ is the obstruction space for lifting a first‑order deformation to a formal one. This cokernel is precisely $\mathrm{HH}^3(A,A)$ because $\operatorname{Coker}(dF_{\mu_0}) = \frac{C^3(A,A)}{\operatorname{Im}(\delta^2)} = \mathrm{HH}^3(A,A).$

Thus the low‑degree Hochschild cohomology groups acquire a natural geometric meaning:

$\mathrm{HH}^1(A,A) \cong \operatorname{Der}(A)/\operatorname{InnDer}(A)$ is the space of infinitesimal automorphisms of the algebra (tangent space to the orbit of $\operatorname{GL}(A)$ modulo inner derivations).
$\mathrm{HH}^2(A,A)$ is the tangent space to the moduli space of associative structures at $[\mu_0]$ ; it classifies infinitesimal deformations of the algebra.
$\mathrm{HH}^3(A,A)$ appears as the obstruction space: a first‑order deformation can be extended to a formal deformation if and only if its associated obstruction class in $\mathrm{HH}^3(A,A)$ vanishes.

The whole Hochschild complex can be seen as the linearisation of the infinite‑dimensional map that encodes all higher associativity constraints; this is the core of the deformation theory via differential graded Lie algebras . For a detailed exposition of the geometric inverse function theorem in this context, see . The classical paper of Gerstenhaber remains the foundational reference, while places the construction in the broader framework of formal geometry. provide complementary textbook accounts on other contexts.

§5.The Hochschild Complex of Cochains as a DGLA

The Gerstenhaber Bracket and Graded Lie Algebra Structure

The purpose of this section, strongly based on Witherspoon's work, is to present the Gerstenhaber bracket and to introduce a DGLA structure on the Hochschild complex of cochains. As we defined in the previous sections, for a $k$ -algebra $A$ and an $A$ -bimodule $M$ , the corresponding Hochschild cohomology is $\begin{equation*} HH^n(A,M) = H^n (\operatorname{Hom}(A^{\otimes \bullet}, A), d^*) \end{equation*}$ where, if we take $M = A$ , can be rewritten as $\begin{equation} HH^\bullet (A) = \bigoplus_{n \geq 0} HH^n(A) \end{equation}$ which characterizes Hochschild cohomology as a graded $k$ -module.

Remark 5. The Hochschild complex of cochains has a natural $\mathbb{Z}$ -graded vector space structure.

Definition 31. Let $f \in \operatorname{Hom} (A^{\otimes m}, A)$ and $g \in \operatorname{Hom} (A^{\otimes n}, A)$ . The cup product $f \smile g$ is the element of $\operatorname{Hom}(A^{\otimes (m + n)}, A)$ defined as $\begin{equation} (f \smile g)(a_1 \otimes \dots \otimes a_{n+m}) = (-1)^{mn} f(a_1 \otimes \dots \otimes a_m) g(a_{m+1} \otimes \dots \otimes a_{m + n}). \end{equation}$

This map induces a graded associative product on Hochschild cohomology, denoted, by abuse of notation, as $\begin{equation} \smile: HH^n(A) \times HH^m(A) \rightarrow HH^{m+n}(A). \end{equation}$

With that, we can now introduce a second binary operation on the Hochschild cochain complex, known as the Gerstenhaber bracket. While the cup product endows $\mathrm{HH}^\bullet(A)$ with an associative graded commutative algebra structure, the Gerstenhaber bracket endows the shifted Hochschild cochain complex with a graded Lie algebra structure. Together, these operations form a Gerstenhaber algebra.

However, before we formally introduce the bracket, we must define an auxiliary operation, called the circle product, which generalizes the composition of functions.

Definition 32 (Circle Product). Let $f \in C^m(A)$ and $g \in C^n(A)$ . Define their circle product $f \circ g \in C^{m+n-1}(A)$ by $\begin{aligned} &(f \circ g)(a_1 \otimes \cdots \otimes a_{m+n-1}) \\ &= \sum_{i=1}^{m} (-1)^{(n-1)(i-1)} f\big( a_1 \otimes \cdots \otimes a_{i-1} \otimes g(a_i \otimes \cdots \otimes a_{i+n-1}) \otimes a_{i+n} \otimes \cdots \otimes a_{m+n-1} \big), \end{aligned}$ for all $a_1, \dots, a_{m+n-1} \in A$ . If $m=0$ , we set $f \circ g = 0$ . If $n=0$ , the formula is interpreted by taking the scalar $g(1)$ in place of $g(a_i \otimes \cdots \otimes a_{i+n-1})$ .

In particular, we have $C^n(A) := \operatorname{Hom}_k (A^{\otimes n}, A)$ . Furthermore, the circle product, in essence, inserts the values of $g$ into the arguments of $f$ in all possible positions.

Definition 33 (Gerstenhaber Bracket). Let $f \in C^m(A)$ and $g \in C^n(A)$ . Their Gerstenhaber bracket is the graded commutator of the circle product: $\boxed{ [f, g] := f \circ g - (-1)^{(m-1)(n-1)} g \circ f. }$ This defines an element of $C^{m+n-1}(A)$ .

In a DGLA $(\mathfrak{g}, d, [\cdot, \cdot])$ , the bracket is a map of the form $[\cdot, \cdot]: \mathfrak{g}^p \times \mathfrak{g}^q \rightarrow \mathfrak{g}^{p+q}$ , so we apply a shift in the degree of the elements of $C^\bullet(A)$ , so that $\begin{equation} [-,-]_G: C^\alpha (A) \times C^\beta (A)\rightarrow C^{\alpha + \beta} (A), \end{equation}$ with $\alpha = m - 1$ and $\beta = n - 1$ . Moreover, and in the same spirit, to properly articulate the compatibility of the bracket with the differential, we introduce a sign-modified version of the standard Hochschild differential.

Definition 34 (Shifted Differential). Recall the standard Hochschild coboundary map $d^*_m: C^m(A) \to C^{m+1}(A)$ . For an $m$ -cochain $f \in C^m(A)$ , define a new differential $\partial$ by $\partial(f) := (-1)^{m-1} d^*_m(f).$

Remark 6. It is crucial to observe that $\partial$ is not a new cohomology theory. Since $\partial(f)$ is the standard differential $d^*_m(f)$ multiplied by a non-zero scalar $(-1)^{m-1}$ , we have $\operatorname{Ker}(\partial) = \operatorname{Ker}(d^*), \qquad \operatorname{Im}(\partial) = \operatorname{Im}(d^*).$ Consequently, the cohomology of the complex $(C^*(A), \partial)$ is precisely the usual Hochschild cohomology: $H^*\big(C^\bullet(A), \partial\big) = \mathrm{HH}^\bullet(A).$ The sole purpose of this sign modification is to ensure that the Gerstenhaber bracket satisfies the graded Leibniz rule without extra sign artifacts.

We now state the fundamental properties of the Gerstenhaber bracket. The following lemmas are classical results due to Gerstenhaber .

Lemma 35 (Graded Lie Algebra Properties). Let $f \in C^m(A)$ , $g \in C^n(A)$ , and $h \in C^p(A)$ . The Gerstenhaber bracket satisfies the following identities:

Graded Anticommutativity: $[f, g] = -(-1)^{(m-1)(n-1)} [g, f].$
Graded Jacobi Identity: $(-1)^{(m-1)(p-1)} [f, [g, h]] + (-1)^{(n-1)(m-1)} [g, [h, f]] + (-1)^{(p-1)(n-1)} [h, [f, g]] = 0.$
Derivation Property for the Differential: With the shifted differential $\partial$ defined above, we have $\partial([f, g]) = [\partial(f), g] + (-1)^{m-1} [f, \partial(g)].$

Consequently, the shifted Hochschild cochain complex $\big(C^{\bullet+1}(A), \partial, [-, -]\big)$ is a differential graded Lie algebra (DGLA). Next, we state the crucial compatibility between the Gerstenhaber bracket and the cup product.

Lemma 36 (Derivation with Respect to Cup Product). Let $\alpha \in \mathrm{HH}^m(A)$ , $\beta \in \mathrm{HH}^n(A)$ , and $\gamma \in \mathrm{HH}^p(A)$ . On cohomology, the bracket acts as a graded derivation of the cup product: $[\gamma, \alpha \smile \beta] = [\gamma, \alpha] \smile \beta + (-1)^{m(p-1)} \alpha \smile [\gamma, \beta].$

Combining these properties yields the defining structure of a Gerstenhaber algebra.

Theorem 37 (Gerstenhaber Algebra Structure). Hochschild cohomology $\mathrm{HH}^\bullet(A)$ is a Gerstenhaber algebra. That is:

$(\mathrm{HH}^\bullet(A), \smile)$ is a graded commutative associative algebra.
$(\mathrm{HH}^\bullet(A), [-, -])$ is a graded Lie algebra with bracket of degree $-1$ .
The bracket is a graded derivation of the cup product (Lemma 36).

