Hochschild Cohomology

Table of Contents

• 1. The Hochschild Cochain Complex
  • 1.1 Setup and Definitions
  • 1.2 The Coboundary Operator
  • 1.3 Nilpotency and Cohomology Groups
  • 1.4 Connection to Operadic Cohomology
• 2. Low-Degree Interpretations
  • 2.1 HH0: The Center
  • 2.2 HH1: Outer Derivations
  • 2.3 HH2: Extensions and Infinitesimal Deformations
  • 2.4 HH3: Obstruction Theory
• 3. The Cup Product and Gerstenhaber Algebra
  • 3.1 The Cup Product
  • 3.2 The Pre-Lie Product and Gerstenhaber Bracket
  • 3.3 The Gerstenhaber Algebra Axioms
• 4. The Deligne Conjecture
  • 4.1 Statement
  • 4.2 Proofs and the E2 Structure
  • 4.3 Corollary: Gerstenhaber Structure on Cohomology
• 5. Cyclic Cohomology
  • 5.1 The Cyclic Operator and Cyclic Cochains
  • 5.2 The Connes B-Operator
  • 5.3 The SBI Long Exact Sequence
  • 5.4 Cyclic Cohomology for Commutative Algebras
• 6. The Hochschild-Kostant-Rosenberg Theorem
  • 6.1 Statement
  • 6.2 The HKR Map
  • 6.3 Consequences
  • 6.4 Failure for Singular Varieties
• 7. Hochschild Cohomology and Deformation Theory
  • 7.1 Formal Deformations and the Maurer-Cartan Equation
  • 7.2 Order-by-Order Expansion
  • 7.3 Rigidity and Formality
• 8. Operadic Cohomology as the General Framework
  • 8.1 The Two-Sided Bar Resolution for Ass
  • 8.2 Analogues for Com and Lie
  • 8.3 Closing the Loop: the Deformation Complex of Ass
• References

1. The Hochschild Cochain Complex 📐

1.1 Setup and Definitions

Throughout this note, \(k\) denotes a commutative ring (often a field of characteristic zero for the finer results), \(A\) a unital associative \(k\)-algebra, and \(M\) an \(A\)-bimodule — that is, a \(k\)-module equipped with compatible left and right \(A\)-actions satisfying \((a \cdot m) \cdot b = a \cdot (m \cdot b)\) for all \(a, b \in A\), \(m \in M\).

The central object of this note lives in Koszul Duality and the Bar Construction §8 as the special case \(\mathcal{O} = \mathrm{Ass}\) of operadic cohomology. We give the fully explicit, classical presentation here.

Definition (Hochschild cochain group). For \(n \geq 0\), the \(n\)-th Hochschild cochain group of \(A\) with coefficients in \(M\) is

\[C^n(A, M) := \mathrm{Hom}_k(A^{\otimes n}, M),\]

where \(A^{\otimes 0} := k\) by convention, so \(C^0(A, M) = M\).

An element \(f \in C^n(A, M)\) is called an \(n\)-cochain; it is a \(k\)-linear map \(f : A^{\otimes n} \to M\).

1.2 The Coboundary Operator

Definition (Hochschild coboundary). The coboundary map \(\delta : C^n(A, M) \to C^{n+1}(A, M)\) is defined by

\[(\delta f)(a_1, \ldots, a_{n+1}) = a_1 \cdot f(a_2, \ldots, a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1, \ldots, a_i a_{i+1}, \ldots, a_{n+1}) + (-1)^{n+1} f(a_1, \ldots, a_n) \cdot a_{n+1}.\]

In degrees 0 and 1 this reads explicitly: - Degree 0: \((\delta m)(a) = a \cdot m - m \cdot a\) for \(m \in M = C^0(A, M)\). - Degree 1: \((\delta f)(a_1, a_2) = a_1 \cdot f(a_2) - f(a_1 a_2) + f(a_1) \cdot a_2\) for \(f \in C^1(A,M)\).

1.3 Nilpotency and Cohomology Groups

Proposition (\(\delta^2 = 0\)). The Hochschild coboundary satisfies \(\delta \circ \delta = 0\), making \((C^\bullet(A,M), \delta)\) a cochain complex.

Proof sketch. One computes \((\delta^2 f)(a_1, \ldots, a_{n+2})\) directly. The sum splits into four types of terms: 1. Left action of \(a_1\) on \((\delta f)(a_2, \ldots)\), which introduces terms with consecutive multiplications \(a_i a_{i+1}\) for \(i \geq 2\) and a right-action term. 2. Multiplication terms \((\delta f)(\ldots, a_j a_{j+1}, \ldots)\) for \(j \geq 1\), each producing further left-action, multiplication, and right-action sub-terms. 3. Right-action terms.