For a smooth commutative algebra, the HKR theorem identifies $\mathrm{HH}^\bullet(A)$ with polyvector fields. In this context, the Gerstenhaber bracket corresponds to the classical Schouten-Nijenhuis bracket on polyvector fields. We will return to this geometric interpretation in detail in the HKR theorem and Deformation Quantisation sections. To end this section, however, we are going to define one last thing. Consider a differential graded Lie algebra $(\mathfrak{g},d,[\cdot,\cdot])$ . The elements of degree $1$ that satisfy a certain quadratic equation play a fundamental role in deformation theory. This equation, known as the Maurer–Cartan equation, encodes the condition that a given element defines an integrable deformation of the underlying structure. Its solutions, modulo an appropriate gauge equivalence, are precisely the deformations governed by the DGLA.

Definition 38 (Maurer–Cartan equation). An element $\alpha \in \mathfrak{g}^1$ is called a Maurer–Cartan element if it satisfies $d\alpha + \frac{1}{2}[\alpha,\alpha] = 0. \tag{MC}$ The (generic) set of all such elements is denoted by $\mathrm{MC}(\mathfrak{g})$ .

The Maurer–Cartan equation is preserved under morphisms of DGLAs, that is, if $f : \mathfrak{g} \to \mathfrak{g}'$ is a DGLA morphism and $\alpha \in \mathrm{MC}(\mathfrak{g})$ , then $f(\alpha) \in \mathrm{MC}(\mathfrak{g}')$ . In the context of deformation quantisation, which we shall explore in greater detail in the following sections, we encounter some concrete realisations of this equation. What we can say for now is that the Maurer–Cartan equation provides a unified cohomological framework that governs both classical and quantum deformations.

§6.The Hochschild-Konstant-Rosenberg Theorems

It is desirable, even necessary, that we define what is a smooth algebra before we venture further into the Hochschild-Konstant-Rosenberg Theorems:

Definition 39. Let $S$ be a commutative $k$ -algebra with unit element. A sequence $(x_1,\ldots,x_m)$ of elements of $S$ is called regular if multiplication by $x_i$ in $S/(x_1S+\cdots+x_{i-1}S)$ is injective (i.e., $x_i$ is regular in the quotient) for $i=1,\ldots,m$ . The commutative and unital algebra $A$ is smooth over $k$ if it is flat over $k$ and if, for any maximal ideal $M$ of $A$ , the kernel $J$ of the localized map $\mu_M : (A\otimes_k A)_{\mu^{-1}(M)} \longrightarrow A_M$ is generated by a regular sequence in $(A\otimes_k A)_{\mu^{-1}(M)}.$ Moreover, if the kernel $J$ is $0$ , then $A$ is etale over $k$ .

Although this is the formal definition given in , smoothness of an algebra may also be characterized in terms of Kähler differentials, which is an interesting characterization given the present context:

Definition 40. Let $k$ be a field and $A$ a finitely generated commutative $k$ -algebra. We call $A$ a smooth $k$ -algebra if its module of Kähler differentials $\Omega_{A/k}$ is a projective $A$ -module. Equivalently, for $X = \operatorname{Spec}(A)$ , $X$ is smooth over $k$ iff $\Omega_{X/k}^1$ is locally free.

There is a related, more general notion called formal smoothness, which is defined by a lifting property for nilpotent ideals. For algebras that are locally of finite presentation, being formally smooth (or quasi-free) is equivalent to being smooth. In this case, the projectivity of the Kähler differentials is a key consequence.

>The Hochschild-Konstant-Rosenberg Theorem For Homology

Theorem 41 (Hochschild–Kostant–Rosenberg). Let $A$ be a smooth, commutative algebra over a field $k$ of characteristic zero. Let $HH_n(A)$ denote the $n$ -th Hochschild homology group of $A$ , and let $\Omega^n_{A/k}$ denote the module of Kähler differentials of degree $n$ . Then, for every $n \ge 0$ , there exists a natural isomorphism of $A$ -modules: $\begin{equation*} I_{HKR} \colon HH_n(A) \xrightarrow{\cong} \Omega^n_{A/k}. \end{equation*}$

Proof. The argument is divided into two parts:

the first part establishes the base case where $A = P = k[x_1,\dots,x_n]$ is a commutative polynomial $k$ -algebra;
the second part extends the result to smooth commutative $k$ -algebras in an algebraic approach;
the third part extends the result to smooth commutative $k$ -algebras in a geometric approach.

Base Case. Let $P = k[x_1,\dots,x_n]$ . The enveloping algebra is $P^e = P \otimes_k P$ , which we identify with $k[x_1,\dots,x_n,y_1,\dots,y_n]$ via $x_i = x_i \otimes 1$ and $y_i = 1 \otimes x_i$ . Consider the multiplication morphism $\mu : P^e \to P$ , $\mu(a \otimes b) = ab$ . Its kernel is the ideal $I = \ker(\mu)$ , which is generated by the elements $y_i - x_i$ , so that $I = (y_1 - x_1,\dots,y_n - x_n)$ . Since $P^e$ is a polynomial algebra, the sequence $(y_1 - x_1,\dots,y_n - x_n)$ is regular. Hence the Koszul complex $K_\bullet(\mathbf y - \mathbf x)$ gives a free (and projective) $P^e$ -resolution of $P$ . Its $r$ -th term is $K_r = \bigwedge^r_{P^e}(P^e)^n$ , where $(P^e)^n = (P^e)^{\oplus n}$ . The differential is given on pure wedges by $d_r(e_{i_1}\wedge \cdots \wedge e_{i_r}) = \sum_{j=1}^r (-1)^{j-1}(y_{i_j} - x_{i_j})\, e_{i_1}\wedge \cdots \wedge \widehat{e_{i_j}} \wedge \cdots \wedge e_{i_r}.$

To compute $HH_\bullet(P) = \operatorname{Tor}_\bullet^{P^e}(P,P)$ , we tensor the Koszul complex with $P$ over $P^e$ , obtaining $0 \to \bigwedge^n_{P^e}(P^e)^n \otimes_{P^e} P \to \cdots \to (P^e)^n \otimes_{P^e} P \to P \to 0,$ where each differential is induced by $d_r \otimes \mathrm{id}_P$ . We claim that all differentials vanish. Let $(e_{i_1}\wedge\cdots\wedge e_{i_n})\otimes a \in K_n\otimes_{A^e}A$ . Then

$\begin{align*} (d_n\otimes \mathrm{id}_A)\bigl((e_{i_1}\wedge\cdots\wedge e_{i_n})\otimes a\bigr) &= d_n(e_{i_1}\wedge\cdots\wedge e_{i_n})\otimes a \\ &= \sum_{r=1}^{n}(-1)^{r-1}(y_{i_r}-x_{i_r}) e_{i_1}\wedge\cdots\widehat{e_{i_r}}\cdots\wedge e_{i_n}\otimes a. \end{align*}$

Using the relation $\lambda m\otimes a = m\otimes \lambda a$ for $\lambda\in A^e$ , each summand may be rewritten as

$e_{i_1}\wedge\cdots\widehat{e_{i_r}}\cdots\wedge e_{i_n} \otimes (y_{i_r}-x_{i_r})a.$ Since $A \cong A^e/(y_1-x_1,\ldots,y_n-x_n),$ the elements $y_i-x_i$ act trivially on $A$ , so $(y_{i_r}-x_{i_r})a=0$ . Therefore every term in the above sum vanishes, and hence

$(d_n\otimes \mathrm{id}_A)\bigl((e_{i_1}\wedge\cdots\wedge e_{i_n})\otimes a\bigr)=0.$

It follows that $d_n\otimes \mathrm{id}_A=0$ for all $n$ . Therefore the complex $K_\bullet\otimes_{A^e}A$ has zero differential. Hence the homology of the tensorised complex is the complex itself. We next identify the terms. There is a canonical isomorphism $(P^e)^n \otimes_{P^e} P \cong P^n,$ and more generally $\bigwedge^r_{P^e}(P^e)^n \otimes_{P^e} P \cong \bigwedge^r_P P^n.$ Therefore the complex becomes $0 \to \bigwedge^n_P P^n \to \cdots \to \bigwedge^1_P P^n \to P \to 0,$ with zero differentials, so $HH_r(P) \cong \bigwedge^r_P P^n.$

Now recall that there is a canonical isomorphism $\Omega^1_{P/k} \cong P^n$ , sending $dx_i \mapsto e_i$ . Taking exterior powers gives $\bigwedge^r_P P^n \cong \bigwedge^r_P \Omega^1_{P/k} = \Omega^r_{P/k}$ . Hence $HH_r(P) \cong \Omega^r_{P/k}.$

Smooth Case (Algebraic Approach): Let $A$ be a smooth commutative $k$ -algebra. We consider the enveloping algebra $A^e = A \otimes_k A$ and the multiplication morphism $\mu : A^e \to A$ , defined by $\mu(a \otimes b) = ab$ . Let $I = \ker(\mu)$ be its kernel.

Because $A$ is a smooth $k$ -algebra, the diagonal morphism $\operatorname{Spec}(A) \to \operatorname{Spec}(A^e)$ , with $\operatorname{Spec}(A) = \{ \mathfrak{p} \subset A \mid \mathfrak{p} \; \text{is a prime ideal} \}$ , is a regular immersion . Algebraically, this means that the ideal $I$ is locally generated by a regular sequence. Specifically, for any prime ideal $\mathfrak{p} \subset A$ , let $\mathfrak{p}^e$ be the contraction of $\mathfrak{p}$ in $A^e$ along $\mu$ . In the local ring $(A^e)_{\mathfrak{p}^e}$ , the localized ideal $I_{\mathfrak{p}^e}$ is generated by a regular sequence of length $n$ , where $n$ is the local dimension of $A$ at $\mathfrak{p}$ . Since Hochschild homology is defined via $\operatorname{Tor}$ , which commutes with localization, we can compute $HH_\bullet(A)$ locally: $HH_r(A)_{\mathfrak{p}} \cong \operatorname{Tor}_r^{(A^e)_{\mathfrak{p}^e}}(A_{\mathfrak{p}}, A_{\mathfrak{p}}).$

By the local regularity of $I$ , the ring $(A^e)_{\mathfrak{p}^e}$ admits a Koszul resolution for $A_{\mathfrak{p}}$ constructed from the local regular sequence generating $I_{\mathfrak{p}^e}$ . Exactly as in the base case for polynomial rings, tensoring this Koszul complex with $A_{\mathfrak{p}}$ over $(A^e)_{\mathfrak{p}^e}$ yields a complex with trivial differentials. The homology of this tensorised complex is simply the exterior algebra of the conormal module: $\operatorname{Tor}_r^{(A^e)_{\mathfrak{p}^e}}(A_{\mathfrak{p}}, A_{\mathfrak{p}}) \cong \bigwedge^r_{A_{\mathfrak{p}}} (I_{\mathfrak{p}^e} / I_{\mathfrak{p}^e}^2).$

Because this isomorphism is canonical and holds for all prime ideals $\mathfrak{p}$ , it glues to a global isomorphism of $A$ -modules: $HH_r(A) = \operatorname{Tor}_r^{A^e}(A, A) \cong \bigwedge^r_A (I/I^2).$

To conclude the proof, we relate the global conormal module $I/I^2$ to the module of Kähler differentials $\Omega^1_{A/k}$ . By the universal property of Kähler differentials, there is a canonical $A$ -module isomorphism sending the class of $1 \otimes a - a \otimes 1 \pmod{I^2}$ to $da \in \Omega^1_{A/k}$ . This provides the identification $I/I^2 \cong \Omega^1_{A/k}$ . Taking the $r$ -th exterior power over $A$ yields: $\bigwedge^r_A (I/I^2) \cong \bigwedge^r_A \Omega^1_{A/k} = \Omega^r_{A/k}.$

Stringing these isomorphisms together, we obtain the HKR theorem for the smooth commutative case: $HH_r(A) \cong \Omega^r_{A/k}.$

Smooth Case (Geometric Approach): First of all, we must relate the Koszul complex to the standard Hochschild complex $C_\bullet(P)$ , where $C_n(P) = P^{\otimes (n+1)}$ . That relation exists due to theorem (6), as both resolutions are free resolutions of $P$ as a $P^e$ -module. The theorem asserts that the isomorphism is induced by the projection: $\pi_n \colon C_n(P) \to \Omega^n_{P/k}, \quad a_0 \otimes a_1 \otimes \dots \otimes a_n \mapsto a_0 \, da_1 \wedge \dots \wedge da_n.$ Because $k$ is a field of characteristic zero, we can define the antisymmetrisation map $\epsilon_P \colon \Omega^n_{P/k} \to C_n(P)$ by $\epsilon_P(a_0 \, dx_{i_1} \wedge \dots \wedge dx_{i_n}) = \frac{1}{n!} \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \, a_0 \otimes x_{i_{\sigma(1)}} \otimes \dots \otimes x_{i_{\sigma(n)}}.$ The map $\epsilon_P$ lifts the identity map on $P$ to a chain map, acting as a quasi-isomorphism. By direct computation, $\pi_n \circ \epsilon_P = \mathrm{id}$ on $\Omega^n_{P/k}$ . Because both complexes are projective resolutions, this establishes that the explicit map $I_{HKR}$ is a well-defined canonical isomorphism $HH_n(P) \xrightarrow{\cong} \Omega^n_{P/k}$ . Now let $A$ be an arbitrary smooth commutative $k$ -algebra. A fundamental property of smooth algebras is that they are locally étale over polynomial rings. Since the theorem’s statement is local in nature, we may localize $A$ and assume without loss of generality that there exists a polynomial ring $P = k[x_1, \dots, x_m]$ and a morphism of algebras $f: P \to A$ that is étale.

We rely on two base change properties of étale morphisms:

Differentials Base Change: Because $f$ is étale, it induces an isomorphism of $A$ -modules on the module of Kähler differentials: $\Omega^n_{P/k} \otimes_P A \xrightarrow{\cong} \Omega^n_{A/k}$
Hochschild Base Change: Because étale morphisms are flat, the Tor functor commutes with base change. In other words, the flatness of the morphism $P \to A$ guarantees that $A$ is a flat $P$ -module; one knows that the functor $\operatorname{Tor}$ measures exactly how much the tensor product “fails” to be exact, so tensoring a resolution by a flat module preserves exactness, allowing the functor $\operatorname{Tor}$ to change sides with the tensorial product. This yields a canonical isomorphism: $HH_n(P) \otimes_P A \xrightarrow{\cong} HH_n(A)$

Let $\epsilon_P : HH_n(P) \xrightarrow{\cong} \Omega^n_{P/k}$ be the isomorphism established above. Applying the functor $- \otimes_P A$ to $\epsilon_P$ induces an isomorphism $(\epsilon_P \otimes \mathrm{id}_A)$ . We can construct the following commutative diagram:

$$\begin{tikzcd}[row sep=large, column sep=huge] HH_n(P) \otimes_P A \arrow[r, "\epsilon_P \otimes \mathrm{id}_A", "\sim"'] \arrow[d, "\cong"', "(2)"] & \Omega^n_{P/k} \otimes_P A \arrow[d, "\cong", "(1)"'] \\ HH_n(A) \arrow[r, "\epsilon_A", dashed] & \Omega^n_{A/k} \end{tikzcd}$$

Because the top, left, and right arrows are all isomorphisms, there exists a unique induced map $\epsilon_A$ that makes the diagram commute, which must also be an isomorphism. Because all maps are canonical, these local isomorphisms glue globally over the spectrum of $A$ , proving that $I_{HKR} : HH_n(A) \xrightarrow{\cong} \Omega^n_{A/k}$ for any smooth commutative $k$ -algebra $A$ . ◻

There were some implicitly used results and definitions in the previous proof. The first of these are the Kähler Differentials: Let $A$ be a commutative $k$ -algebra. A $k$ -derivation of $A$ with values in an $A$ -module $M$ is a $k$ -linear map $\delta:A\to M$ satisfying

$\delta(ab)=a\delta(b)+b\delta(a)$

for all $a,b\in A$ . The module of Kähler differentials of $A$ over $k$ , denoted by $\Omega^1_{A/k}$ , is an $A$ -module together with a $k$ -derivation

$d:A\to\Omega^1_{A/k}$ such that for every $A$ -module $M$ and every derivation $\delta:A\to M$ , there exists a unique $A$ -linear map $\varphi:\Omega^1_{A/k}\to M$ for which the diagram \[ \begin{array}{ccc} A & \xrightarrow{\;d\;} & \Omega^1_{A/k} \\ {\scriptstyle \delta}\searrow & & \downarrow{\scriptstyle \exists!\,\varphi} \\ & M & \end{array} \] commutes. This universal property characterizes $\Omega^1_{A/k}$ uniquely up to unique isomorphism. The second, are two key algebraic facts relied upon in the smooth case:

Regular Immersions of Smooth Schemes: For any smooth $k$ -algebra $A$ , the diagonal morphism is a regular immersion. Equivalently, the kernel $I = \ker(A \otimes_k A \to A)$ is locally generated by a regular sequence. This structural property is what permits the use of the Koszul complex to locally resolve $A$ as an $A^e$ -module.
The Conormal Module Isomorphism: The module of Kähler differentials $\Omega^1_{A/k}$ represents universal derivations. The map $\delta: A \to I/I^2$ given by $a \mapsto 1 \otimes a - a \otimes 1 \pmod{I^2}$ is a $k$ -derivation. By the universal property of Kähler differentials, any such derivation factors uniquely through the universal derivation $d: A \to \Omega^1_{A/k}$ , yielding a commutative diagram: \[ \begin{array}{ccc} A & \xrightarrow{\;d\;} & \Omega^1_{A/k} \\ {\scriptstyle \delta}\searrow & & \downarrow{\scriptstyle \cong\ \text{(canonical)}} \\ & I/I^2 & \end{array} \] This unique $A$ -module homomorphism induces the canonical isomorphism $I/I^2 \xrightarrow{\sim} \Omega^1_{A/k}$ , linking the Hochschild homology to differential forms.