Each pair of consecutive sign indices produces a cancelling pair with opposite signs, by the standard simplicial identity \(d_i d_j = d_{j+1} d_i\) for \(i \leq j\) (here \(d_i\) is the \(i\)-th face map). Precisely, for \(0 \leq i < j \leq n+1\) the contribution from applying face \(d_j\) then \(d_i\) cancels against the contribution from applying \(d_{i+1}\) then \(d_j\), since \((-1)^i (-1)^j + (-1)^{j+1} (-1)^i = 0\). The full telescoping yields \(\delta^2 = 0\). \(\square\)

Definition (Hochschild cohomology). The \(n\)-th Hochschild cohomology group of \(A\) with coefficients in \(M\) is

\[HH^n(A, M) := H^n(C^\bullet(A, M), \delta) = \frac{\ker(\delta : C^n \to C^{n+1})}{\mathrm{im}(\delta : C^{n-1} \to C^n)}.\]

Elements of \(\ker \delta\) are \(n\)-cocycles; elements of \(\mathrm{im} \delta\) are \(n\)-coboundaries.

1.4 Connection to Operadic Cohomology

The Hochschild complex is the prototypical example of operadic cohomology for the operad \(\mathrm{Ass}\). Recall from Koszul Duality and the Bar Construction §8.1 that for an operad \(\mathcal{O}\), an \(\mathcal{O}\)-algebra \(A\), and an \(A\)-module \(M\), the operadic cohomology \(H^\bullet_\mathcal{O}(A, M)\) is computed by the two-sided bar resolution \(B(\mathcal{O}^!, A, M)\), where \(\mathcal{O}^!\) is the Koszul dual cooperad.

Theorem (Hochschild = operadic cohomology for Ass). There is a natural isomorphism

\[HH^\bullet(A, M) \cong H^\bullet_{\mathrm{Ass}}(A, M),\]

where the right-hand side is computed by the two-sided bar complex \(B(\mathrm{Ass}^!, A, M) = B(\mathrm{Com}, A, M)\), using the fact that \(\mathrm{Ass}^! \cong \mathrm{Ass}\) (the Koszul dual of \(\mathrm{Ass}\) as a cooperad is \(\mathrm{Com}\), reflecting the duality \(\mathrm{Ass}^\vee \cong \mathrm{Ass}\)). The explicit generators of the bar complex in arity \(n\) are \(k\)-linear maps \(A^{\otimes n} \to M\), recovering \(C^n(A,M)\), and the bar differential recovers \(\delta\) precisely.

This is the crucial point: every special feature of Hochschild cohomology — the Gerstenhaber algebra structure, the deformation interpretation, the HKR theorem — is a manifestation of the rich structure of the operad \(\mathrm{Ass}\).

2. Low-Degree Interpretations 🔑

2.1 HH0: The Center

Proposition (HH0 = center). There is a natural identification

\[HH^0(A, M) = \{m \in M : a \cdot m = m \cdot a \text{ for all } a \in A\},\]

the center \(Z(A, M)\) of \(M\) as an \(A\)-bimodule. In particular, \(HH^0(A, A) = Z(A)\), the center of the algebra \(A\).

Proof. Since \(C^{-1}(A,M) = 0\), every \(m \in C^0(A,M) = M\) is trivially a coboundary-free cochain; the cocycle condition \(\delta m = 0\) reads \(a \cdot m - m \cdot a = 0\) for all \(a \in A\). \(\square\)

2.2 HH1: Outer Derivations

Definition (Derivation). A derivation of \(A\) with values in \(M\) is a \(k\)-linear map \(d : A \to M\) satisfying the Leibniz rule:

\[d(ab) = a \cdot d(b) + d(a) \cdot b \quad \text{for all } a, b \in A.\]

The \(k\)-module of all such derivations is denoted \(\mathrm{Der}(A, M)\). An inner derivation is one of the form \(d_m(a) = a \cdot m - m \cdot a\) for some fixed \(m \in M\).

Proposition (HH1 = outer derivations). There is a natural isomorphism

\[HH^1(A, M) \cong \mathrm{Der}(A, M) / \mathrm{InnDer}(A, M).\]

Proof. We show cocycles \(= \mathrm{Der}(A,M)\) and coboundaries \(= \mathrm{InnDer}(A,M)\).

Cocycles are derivations. Let \(f \in C^1(A,M) = \mathrm{Hom}_k(A, M)\). Then

\[(\delta f)(a_1, a_2) = a_1 \cdot f(a_2) - f(a_1 a_2) + f(a_1) \cdot a_2.\]

Setting \(\delta f = 0\) gives exactly \(f(a_1 a_2) = a_1 \cdot f(a_2) + f(a_1) \cdot a_2\), which is the Leibniz rule.

Coboundaries are inner. An element \(\delta m \in C^1(A,M)\) for \(m \in M\) satisfies \((\delta m)(a) = a \cdot m - m \cdot a = d_m(a)\). This is precisely the inner derivation associated to \(m\).

Combining: \(HH^1(A,M) = \ker \delta / \mathrm{im} \delta = \mathrm{Der}(A,M) / \mathrm{InnDer}(A,M)\). \(\square\)

2.3 HH2: Extensions and Infinitesimal Deformations

Definition (Singular extension). A singular extension of \(A\) by \(M\) is a short exact sequence of \(k\)-algebras

\[0 \to M \to E \xrightarrow{\pi} A \to 0,\]

where \(M\) is given the zero-product algebra structure (\(M^2 = 0\)), and the \(A\)-bimodule structure on \(M\) is induced by choosing a \(k\)-linear section \(s : A \to E\) of \(\pi\).

Two extensions are equivalent if there is an algebra isomorphism \(\phi : E \to E'\) commuting with the inclusion of \(M\) and projection to \(A\).

Theorem (HH2 classifies extensions). There is a bijection

\[HH^2(A, M) \xrightarrow{\sim} \{\text{equivalence classes of singular extensions of } A \text{ by } M\}.\]

Proof sketch. Given a \(k\)-linear section \(s : A \to E\), the failure of \(s\) to be an algebra map is measured by

\[\mu_1(a, b) := s(a) s(b) - s(ab) \in M \quad (M \hookrightarrow E),\]

since \(\pi(s(a)s(b) - s(ab)) = ab - ab = 0\). One checks that \(\mu_1 \in C^2(A,M)\).

Associativity of \(E\) forces \(\delta \mu_1 = 0\): indeed, \((s(a)s(b))s(c) = s(a)(s(b)s(c))\) implies

\[\mu_1(a,b) \cdot c - \mu_1(ab, c) + \mu_1(a, bc) - a \cdot \mu_1(b,c) = 0,\]

which is \((\delta \mu_1)(a,b,c) = 0\). A different choice of section \(s' = s + \phi\) (where \(\phi : A \to M\)) changes \(\mu_1\) by \(\delta\phi\). Thus the cohomology class \([\mu_1] \in HH^2(A,M)\) is well-defined and classifies the extension. One constructs an inverse: given a 2-cocycle \(\mu_1\), set \(E = M \oplus A\) as \(k\)-modules with multiplication \((m, a)(n, b) = (m \cdot b + a \cdot n + \mu_1(a,b), ab)\). Associativity of this product is equivalent to \(\delta \mu_1 = 0\). \(\square\)

Corollary (HH2 classifies infinitesimal deformations). An infinitesimal deformation of \(A\) is a \(k[\epsilon]/(\epsilon^2)\)-algebra \(A_\epsilon = (A[\epsilon]/(\epsilon^2), \star)\) with

\[a \star b = ab + \epsilon \mu_1(a, b), \quad \mu_1 \in C^2(A, A).\]

Associativity of \(\star\) forces \(\delta \mu_1 = 0\). Two deformations \(\mu_1, \mu_1'\) are isomorphic (via a \(k[\epsilon]/(\epsilon^2)\)-algebra isomorphism \(\phi_\epsilon = \mathrm{id} + \epsilon \phi_1\)) if and only if \(\mu_1 - \mu_1' = \delta \phi_1\). Thus \(HH^2(A,A)\) is in bijection with isomorphism classes of infinitesimal deformations of \(A\).

2.4 HH3: Obstruction Theory

Theorem (HH3 = obstructions). Let \(\mu_1 \in Z^2(A,A)\) be a 2-cocycle representing an infinitesimal deformation. An extension to a second-order deformation \(\star = ab + t\mu_1(a,b) + t^2 \mu_2(a,b) + O(t^3)\) over \(k[t](/notes/t/)/(t^3)\) exists if and only if a certain obstruction class \(\mathrm{obs}(\mu_1) \in HH^3(A,A)\) vanishes.