Next, we give a few examples of applications of the theorem.

Example 41 (Non-commutative Geometry and Cyclic Homology). While the bimodule of Kähler 1-forms $\Omega^1_{A/k}$ can be defined for non-commutative algebras, the lack of commutativity prevents the construction of well-behaved higher exterior powers $\bigwedge^n_A \Omega^1_{A/k}$ . Consequently, the classical de Rham complex collapses, making it impossible to perform differential calculus in the standard way. However, Alain Connes sought to develop differential calculus over non-commutative spaces, such as Von Neumann algebras¹. His solution was to use Hochschild homology as a substitute for differential forms.

The HKR theorem justifies this substitution. Because $I_{HKR} \colon HH_n(A) \xrightarrow{\cong} \Omega^n_{A/k}$ holds for smooth commutative algebras and because the Hochschild homology is always defined for associative algebras, using the HKR theorem, Connes simply defined $\Omega^n_A$ as $HH_n(A)$ for the non-commutative case. Furthermore, to mimic the classical exterior derivative $d_{dR}$ , Connes introduced a purely algebraic boundary operator $B \colon C_n(A) \to C_{n+1}(A)$ . In low degrees, $B$ acts on elementary tensors as: $B(a_0) = 1 \otimes a_0$ $B(a_0 \otimes a_1) = 1 \otimes a_0 \otimes a_1 - 1 \otimes a_1 \otimes a_0$

To see why this works, we can apply the projection map $\pi_n \colon C_n(A) \to \Omega^n_{A/k}$ (which induces $I_{HKR}$ on homology) to these expressions in the commutative setting: $\begin{aligned} \pi_1(B(a_0)) &= \pi_1(1 \otimes a_0) = 1 \, da_0 = d_{dR}(a_0), \\ \pi_2(B(a_0 \otimes a_1)) &= \pi_2(1 \otimes a_0 \otimes a_1 - 1 \otimes a_1 \otimes a_0) \\ &= 1 \, da_0 \wedge da_1 - 1 \, da_1 \wedge da_0 \\ &= 2 \, da_0 \wedge da_1 \\ &= 2 \, d_{dR}(a_0 \, da_1). \end{aligned}$

Up to factorial constants depending on the degree, this explicit calculation shows that $\pi \circ B = d_{dR} \circ \pi$ . Thus, the HKR theorem ensures that Connes’ algebraic operator $B$ precisely recovers the classical de Rham exterior derivative in the smooth commutative case, validating its use as a generalized differential in the non-commutative realm.

Example 42 (Algebraic K-Theory and the Dennis Trace). Algebraic K-theory groups, $K_n(A)$ , extract deep structural invariants from an algebra $A$ , but they are notoriously difficult to compute. To study them, mathematicians use the Dennis trace map, a natural homomorphism $\operatorname{tr}_n \colon K_n(A) \to HH_n(A)$ that links K-theory to Hochschild homology . The HKR theorem makes this link geometrically computable for smooth commutative algebras.

In degree 1, $K_1(A)$ is constructed from the invertible matrices over $A$ . If we restrict to the group of units $A^\times$ , the Dennis trace map $\operatorname{tr}_1 \colon A^\times \to HH_1(A)$ is defined by evaluating the homology class of the elementary tensor: $\operatorname{tr}_1(u) = [u^{-1} \otimes u]$

To see that this is a well-defined homomorphism from a multiplicative group to an additive one, we evaluate the trace of a product $uv$ . The Hochschild boundary relations in $HH_1(A)$ state that $[a \otimes bc] = [ab \otimes c] + [ca \otimes b]$ . Therefore: $\begin{aligned} \operatorname{tr}_1(uv) &= [(uv)^{-1} \otimes uv] \\ &= [v^{-1}u^{-1} \otimes uv] \\ &= [v^{-1}u^{-1}u \otimes v] + [uv v^{-1}u^{-1} \otimes u] \\ &= [v^{-1} \otimes v] + [u^{-1} \otimes u] \\ &= \operatorname{tr}_1(v) + \operatorname{tr}_1(u). \end{aligned}$

Applying the HKR isomorphism $I_{HKR}$ to the trace of $u$ , which is induced by the projection $\pi_1(a_0 \otimes a_1) = a_0 \, da_1$ , yields: $I_{HKR}(\operatorname{tr}_1(u)) = u^{-1} \, du = \frac{du}{u}$

So, through the lens of the HKR theorem, the Dennis trace in degree 1 is precisely the classical logarithmic derivative $d\log(u)$ .

The Hochschild-Konstant-Rosenberg Theorem For Cohomology

Theorem 42 (Hochschild–Kostant–Rosenberg). Let $A$ be a smooth, commutative algebra over a field $k$ of characteristic zero. Let $HH^n(A)$ denote the $n$ -th Hochschild cohomology group of $A$ , and let $\bigwedge^n_A \operatorname{Der}_k(A)$ denote the $n$ -th exterior power of the module of $k$ -derivations of $A$ . Then, for every $n \ge 0$ , there exists a natural isomorphism of $A$ -modules: $\begin{equation*} I^{HKR} \colon HH^n(A) \xrightarrow{\cong} \bigwedge^n_A \operatorname{Der}_k(A). \end{equation*}$

note 43. In anticipation of the geometric interpretation developed in later sections, we shall sometimes denote $\bigwedge^n_A \operatorname{Der}_k(A)$ by $\mathfrak{X}^n(M)$ , where $M = \operatorname{Spec}(A)$ (or, in the smooth real setting, $M$ is the manifold with $A = C^\infty(M)$ ); see Remark 7 below for the precise convention.

Proof. As we did for homology, the argument for this proof is divided into two parts :

the first part establishes the base case where $A = P = k[x_1,\dots,x_m]$ is a commutative polynomial $k$ -algebra;
the second part extends the result to arbitrary smooth commutative $k$ -algebras using a local-to-global algebraic approach.

Recall preliminarily that, by definition, the Hochschild cohomology of $A$ with coefficients in itself is given by the global derived functor: $HH^n(A) = \operatorname{Ext}^n_{A^e}(A, A)$ where $A^e = A \otimes_k A$ is the enveloping algebra and $A$ is viewed as an $A^e$ -module via the multiplication morphism $\mu(a \otimes b) = ab$ .

Base Case: Consider $A = P = k[x_1, \dots, x_m]$ . The enveloping algebra is $P^e \cong k[x_1, \dots, x_m, y_1, \dots, y_m]$ , where $x_i = x_i \otimes 1$ and $y_i = 1 \otimes x_i$ . The ideal $I = \ker(\mu)$ is generated by the regular sequence $(y_1 - x_1, \dots, y_m - x_m)$ . Since the sequence is regular, the $P^e$ -module $P$ admits a global free (hence projective) resolution given by the Koszul complex $K_\bullet \to P$ . The $r$ -th term of this complex is: $K_r = P^e \otimes_k \bigwedge^r V$ where $V$ is the $m$ -dimensional vector space generated by the elements $(y_i - x_i)$ .