Proof sketch. Associativity of \(a \star (b \star c) = (a \star b) \star c\) at order \(t^2\) yields the equation

\[\delta \mu_2(a,b,c) = \frac{1}{2} [\mu_1, \mu_1]_G(a,b,c),\]

where \([-,-]_G\) is the Gerstenhaber bracket defined in §3. The right-hand side is always a 3-cocycle (by the Jacobi identity for \([-,-]_G\)). A solution \(\mu_2\) exists if and only if \([\mu_1, \mu_1]_G\) is a coboundary, i.e., \([[\mu_1, \mu_1]_G] = 0 \in HH^3(A,A)\). If \(HH^3(A,A) = 0\) then every infinitesimal deformation extends to all orders. \(\square\)

3. The Cup Product and Gerstenhaber Algebra 📐

3.1 The Cup Product

The special case \(M = A\) (coefficients in \(A\) itself) endows \(C^\bullet(A,A)\) with additional multiplicative structure. Here \(A\) is simultaneously the coefficient bimodule and the algebra, and the product in \(A\) can be used to compose cochains.

Definition (Cup product). For \(f \in C^p(A,A)\) and \(g \in C^q(A,A)\), define the cup product \(f \smile g \in C^{p+q}(A,A)\) by

\[(f \smile g)(a_1, \ldots, a_{p+q}) = f(a_1, \ldots, a_p) \cdot g(a_{p+1}, \ldots, a_{p+q}),\]

where the dot on the right is the product in \(A\).

Proposition (Leibniz rule for the cup product). The coboundary satisfies

\[\delta(f \smile g) = (\delta f) \smile g + (-1)^p f \smile (\delta g).\]

Proof sketch. Expand \((\delta(f \smile g))(a_1, \ldots, a_{p+q+1})\) using the definition of \(\delta\). The left-action term \(a_1 \cdot (f \smile g)(a_2, \ldots)\) gives rise to \(a_1 \cdot f(a_2, \ldots, a_{p+1}) \cdot g(a_{p+2}, \ldots)\), which belongs to the \((\delta f) \smile g\) expansion. The middle multiplication terms split at the break point \(i = p\): for \(i < p\) they contribute to \((\delta f) \smile g\), and for \(i > p\) they contribute to \((-1)^p f \smile (\delta g)\). The term at \(i = p\) produces \(f(a_1, \ldots, a_{p-1}, a_p) \cdot g(a_{p+1}, \ldots)\), which from the left factor side gives \(f(\ldots, a_{p-1} a_p)\) times \(g\) and from the right factor side gives \(f(\ldots, a_p) \cdot g(a_{p+1} a_{p+2}, \ldots)\) — these appear with signs that match the Leibniz sign. The right-action term completes the \((-1)^p f \smile (\delta g)\) side. \(\square\)

Corollary. The cup product descends to cohomology, making \(HH^\bullet(A,A)\) into a graded ring. Moreover, the cup product is graded-commutative on cohomology: \([f \smile g] = (-1)^{pq} [g \smile f]\).

3.2 The Pre-Lie Product and Gerstenhaber Bracket

To capture the full Lie-algebraic structure, one introduces a finer operation on cochains.

Definition (Pre-Lie product). For \(f \in C^p(A,A)\) and \(g \in C^q(A,A)\), the pre-Lie product (or circle product) \(f \circ g \in C^{p+q-1}(A,A)\) is

\[(f \circ g)(a_1, \ldots, a_{p+q-1}) = \sum_{i=0}^{p-1} (-1)^{i(q-1)} f(a_1, \ldots, a_i, g(a_{i+1}, \ldots, a_{i+q}), a_{i+q+1}, \ldots, a_{p+q-1}).\]

This inserts \(g\) into each of the \(p\) slots of \(f\), with the sign \((-1)^{i(q-1)}\) accounting for the graded nature of the insertion.

Definition (Gerstenhaber bracket). The Gerstenhaber bracket of \(f \in C^p(A,A)\) and \(g \in C^q(A,A)\) is

\[[f, g]_G = f \circ g - (-1)^{(p-1)(q-1)} g \circ f \in C^{p+q-1}(A,A).\]

Proposition (Properties of the bracket). On \(C^\bullet(A,A)\), the bracket \([-,-]_G\) satisfies: 1. Graded skew-symmetry: \([f,g]_G = -(-1)^{(p-1)(q-1)} [g,f]_G\). 2. Graded Jacobi identity: \([f, [g,h]_G]_G = [[f,g]_G, h]_G + (-1)^{(p-1)(q-1)}[g, [f,h]_G]_G\). 3. Bracket vs. coboundary: \(\delta[f,g]_G = [\delta f, g]_G + (-1)^{p-1} [f, \delta g]_G\) (i.e., \(\delta\) is a derivation of the bracket).

Proof sketch for (2). This follows from the pre-Lie associator identity: \((f \circ g) \circ h - f \circ (g \circ h) = (f \circ h) \circ g - f \circ (h \circ g)\) (symmetry of the “failure of associativity”). Antisymmetrizing recovers the Jacobi identity. \(\square\)

3.3 The Gerstenhaber Algebra Axioms

Definition (Gerstenhaber algebra). A Gerstenhaber algebra is a graded \(k\)-module \(G = \bigoplus_{n \geq 0} G^n\) equipped with: - A graded-commutative associative product \(\smile : G^p \otimes G^q \to G^{p+q}\). - A degree \((-1)\) Lie bracket \([-,-]_G : G^p \otimes G^q \to G^{p+q-1}\), satisfying graded skew-symmetry and the graded Jacobi identity. - A Poisson compatibility (graded Leibniz rule): for \(f \in G^p\),

\[[f, g \smile h]_G = [f, g]_G \smile h + (-1)^{(p-1)|g|} g \smile [f, h]_G.\]

Theorem (Gerstenhaber, 1963). The triple \((HH^\bullet(A,A), \smile, [-,-]_G)\) is a Gerstenhaber algebra.

Proof sketch. The graded-commutativity of \(\smile\) and graded Jacobi identity for \([-,-]_G\) both hold on cohomology (they hold up to coboundary at chain level, and coboundaries vanish in cohomology). The Poisson compatibility follows from the Leibniz property of \(\delta\) with respect to both operations. \(\square\)

Remark. The degree shift is conventional: if one sets \(\tilde{G}^n = G^{n+1}\), then \([-,-]_G\) becomes a degree \(+1\) Lie bracket on \(\tilde{G}^\bullet\), matching the convention in which a Gerstenhaber algebra is a graded-commutative algebra with a degree \(+1\) Poisson bracket. Both conventions appear in the literature.

4. The Deligne Conjecture 💡

4.1 Statement

In a 1993 letter, Pierre Deligne asked whether the Hochschild cochain complex \(C^\bullet(A,A)\) carries an action of the little 2-disks operad \(\mathcal{D}_2\) (also called the \(E_2\) operad) at the chain level — not merely the Gerstenhaber structure visible on cohomology. This became known as the Deligne conjecture.

Conjecture/Theorem (Deligne, proven 1998–2000). For any associative \(k\)-algebra \(A\), the Hochschild cochain complex \(C^\bullet(A,A)\) admits a natural structure of an algebra over the little 2-disks operad \(\mathcal{D}_2\) (equivalently, an \(E_2\)-algebra structure). This chain-level structure descends on passing to cohomology to the Gerstenhaber algebra structure on \(HH^\bullet(A,A)\).

The little \(n\)-disks operad \(\mathcal{D}_n\) has \(\mathcal{D}_n(k)\) = the space of configuration of \(k\) non-overlapping disks in the unit \(n\)-disk. An \(E_n\)-algebra is an algebra over (a chain-level model of) \(\mathcal{D}_n\). An \(E_1\)-algebra is an \(A_\infty\)-algebra; an \(E_2\)-algebra has both an \(A_\infty\)-structure and a compatible (homotopy) commutativity one degree down.

4.2 Proofs and the E2 Structure

The Deligne conjecture has been proven by multiple teams:

• McClure–Smith (2002): Using the surjection operad, a combinatorial model for \(E_2\), acting on the Hochschild complex via explicit operations on cochains.
• Tamarkin (1998): Using the formality of \(\mathcal{D}_2\) over \(\mathbb{Q}\) (equivalently, the fact that \(\mathcal{D}_2\) is formal as an operad) to transport the \(E_2\)-structure to \(C^\bullet(A,A)\).
• Kontsevich–Soibelman: Via explicit formulas using configuration spaces and compactifications.
• Voronov: Using the Swiss-cheese operad.

The key mechanism: \(C^\bullet(A,A)\) is a brace algebra, which is an algebra over a certain combinatorial operad \(\mathrm{Br}\) whose spaces of operations \(\mathrm{Br}(k)\) are spanned by rooted bipartite planar trees. The brace operations encode all ways to “insert” cochains into one another with correct signs, and the operad \(\mathrm{Br}\) receives a quasi-isomorphism from the chains \(C_\bullet(\mathcal{D}_2)\) on the little 2-disks operad.