To compute $\operatorname{Ext}^\bullet_{P^e}(P, P)$ , we do as we always do and apply the contravariant functor $\operatorname{Hom}_{P^e}(-, P)$ to the complex $K_\bullet$ . Let us analyze the structure of the resulting terms. By the Tensor-Hom adjunction property and considering that the action of $P^e$ fixes the basis over $k$ , we have the canonical isomorphism: $\operatorname{Hom}_{P^e}\left(P^e \otimes_k \bigwedge^r V, P\right) \cong \operatorname{Hom}_k\left(\bigwedge^r V, P\right)$ Since the spaces are finite-dimensional, extending the linear maps to the ring $P$ yields: $\operatorname{Hom}_k\left(\bigwedge^r V, P\right) \cong P \otimes_k \bigwedge^r V^* \cong \bigwedge^r_P (P \otimes_k V^*)$ The dual space $V^*$ has a natural basis given by the partial differential operators $\partial/\partial x_i$ , so that $P \otimes_k V^* \cong \operatorname{Der}_k(P)$ . More precisely, this isomorphism is given by the bijective map $\Phi: P \otimes_k V^* \rightarrow \operatorname{Der}_k(P)$ , $\sum_i p_i \otimes \partial/\partial x_i \mapsto \sum_i p_i \partial/\partial x_i$ . Consequently, the $r$ -th term of our cochain complex is exactly $\bigwedge^r_P \operatorname{Der}_k(P)$ . We need to justify that the resulting complex has trivial differentials. The original differential $d_r$ of the Koszul complex acts by multiplying elements by linear combinations of the generators $(y_i - x_i)$ . Upon applying the $\operatorname{Hom}$ functor, the induced differential $d^*$ acts on a cochain $f$ by precomposition: $d^*(f)(m) = f(d(m))$ . Since $f$ is a morphism of $P^e$ -modules, scalars in $P^e$ can be pulled out of the function: $f((y_i - x_i) \cdot m) = (y_i - x_i) \cdot f(m)$ However, the image $f(m)$ resides in $P$ . The bimodule action of $P^e$ on $P$ forces both $x_i \otimes 1$ and $1 \otimes x_i$ to act as usual multiplication by $x_i$ . Hence: $(y_i - x_i) \cdot f(m) = x_i f(m) - x_i f(m) = 0$ This implies that the image of every differential is zero ( $d^* = 0$ ). Since the cohomology of a complex with identically zero differentials is the complex itself, we conclude that: $HH^n(P) \cong \bigwedge^n_P \operatorname{Der}_k(P)$

General Smooth Case: Let $A$ be an arbitrary smooth commutative algebra over $k$ . Geometrically, the smoothness of $A$ guarantees that the diagonal morphism $\operatorname{Spec}(A) \to \operatorname{Spec}(A^e)$ is a regular immersion . Algebraically, this means that, locally for any prime ideal $\mathfrak{p}$ , the kernel $I = \ker(A^e \to A)$ is generated by a regular sequence in the localized ring. This is exactly the same argument used for homology.

Thanks to the local existence of this regular sequence, the localized ring $(A^e)_{\mathfrak{p}^e}$ also admits a Koszul complex resolving $A_{\mathfrak{p}}$ . For exactly the same reason detailed in Step 1—the fact that the ideal $I$ annihilates the ring $A$ itself—the differentials of the dual cochain complex vanish locally. The computation of the local $\operatorname{Ext}$ thus reduces to the exterior power of the dual of the conormal module: $\operatorname{Ext}^n_{(A^e)_{\mathfrak{p}^e}}(A_{\mathfrak{p}}, A_{\mathfrak{p}}) \cong \bigwedge^n_{A_{\mathfrak{p}}} \operatorname{Hom}_{A_{\mathfrak{p}}}(I_{\mathfrak{p}^e} / I_{\mathfrak{p}^e}^2, A_{\mathfrak{p}})$

Because this isomorphism is built over the diagonal of an affine space that behaves locally like a polynomial ring, it is canonical and glues perfectly at the global level, allowing us to rewrite the global cohomology as: $HH^n(A) = \operatorname{Ext}^n_{A^e}(A, A) \cong \bigwedge^n_A \operatorname{Hom}_A(I/I^2, A)$

To conclude the proof, we use the universal property of Kähler differentials. There is a natural equivalence $I/I^2 \cong \Omega^1_{A/k}$ . Applying the functor $\operatorname{Hom}_A(-, A)$ to both sides translates the module of forms into the module of derivations: $\operatorname{Hom}_A(I/I^2, A) \cong \operatorname{Hom}_A(\Omega^1_{A/k}, A) \cong \operatorname{Der}_k(A)$ The last isomorphism is given by the map $\Phi: \operatorname{Hom}_A (\Omega^1_{A/k}, A) \rightarrow \operatorname{Der}_k(A)$ , $\varphi \mapsto \varphi \circ d$ , with inverse $\Psi: \operatorname{Der}_k(A) \rightarrow \operatorname{Hom}_A (\Omega^1_{A/k}, A)$ , $D \mapsto \varphi_D$ , where $\varphi_D (da) = D(a)$ . Since $A$ is smooth, the module of differentials $\Omega^1_{A/k}$ is a finitely generated projective $A$ -module. This guarantees that tensorial operations (such as exterior powers) commute perfectly with dualization . Substituting this fact into the exterior power functor, we arrive at the final result: $HH^n(A) \cong \bigwedge^n_A \operatorname{Der}_k(A)$ ◻

As one can notice, the proof is entirely analogous to the proof for homology, which is to be expected. We end this section with some examples of applications of the theorem.

Example 43 (Multivector Fields and Poisson Structures). For a smooth commutative algebra $A$ , the HKR theorem for cohomology identifies $HH^n(A)$ with the $n$ -th exterior power of derivations: $I^{HKR} \colon HH^n(A) \xrightarrow{\cong} \bigwedge^n_A \operatorname{Der}_k(A)$ In geometry, elements of $\bigwedge^n_A \operatorname{Der}_k(A)$ are precisely the multivector fields on the manifold $M = \operatorname{Spec}(A)$ . When $n=2$ , an element $\pi \in HH^2(A)$ corresponds to a bivector field. The condition for $\pi$ to define a Poisson structure on $M$ is the vanishing of the Gerstenhaber bracket, $[\pi, \pi] = 0$ . Since the Gerstenhaber bracket on $HH^\bullet(A)$ corresponds to the Schouten-Nijenhuis bracket on multivector fields under the HKR isomorphism, Hochschild cohomology provides the natural framework for the algebraic study of Poisson geometry .

§7.A Faster-than-Light Course in Deformation Theory

Consider $k$ a commutative ring and $A$ a $k$ -algebra. Through this subsection, let $k[[t]]$ denote the ring of power series in the formal parameter $t$ with coefficients in $k$ and $A[[t]]$ the $k[[t]]$ -algebra of formal power series with coefficients in $A$ .

Definition 44. The formal deformation of the product $\cdot: A \times A \rightarrow A$ is the $k[[t]]$ -linear map $\star: A[[t]] \times A[[t]] \rightarrow A[[t]]$ such that, for any $w,v \in A$ , we have that $\begin{equation} w \star v = w \cdot v \mod t \end{equation}$

Definition (44) simply means that we associate to the original product of $A$ a new product that is given by a formal power series, with the first (or zeroth order) term being the original product:

$\begin{equation} w \star b = w \cdot b + \sum_{n = 1}^\infty t^i \mu_n (w, v) \end{equation}$ with each $\mu_i$ $k$ -linear maps on $A$ . Given $f_k, g_l \in A$ , the extension to formal power series is given by $\begin{equation} \label{eq:ext} \left(\sum_{k=0}^{\infty} f_k t^k\right) \star \left(\sum_{l=0}^{\infty} g_l t^l\right) = \sum_{n=0}^{\infty} \left( \sum_{k+l+m=n} \mu_m(f_k,g_l) \right) t^n. \end{equation}$ Moreover, $\begin{equation} \label{eq:ass} \sum_{i + j = n} \mu_i (\mu_j(a,b),c) = \sum_{i + j = n} \mu_i(a, \mu_j(b,c)) \end{equation}$ for all $a,b,c\in A$ , defines the associative condition for $\star$ and can be formally extended with the same philosophy used in ([eq:ext]). The following result is particularly important for the context in which this section is situated.

Definition 45. Consider $(M, \pi)$ a Poisson manifold. A star product on $M$ is a formal deformation of $(C^{\infty}(M), \cdot)$ , denoted by $\star: C^{\infty}(M)[[t]] \times C^{\infty}(M)[[t]] \rightarrow C^{\infty}(M)[[t]]$ and given by $\begin{equation} f \star g = f \cdot g + \sum_{n = 0}^\infty \mu_n(f,g) t^n \end{equation}$ with $\mu_n: C^{\infty}(M) \times C^{\infty}(M) \rightarrow C^{\infty}(M)$ bi-differential operators and $\mathbb{R}$ -linear maps.

The associativity condition ([eq:ass]) for the star product imposes a hierarchy of constraints on the bilinear operators $\mu_n$ . In particular, at first order in $t$ , these constraints imply that the antisymmetric part of $\mu_1$ satisfies the Jacobi identity and the Leibniz rule with respect to the original commutative product. Therefore, it defines a Poisson bracket on the algebra $A$ . This result is stated in the following proposition.

Proposition 8. The operation $\{a,b\}= \mu_1(a,b)-\mu_1(b,a)$ is a Poisson bracket on $A$ .