4.3 Corollary: Gerstenhaber Structure on Cohomology

Corollary. Since the homology of an \(E_2\)-algebra is a Gerstenhaber algebra (this is a formal consequence of the operad theory: the homology of \(C_\bullet(\mathcal{D}_2)\) is the Gerstenhaber operad), the Deligne conjecture implies that \(HH^\bullet(A,A)\) is a Gerstenhaber algebra.

This recovers Gerstenhaber’s 1963 theorem as a corollary of the operadic perspective.

The Deligne conjecture also generalizes: for any \(A_\infty\)-algebra \((A, \{m_n\})\), its Hochschild cochain complex is an \(E_2\)-algebra (Kontsevich–Soibelman, Tamarkin). This is the starting point for the formality theorem in deformation quantization.

5. Cyclic Cohomology 📐

Cyclic cohomology, introduced independently by Alain Connes and Boris Tsygan in the 1980s, is a refinement of Hochschild cohomology that captures additional information related to traces, index theory, and the de Rham cohomology of the “noncommutative space” represented by \(A\).

5.1 The Cyclic Operator and Cyclic Cochains

We specialize to \(M = A^* := \mathrm{Hom}_k(A, k)\) (the linear dual of \(A\)), identifying \(C^n(A, A^*) \cong \mathrm{Hom}_k(A^{\otimes n+1}, k)\) via \(f(a_0, \ldots, a_n) = (f(a_1, \ldots, a_n))(a_0)\).

Definition (Cyclic operator). The cyclic operator \(\lambda : C^n(A, A^*) \to C^n(A, A^*)\) (equivalently, on \(\mathrm{Hom}_k(A^{\otimes n+1}, k)\)) is defined by

\[(\lambda f)(a_0, a_1, \ldots, a_n) = (-1)^n f(a_n, a_0, a_1, \ldots, a_{n-1}).\]

This implements a cyclic rotation of the inputs, with the sign \((-1)^n\) ensuring compatibility with the coboundary.

Definition (Cyclic cochains and cyclic cohomology). The cyclic cochains are the fixed-points of \(\lambda\):

\[C^n_\lambda(A) := \ker(1 - \lambda) \subset C^n(A, A^*).\]

These are functions \(f : A^{\otimes n+1} \to k\) satisfying \(f(a_0, \ldots, a_n) = (-1)^n f(a_n, a_0, \ldots, a_{n-1})\) (cyclic symmetry). The key fact is that \(\delta\) preserves the cyclic eigenspaces, so \((C^\bullet_\lambda(A), \delta)\) is a sub-cochain complex. The cyclic cohomology of \(A\) is

\[HC^n(A) := H^n(C^\bullet_\lambda(A), \delta).\]

5.2 The Connes B-Operator

Definition (Connes’ \(B\)-operator). The operator \(B : C^n(A, A^*) \to C^{n-1}(A, A^*)\) (a map decreasing degree by 1) is defined by

\[(Bf)(a_0, \ldots, a_{n-1}) = \sum_{i=0}^{n-1} (-1)^{ni} f(1, a_i, a_{i+1}, \ldots, a_{n-1}, a_0, \ldots, a_{i-1}),\]

where \(1 \in A\) is the unit. Equivalently, \(B = (1 - (-1)^n \lambda) \circ s \circ N\) where \(s\) is the extra degeneracy (insertion of 1) and \(N\) is the cyclic norm.

Proposition. The operators \(b = \delta\) (Hochschild coboundary) and \(B\) satisfy: 1. \(B^2 = 0\). 2. \(b B + B b = 0\).

These make \((C^\bullet(A, A^*), b, B)\) a mixed complex — a cochain complex for \(b\) equipped with a chain map \(B\) satisfying \(B^2 = 0\).

Proof sketch. \(B^2 = 0\) follows from \(N(1 - \lambda) = 0\). For \(bB + Bb = 0\): this is a direct consequence of the simplicial identities for the extra degeneracy \(s\) and the Hochschild differential \(b\). \(\square\)

5.3 The SBI Long Exact Sequence

The deepest structural result relating Hochschild and cyclic cohomology is Connes’ long exact sequence, often called the SBI sequence (Periodicity–\(B\)–Inclusion).

Theorem (Connes’ SBI sequence). There is a long exact sequence

\[\cdots \to HC^{n-1}(A) \xrightarrow{S} HC^{n+1}(A) \xrightarrow{B} HH^n(A, A^*) \xrightarrow{I} HC^n(A) \xrightarrow{S} HC^{n+2}(A) \to \cdots\]

where: - \(I : HH^\bullet(A, A^*) \to HC^\bullet(A)\) is the natural map induced by the inclusion of Hochschild cochains with the zero \(B\)-differential. - \(B : HC^{n+1}(A) \to HH^n(A, A^*)\) is induced by the Connes \(B\)-operator. - \(S : HC^{n-1}(A) \to HC^{n+1}(A)\) is the periodicity operator (cup product with a canonical class in \(HC^2(k)\)).

Proof sketch. Consider the short exact sequence of mixed complexes \(0 \to C^\bullet_\lambda \to C^\bullet(A, A^*) \to \Sigma^{-2} C^\bullet_\lambda \to 0\) (where \(\Sigma^{-2}\) is a degree shift), whose connecting homomorphism is \(S\). The associated long exact sequence in cohomology is the SBI sequence. \(\square\)

The periodicity operator \(S\) has degree \(+2\): it relates cyclic cohomology groups two steps apart. Taking the direct limit along \(S\) gives periodic cyclic cohomology \(HP^\bullet(A) = \varinjlim_S HC^\bullet(A)\), which is \(\mathbb{Z}/2\mathbb{Z}\)-graded.

5.4 Cyclic Cohomology for Commutative Algebras

For a smooth commutative \(k\)-algebra \(A\), the Hochschild–Kostant–Rosenberg theorem (§6) and the Connes \(B\)-operator together give a complete picture.

Theorem (HKR + cyclic cohomology). For \(A\) smooth commutative over \(k \supset \mathbb{Q}\):

\[HC^n(A) \cong \frac{\Omega^n_{A/k}}{d\Omega^{n-1}_{A/k}} \oplus H^{n-2}_{\mathrm{dR}}(A) \oplus H^{n-4}_{\mathrm{dR}}(A) \oplus \cdots\]

where \(H^\bullet_{\mathrm{dR}}(A) = H^\bullet(\Omega^\bullet_{A/k}, d)\) is the algebraic de Rham cohomology. The Connes \(B\)-operator corresponds to the de Rham differential \(d : \Omega^n \to \Omega^{n+1}\) under the HKR identification.

This is the algebraic incarnation of the statement that cyclic cohomology of the algebra of functions on a manifold recovers the de Rham cohomology of the manifold.

6. The Hochschild-Kostant-Rosenberg Theorem 🔑

6.1 Statement

The HKR theorem is one of the most important structural results in algebra: it identifies Hochschild cohomology with polyvector fields and Hochschild homology with differential forms, for smooth commutative algebras. It is also the theorem that fails in the noncommutative or singular setting, forcing the use of the full machinery of deformation theory.

Theorem (Hochschild–Kostant–Rosenberg, 1962). Let \(k\) be a field of characteristic zero and let \(A\) be a smooth, finitely generated commutative \(k\)-algebra (equivalently, the coordinate ring \(\mathcal{O}(X)\) of a smooth affine variety \(X = \mathrm{Spec}(A)\)). Then there are natural isomorphisms of graded \(A\)-modules:

\[HH^n(A, A) \cong \bigwedge^n_A \mathrm{Der}(A, A) \qquad \text{(polyvector fields)},\]

\[HH^n(A, A^*) \cong \Omega^n_{A/k} \qquad \text{(Kähler differential forms)}.\]

6.2 The HKR Map

The HKR isomorphism is given by an explicit quasi-isomorphism at chain level.

Definition (HKR map). Define \(\mathrm{HKR} : \bigwedge^n_A \mathrm{Der}(A) \to C^n(A,A)\) by

\[\mathrm{HKR}(\partial_1 \wedge \cdots \wedge \partial_n)(a_1, \ldots, a_n) = \frac{1}{n!} \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) \, \partial_{\sigma(1)}(a_1) \cdots \partial_{\sigma(n)}(a_n),\]

where \(\partial_1, \ldots, \partial_n \in \mathrm{Der}(A,A)\) are derivations and the product is in \(A\).

Proposition (HKR is a quasi-isomorphism). The map \(\mathrm{HKR}\) is a morphism of cochain complexes (where \(\bigwedge^\bullet_A \mathrm{Der}(A)\) has zero differential) and induces the HKR isomorphism on cohomology.