Proof. Bilinearity and skew‑symmetry are immediate from the bilinearity of $\mu_1$ and the definition. It remains to verify the Jacobi identity and the Leibniz rule. Consider the commutator with respect to the deformed product: $[a,b]_\star := a\star b - b\star a.$ Since $\star$ is associative, the commutator satisfies the Jacobi identity and the derivation property: $[a,[b,c]_\star]_\star + [b,[c,a]_\star]_\star + [c,[a,b]_\star]_\star = 0, \tag{1}$ $[a, b\star c]_\star = [a,b]_\star \star c + b\star [a,c]_\star. \tag{2}$

Now expand $[a,b]_\star$ in powers of $t$ : $[a,b]_\star = (a\star b - b\star a) = t\bigl(\mu_1(a,b)-\mu_1(b,a)\bigr) + O(t^2) = t\,\{a,b\} + O(t^2). \tag{3}$

Similarly, $a\star b = ab + t\,\mu_1(a,b) + O(t^2). \tag{4}$

Substitute (3) and (4) into (1) and (2), then compare the coefficients of $t$ after dividing by $t$ and taking the limit $t\to 0$ .

Jacobi identity.

Using (3) in (1), the coefficient of $t$ in (1) gives $\{a,\{b,c\}\} + \{b,\{c,a\}\} + \{c,\{a,b\}\} = 0,$ which is exactly the Jacobi identity for $\{\cdot,\cdot\}$ .

Leibniz rule.

Using (3) and (4) in (2), the coefficient of $t$ in (2) yields $\{a, bc\} = \{a,b\}\,c + b\,\{a,c\}.$ Similarly, by skew‑symmetry, the rule in the second argument follows. Thus $\{\cdot,\cdot\}$ is a derivation in each argument. Hence all requirements for a Poisson bracket are satisfied. ◻

Having introduced the DGLA structure in the preceding sections, culminating in the Hochschild cochain complex, the motivating premise has been that the cohomology of this complex, endowed with the structure of a DGLA, would govern the deformation problem of the underlying algebra. It is now time to make this assertion precise and to elucidate exactly how the cohomology groups of the complex classify infinitesimal deformations, control obstructions to extending them to higher orders, and encode the infinitesimal automorphisms of the deformed structures. To do so, we start by considering a first-order deformation of the multiplication: $\mu = \mu_0 + t \phi, \qquad \phi \in C^1(A),$ with $C^n(A) := \operatorname{Hom}_k (A^{\otimes n}, A)$ and $t^2 = 0$ . Associativity requires $[\mu, \mu] = 0 \quad \Longleftrightarrow \quad d\phi = 0,$ since $[\mu_0,\mu_0]=0$ and $t^2=0$ . Thus $\phi$ is a Hochschild 2-cocycle. Two infinitesimal deformations $\mu_0 + t\phi$ and $\mu_0 + t\phi'$ are equivalent via a formal automorphism $T = \operatorname{Id} + t\psi$ , with $\psi \in C^0(A) = A$ , if $\begin{equation} T(\mu(a,b)) = \mu'(T(a), T(b)). \end{equation}$ At first order this yields $\phi - \phi' = d\psi.$ Hence $\phi$ and $\phi'$ differ by a coboundary. We conclude:

Proposition 9. The space of equivalence classes of infinitesimal deformations of an associative algebra $A$ is naturally isomorphic to the second Hochschild cohomology group $HH^2(A)$ .

Suppose we have a deformation to order $n$ : $\mu_t = \mu_0 + t\mu_1 + \cdots + t^n \mu_n,$ satisfying associativity modulo $t^{n+1}$ . We ask whether it can be extended to order $n+1$ , i.e. whether there exists $\mu_{n+1}$ such that the associativity condition holds modulo $t^{n+2}$ .

Writing $[\mu_t, \mu_t] = 0 \pmod{t^{n+1}}$ , the obstruction to extending is the coefficient of $t^{n+1}$ in $[\mu_t, \mu_t]$ , namely $\operatorname{Obs}_n = \sum_{i+j=n+1} [\mu_i, \mu_j] \in C^3(A).$ It can be shown that this element is a 3-cocycle, and its cohomology class $[\operatorname{Obs}_n] \in HH^3(A)$ is independent of the choices made. The deformation extends to order $n+1$ if and only if this obstruction class vanishes in $HH^3(A)$ . If $HH^3(A) = 0$ , then every deformation is unobstructed and can be extended to a formal deformation.

Proposition 10. The obstruction to extending a deformation of an associative algebra to the next order lies in the third Hochschild cohomology group $HH^3(A)$ . In particular, if $HH^3(A) = 0$ , the algebra is formally rigid in the sense that every formal deformation is equivalent to a trivial one (or at least unobstructed).

The first cohomology group controls infinitesimal automorphisms of the structure. Indeed, a derivation of the algebra is given by $D \in C^1(A)$ . The inner derivations, which are of the form $d a$ for $a \in A$ , correspond to coboundaries in the Hochschild complex. The quotient $HH^1(A) = \frac{\text{Derivations}}{\text{Inner derivations}}$ classifies the outer derivations. More generally, for a deformation problem governed by $(\mathfrak{g}, d, [-,-])$ a DGLA, the tangent space to the automorphism group of a given deformed structure is precisely $H^0(\mathfrak{g})$ , while the infinitesimal automorphisms of the deformation are controlled by $H^1(\mathfrak{g})$ .

§8. The Culminating Point: Deformation Quantisation

In the context of mathematical physics — and in particular in the context of deformation quantisation —, a consequence of the HKR theorem is an explicit connection between the underlying geometry of the system, which controls its dynamics, and its algebraic structure, which controls its deformation (quantisation) and observables. We now examine this relation in depth. This section is strongly based on .

The Cohomological Bridge: HKR, DGLA, and Deformation Quantization

The purpose of this subsection is to introduce the two differential graded Lie algebras that govern the deformation problems relevant to quantization: the DGLA $T_{\mathrm{poly}}(M)$ of multivector fields, which controls the deformations of Poisson structures, and the DGLA $D_{\mathrm{poly}}(M)$ of multidifferential operators, which controls the deformations of the associative algebra of functions. The Hochschild–Kostant–Rosenberg (HKR) theorem then provides a crucial cohomological link between these two objects, revealing that their underlying cohomologies are isomorphic. However, as we shall see, this isomorphism is not a morphism of DGLAs; it fails to preserve the Lie brackets. This failure is precisely what necessitates the introduction of $\mathsf{L}_\infty$ -structures and constitutes the central insight of Kontsevich’s formality theorem. In this subsection, we elucidate how these three ingredients — the HKR theorem, the DGLA structure of the Hochschild complex, and deformation quantization — converge to provide a complete cohomological classification of star products, thereby establishing a profound bridge between algebraic, geometric, and physical structures.

Remark 7 (Notation for multivector fields). Throughout this section, $M$ denotes a smooth manifold and $A = C^\infty(M)$ . Building on the notation introduced in section (4), we write $\mathfrak{X}^k(M) \;:=\; \bigwedge^k_A \operatorname{Der}_k(A)$ for the $A$ -module of $k$ -vector fields on $M$ , so that $\mathfrak{X}^1(M) = \operatorname{Der}_k(A)$ is the module of vector fields and $\mathfrak{X}^0(M) = A = C^\infty(M)$ . We also write $\mathfrak{X}^\bullet(M) := \bigoplus_{k \geq 0} \mathfrak{X}^k(M)$ . The degree conventions adopted below are those of section (4): within the shifted Hochschild DGLA $\bigl(C^{*+1}(A),\,\partial,\,[-,-]_G\bigr)$ , the degree- $n$ component is $C^{n+1}(A) = \operatorname{Hom}_k(A^{\otimes(n+1)}, A)$ .

The DGLA of Multivector Fields and Poisson Structures

Definition 46 (The DGLA $T_{\mathrm{poly}}(M)$ ). The graded vector space of formal multivector fields on $M$ is $T_{\mathrm{poly}}(M) \;:=\; \bigoplus_{n \geq 0} T_{\mathrm{poly}}^n(M), \qquad T_{\mathrm{poly}}^n(M) \;:=\; \mathfrak{X}^{n+1}(M)[[t]],$ where an element $X \in \mathfrak{X}^{n+1}(M)$ is assigned DGLA degree $n$ (consistent with the degree shift of section (4), so that bivectors $\pi \in \mathfrak{X}^2(M)$ sit at degree $1$ ). This graded vector space carries a natural DGLA structure:

Differential: $d_T = 0$ (the trivial differential);
Bracket: the Schouten–Nijenhuis bracket $[\cdot,\cdot]_S$ , extended to formal power series by $\mathbb{R}[[t]]$ -bilinearity.

The Schouten–Nijenhuis bracket is the unique extension of the Lie bracket of vector fields to all of $T_{\mathrm{poly}}(M)$ , satisfying the graded Jacobi identity and the graded Leibniz rule with respect to the exterior product of multivector fields. A bivector field $\pi \in \mathfrak{X}^2(M) = T_{\mathrm{poly}}^1(M)$ defines a Poisson structure precisely when $[\pi,\pi]_S = 0$ . More generally, an element $\pi_t = \sum_{n \geq 1} t^n \pi_n \in t\,T_{\mathrm{poly}}^1(M)$ is a formal Poisson structure if and only if it satisfies the Maurer–Cartan equation $[\pi_t,\pi_t]_S = 0.$ Thus, the set of solutions to the Maurer–Cartan equation in the DGLA $T_{\mathrm{poly}}(M)$ is precisely the set of formal Poisson structures on $M$ . Moreover, the gauge action of the group $\exp(t\,\mathfrak{X}^1(M)[[t]])$ on $T_{\mathrm{poly}}(M)$ corresponds exactly to the action of formal diffeomorphisms on Poisson structures. Consequently, the deformation space $\mathrm{Def}(T_{\mathrm{poly}}(M)) = \frac{\mathrm{MC}(T_{\mathrm{poly}}(M))}{\mathrm{G}^0(T_{\mathrm{poly}}(M))}$ is in bijection with the set of equivalence classes of formal Poisson structures on $M$ . This is a purely geometric deformation problem, governed by the Schouten–Nijenhuis bracket.

The DGLA of Multidifferential Operators and Star Products

On the algebraic side, consider the associative algebra $A = C^\infty(M)$ with pointwise multiplication. As established in section (4), the Hochschild cochain complex $C^\bullet(A) = \bigoplus_{n=-1}^\infty \mathrm{Hom}(A^{\otimes n+1}, A)$ carries the structure of a DGLA, called the Hochschild DGLA, with:

Differential: the Hochschild differential $d_H = [\mu_0,\cdot]_G$ , where $\mu_0$ denotes the pointwise multiplication;
Bracket: the Gerstenhaber bracket $[\cdot,\cdot]_G$ .

The subcomplex of multidifferential operators vanishing on constants, denoted $D_{\mathrm{poly}}(M)$ , is a DGL subalgebra of the Hochschild DGLA. A star product on $M$ is a formal deformation of the pointwise product, given by $\mu_t = \mu_0 + \sum_{n=1}^\infty t^n \mu_n,$ where each $\mu_n$ is a bidifferential operator. The associativity of $\mu_t$ is equivalent to the Maurer–Cartan equation in the Hochschild DGLA: $d_H \mu_t + \frac{1}{2}[\mu_t,\mu_t]_G = 0.$ Indeed, since $[\mu_0,\mu_0]_G = 0$ and $d_H = [\mu_0,\cdot]_G$ , we have $[\mu_t,\mu_t]_G = 2d_H M + [M,M]_G,$ where $M = \mu_t - \mu_0 \in t\,D_{\mathrm{poly}}^1(M)$ . Hence, the associativity of the star product is precisely the Maurer–Cartan equation $d_H M + \frac{1}{2}[M,M]_G = 0.$ The gauge equivalence of star products corresponds exactly to the gauge action of the group $\exp(t\,D_{\mathrm{poly}}^0(M))$ on the space of Maurer–Cartan elements. Therefore, the deformation space $\mathrm{Def}(D_{\mathrm{poly}}(M)) = \frac{\mathrm{MC}(D_{\mathrm{poly}}(M))}{\mathrm{G}^0(D_{\mathrm{poly}}(M))}$ is in bijection with the set of equivalence classes of star products on $M$ . This is a purely algebraic deformation problem, governed by the Hochschild cohomology and the Gerstenhaber bracket.

The Hochschild–Kostant–Rosenberg Theorem: A Cohomological Bridge

The HKR theorem establishes a fundamental connection between the two complexes just described. It states that the map $U_1 : T_{\mathrm{poly}}(M) \longrightarrow D_{\mathrm{poly}}(M)$ defined on homogeneous elements $X_0 \wedge \cdots \wedge X_n \in \mathfrak{X}^{n+1}(M)$ by $(U_1(X_0 \wedge \cdots \wedge X_n))(f_0,\dots,f_n) = \frac{1}{(n+1)!} \sum_{\sigma \in S_{n+1}} \epsilon(\sigma) X_{\sigma(0)}(f_0) \cdots X_{\sigma(n)}(f_n)$ is a quasi-isomorphism of complexes. That is, $U_1$ induces an isomorphism in cohomology: $H^\bullet(T_{\mathrm{poly}}(M)) \cong HH^\bullet(C^\infty(M)).$ The cohomology of $T_{\mathrm{poly}}(M)$ is simply $T_{\mathrm{poly}}(M)$ itself (since the differential is trivial), while the cohomology of $D_{\mathrm{poly}}(M)$ is the Hochschild cohomology $HH^\bullet(C^\infty(M))$ . Thus, the HKR theorem provides an isomorphism $\mathfrak{X}^\bullet(M) \cong HH^\bullet(C^\infty(M)).$ In particular, $HH^2(C^\infty(M)) \cong \mathfrak{X}^2(M)$ , meaning that bivector fields — and hence Poisson structures — correspond to infinitesimal deformations of the algebra of functions. Similarly, $HH^3(C^\infty(M)) \cong \mathfrak{X}^3(M)$ , so obstructions to extending deformations correspond to trivector fields.

This cohomological identification tells us that the geometric data of a Poisson structure lives in the same cohomological degree as the algebraic data of an infinitesimal deformation of the function algebra. The Poisson bracket itself corresponds, under the HKR isomorphism, to the first-order term of a star product. More precisely, if $\pi \in \mathfrak{X}^2(M)$ is a Poisson bivector, then its image under $U_1$ is a Hochschild 2-cocycle, and the associated infinitesimal deformation of the product is $\mu_0 + t\,U_1(\pi)$ . The antisymmetric part of this cocycle gives the Poisson bracket.

The Failure of HKR as a DGLA Morphism and the Need for $\mathsf{L}_\infty$

Despite its cohomological significance, the map $U_1$ is not a morphism of DGLAs. Indeed, it fails to preserve the Lie brackets: $U_1([X,Y]_S) \neq [U_1(X), U_1(Y)]_G.$ The discrepancy already appears at order 2. For vector fields $X_1, X_2, Y_1, Y_2$ , one finds that $U_1([X_1 \wedge X_2, Y_1 \wedge Y_2]_S) - [U_1(X_1 \wedge X_2), U_1(Y_1 \wedge Y_2)]_G \neq 0.$ This failure reflects a fundamental difference between the Schouten–Nijenhuis bracket on multivector fields and the Gerstenhaber bracket on multidifferential operators. The former is a purely geometric bracket encoding the Poisson structure, while the latter is a purely algebraic bracket encoding the deformation of the product. The HKR isomorphism identifies the underlying cohomologies, but it does not identify the deformation problems themselves.

This is precisely where Kontsevich’s formality theorem enters. The theorem asserts that there exists an $\mathsf{L}_\infty$ -quasi-isomorphism $U : T_{\mathrm{poly}}(M) \longrightarrow D_{\mathrm{poly}}(M)$ whose first component is precisely the HKR map $U_1$ . An $\mathsf{L}_\infty$ -morphism is a sequence of multilinear maps $U_n : \wedge^n T_{\mathrm{poly}}(M) \to D_{\mathrm{poly}}(M)$ that together preserve the full $\mathsf{L}_\infty$ -structure, including the brackets at all orders. The higher components $U_n$ for $n \ge 2$ correct the failure of $U_1$ to be a Lie algebra homomorphism. The existence of such an $\mathsf{L}_\infty$ -quasi-isomorphism is a highly nontrivial result, proved by Kontsevich through explicit combinatorial formulas involving graphs and integrals.

The Cohomological Control of Deformations

The power of the DGLA formalism lies in its cohomological control of the deformation problem. For the Hochschild DGLA $D_{\mathrm{poly}}(M)$ , the cohomology groups $HH^\bullet(C^\infty(M))$ classify the deformation problem of the algebra of functions:

$HH^0(C^\infty(M)) \cong C^\infty(M)$ controls infinitesimal automorphisms of the deformed algebra.
$HH^1(C^\infty(M)) \cong \mathfrak{X}^1(M)$ classifies infinitesimal automorphisms of a given deformation.
$HH^2(C^\infty(M)) \cong \mathfrak{X}^2(M)$ classifies infinitesimal deformations of the multiplication. A Poisson bivector $\pi \in \mathfrak{X}^2(M)$ satisfying $[\pi,\pi]_S = 0$ corresponds to an infinitesimal deformation $\mu_0 + t\,\pi$ of the product.
$HH^3(C^\infty(M)) \cong \mathfrak{X}^3(M)$ governs obstructions to extending an infinitesimal deformation to a formal one. The Jacobi identity for the Poisson bracket, $[\pi,\pi]_S = 0$ , is precisely the condition that the obstruction vanishes.

This cohomological picture provides a beautiful unified framework. The Poisson structure $\pi$ , which geometrically is a bivector field satisfying $[\pi,\pi]_S = 0$ , is cohomologically a Hochschild 2-cocycle whose square (under the Gerstenhaber bracket) vanishes. The associativity of the star product is, in the Hochschild DGLA, precisely the Maurer–Cartan equation. The equivalence of star products is the gauge equivalence of Maurer–Cartan elements. Thus, the entire problem of deformation quantization is reformulated as a deformation problem in a DGLA.

The Bridge Between Algebraic, Geometric, and Physical Structures

Reinforcing the previous section, we state that the convergence of the HKR theorem, the DGLA structures, and deformation quantization establishes a “ $3$ -fold” dictionary:

Algebraic $\leftrightarrow$ Geometric: The HKR theorem identifies the algebraic Hochschild cohomology of the function algebra with the geometric space of multivector fields. This means that:
- Poisson structures (geometric objects) correspond to infinitesimal deformations of the commutative algebra of functions (algebraic objects).
- The Jacobi identity of the Poisson bracket corresponds to the vanishing of the obstruction to extending an infinitesimal deformation to a formal one.
- Equivalence classes of Poisson structures under diffeomorphisms correspond to equivalence classes of star products under formal automorphisms.
Geometric $\leftrightarrow$ Physical: The Poisson bracket governs the classical mechanics of a physical system. The dynamics of classical observables is given by Hamilton’s equations, which are expressed in terms of the Poisson bivector by $\begin{equation} X_H = \pi(dH, \cdot) , \end{equation}$ so that the Hamiltonian vector field $X_H$ is recovered from the bivector $\pi$ applied to $dH$ . Deformation quantization promotes this geometric Poisson structure to a noncommutative star product, which describes the quantum algebra of observables $(C^\infty(M)[[\hbar]], \star_\hbar)$ . The classical limit $\hbar \to 0$ recovers the Poisson bracket from the commutator of the star product: $\lim_{\hbar \to 0} \frac{1}{\hbar}[f,g]_\star = \{f,g\}.$ Thus, the geometric Poisson structure is the classical limit of the quantum commutator.
Algebraic $\leftrightarrow$ Physical: The star product is an associative deformation of the algebra of functions and provides a perfect algebraic framework for quantum mechanics, as it encodes all the properties needed for basic quantum theory. Its noncommutativity, for example, encodes the uncertainty principle: the Moyal star product, as sketched in Appendix 9, deforms the pointwise product of phase-space functions: $f \star g = fg + \frac{i\hbar}{2}\{f,g\} + \mathcal{O}(\hbar^2).$ Its noncommutativity is measured by the star-commutator: $\left[f\,,\, g\right]_{\star} = f\star g - g\star f = i\hbar\,\{f,g\} + \mathcal{O}(\hbar^3).$ For the canonical coordinates on the manifold $M$ , we have that $\{q,p\}=1$ , hence $\left[q\,,\, p\right]_{\star} = i\hbar,$ which is exactly the canonical commutation relation $[\hat q,\hat p]=i\hbar$ . In quantum theory, the Robertson–Schrödinger inequality states that for any two observables $\hat A,\hat B$ , $\Delta A \, \Delta B \ge \frac{1}{2}|\langle [\hat{A}, \hat{B}]\rangle|.$ Thus, if $\left[A\,,\, B\right]_{\star}\neq 0$ , the corresponding uncertainties have a strictly positive lower bound. In particular, $\Delta q \, \Delta p \ge \frac{\hbar}{2},$ which is Heisenberg’s uncertainty principle. Therefore, the noncommutativity of the star product directly encodes the physical limitation that noncommuting observables cannot be simultaneously measured with arbitrary precision. Moreover, the associativity of the star product ensures the consistency of the quantum algebra of observables; the equivalence classes of star products classify inequivalent quantizations of the same classical system, up to formal redefinitions of the observables and so on.

The formality theorem, which provides the $\mathsf{L}_\infty$ -quasi-isomorphism $U : T_{\mathrm{poly}}(M) \to D_{\mathrm{poly}}(M)$ , completes this bridge by showing that the geometric deformation problem of Poisson structures and the algebraic deformation problem of star products are isomorphic as deformation problems, going well beyond their mere cohomological equivalence. The $\mathsf{L}_\infty$ -morphism $U$ maps Maurer–Cartan elements of $T_{\mathrm{poly}}(M)$ (formal Poisson structures) to Maurer–Cartan elements of $D_{\mathrm{poly}}(M)$ (star products) in a way that preserves the gauge equivalence. This establishes a bijection $\mathrm{Def}(T_{\mathrm{poly}}(M)) \cong \mathrm{Def}(D_{\mathrm{poly}}(M)),$ which is precisely Kontsevich’s classification theorem. We can then summarize the equivalences constructed through this dissertation on the following table:

Cohomological control of deformations for the algebra $C^\infty(M)$ . The Hochschild cohomology groups $HH^\bullet(C^\infty(M))$ are isomorphic, via the HKR theorem, to the spaces of multivector fields. The table summarises their geometric, algebraic, and physical interpretations in the context of deformation quantisation.
Cohomology	Geometric object	Algebraic meaning	Physical interpretation
	(HKR isomorphism)	(Deformation theory)
$HH^0(C^\infty(M))$	$C^\infty(M)$	Infinitesimal automorphisms	Classical observables;
	(functions)	of the deformed algebra	constants of motion
$HH^1(C^\infty(M))$	$\mathfrak{X}^1(M)$	Infinitesimal gauge	Symmetries;
	(vector fields)	transformations	Hamiltonian vector fields
$HH^2(C^\infty(M))$	$\mathfrak{X}^2(M)$	Infinitesimal deformations	Poisson structures;
	(bivector fields)	(first-order deformations)	possible quantisations
$HH^3(C^\infty(M))$	$\mathfrak{X}^3(M)$	Obstructions to extending	Quantum anomalies;
	(trivector fields)	deformations to higher orders	obstructions to quantisation

§6. Conclusion and Further Comments<

Key Results and Conceptual Insights

I hope that, at this point, it is clear that the interplay between Hochschild theory, HKR, DGLA, and deformation quantisation yields several interesting results on three fronts: physical, geometrical, and algebraic. Furthermore, it yields some other key results regarding further approaches that, although we shall not examine them in depth, are interesting enough to list:

Fedosov’s construction: For symplectic manifolds, Fedosov’s geometric construction provides an explicit star product. This can be understood as a particular solution of the Maurer–Cartan equation in the Hochschild DGLA, obtained using a symplectic connection.
Weyl quantization: The Moyal product on $\mathbb{R}^{2n}$ is the simplest example of a star product, obtained from the constant Poisson structure. It corresponds to the symmetric ordering prescription.
Geometric quantization: This approach focuses on constructing a Hilbert space of states rather than deforming the algebra of observables. The relation between geometric and deformation quantization remains an active area of research, with connections via the quantization of symplectic groupoids.
Strict deformation quantization: While formal deformation quantization uses formal power series in $\hbar$ , strict deformation quantization produces genuine $C^*$ -algebras. The relationship between the two is not fully understood in general, though in the case of $\mathbb{R}^{2n}$ with the Moyal product, the formal series converges in suitable topologies.

Moreover, this work has gone through a long journey. In particular, we have demonstrated that the HKR isomorphism bridges the algebraic properties of smooth commutative algebras and their underlying geometric structures and that the DGLA structure on the Hochschild cocomplex acts as a unifying framework that bridges physics, geometry, and algebra, with its cohomologies acting as a control mechanism for deformations. But, beyond the setting discussed herein, the utility of the HKR theorem continues to expand into frontiers such as:

Singularities and Non-commutative Geometry: Extending the HKR perspective to singular varieties where the classical smoothness assumption fails, requiring the use of derived algebraic geometry and complexes of sheaves.
Higher Structures: Investigating whether the HKR correspondence can be fully lifted to $L_\infty$ and $A_\infty$ structures, where the Gerstenhaber bracket is replaced by higher operations that encode the “quantum” nature of space-time more deeply.
Categorification: Looking into the HKR theorem through the lens of categorical representations, where the goal is to view the HKR isomorphism as a reflection of deeper dualities between categories of modules and sheaves.

Ultimately, the framework established in this dissertation (or work, perhaps; the author remains unsure which term is the more appropriate one) is one of the most beautiful in mathematical physics. It gives physical – and, depending on one’s ontological commitment, a material – meaning to otherwise abstract mathematical objects such as cohomology groups.

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This class of algebras, in particular, needs additional structure.↩︎