Proof sketch. One verifies that the image of \(\mathrm{HKR}\) lands in \(\ker \delta\): this uses commutativity of \(A\) and the Leibniz rule for derivations. The key identity is that \(\delta(\mathrm{HKR}(\partial_1 \wedge \cdots \wedge \partial_n)) = 0\) for any tuple of derivations — a consequence of the fact that for commutative \(A\), the “non-flat” terms in the Hochschild differential cancel by symmetry. To show \(\mathrm{HKR}\) is a quasi-isomorphism: filter both sides by the polynomial degree in \(A\), and use the Koszul resolution of \(A\) as an \(A \otimes_k A^{\mathrm{op}}\)-module (which exists and is explicit when \(A\) is smooth/flat over \(k\)). The associated graded computation reduces to an elementary polynomial calculation. \(\square\)

6.3 Consequences

The HKR theorem has structural consequences at the dg level as well.

Theorem (HKR as dg-algebra quasi-isomorphism, Kontsevich). For \(A = k[x_1, \ldots, x_d]\) (or any smooth affine variety over \(k \supset \mathbb{Q}\)), there is a quasi-isomorphism of dg-algebras

\[\left(\bigwedge^\bullet_A \mathrm{Der}(A), 0\right) \xrightarrow{\sim} (C^\bullet(A,A), \delta).\]

Moreover, there is a quasi-isomorphism of \(L_\infty\)-algebras (the Kontsevich formality theorem)

\[\left(\bigwedge^\bullet_A \mathrm{Der}(A), 0, \text{Schouten bracket}\right) \xrightarrow{\sim} (C^\bullet(A,A), \delta, [-,-]_G),\]

where the Schouten bracket on polyvector fields is the natural extension of the Lie bracket on \(\mathrm{Der}(A)\).

Corollary (Classification of formal deformations of smooth commutative algebras). Every formal deformation of a smooth commutative \(k\)-algebra \(A\) (over \(k \supset \mathbb{Q}\)) is classified, up to gauge equivalence, by a Maurer-Cartan element in \((\bigwedge^\bullet_A \mathrm{Der}(A))[t](/notes/t/)\) — that is, by a formal Poisson bivector field \(\pi \in \mathrm{Der}(A)^{\wedge 2}[t](/notes/t/)\) satisfying \([\pi, \pi]_{\text{Schouten}} = 0\). This is the algebraic content of Kontsevich’s deformation quantization theorem.

6.4 Failure for Singular Varieties

Remark (Singular case). When \(A\) is not smooth, the HKR isomorphism fails. The correct replacement involves the cotangent complex \(\mathbb{L}_{A/k}\) of André–Quillen: one has

\[HH^\bullet(A,A) \cong \mathrm{Ext}^\bullet_{A \otimes A^{\mathrm{op}}}(A, A),\]

and this no longer simplifies to polyvector fields unless \(\mathbb{L}_{A/k} \simeq \Omega^1_{A/k}[0]\) (i.e., \(A\) is smooth). For singular \(A\), deformation theory is controlled by the full derived structure of the cotangent complex, not by Kähler differentials alone.

7. Hochschild Cohomology and Deformation Theory 💡

This section connects to Entropy as an Operad Derivation and to the deformation theory portion of the operads curriculum.

7.1 Formal Deformations and the Maurer-Cartan Equation

Definition (Formal deformation). A formal deformation of an associative \(k\)-algebra \(A\) is a \(k[t](/notes/t/)\)-algebra \((A[t](/notes/t/), \star)\) where \(A[t](/notes/t/) = A \otimes_k k[t](/notes/t/)\) as a \(k[t](/notes/t/)\)-module, and the product is

\[a \star b = ab + \sum_{n=1}^\infty t^n \mu_n(a,b), \quad \mu_n \in C^2(A,A),\]

with \(ab\) the original product in \(A\). Write \(\mu = \sum_{n \geq 1} t^n \mu_n \in C^2(A,A)[t](/notes/t/)\).

Associativity of \(\star\) is equivalent to the Maurer-Cartan equation in the differential graded Lie algebra \((C^\bullet(A,A)[t](/notes/t/), \delta, [-,-]_G)\):

\[\delta\mu + \frac{1}{2}[\mu, \mu]_G = 0 \quad \in C^3(A,A)[t](/notes/t/).\]

7.2 Order-by-Order Expansion

Expanding the Maurer-Cartan equation order by order in \(t\):

Order \(t^1\): \(\delta\mu_1 = 0\). Thus \(\mu_1 \in Z^2(A,A)\) is a Hochschild 2-cocycle. Every infinitesimal deformation comes from a 2-cocycle.

Order \(t^2\): \(\delta\mu_2 + \frac{1}{2}[\mu_1, \mu_1]_G = 0\). Since \([\mu_1, \mu_1]_G \in C^3(A,A)\), a solution \(\mu_2\) exists if and only if \(\frac{1}{2}[\mu_1, \mu_1]_G\) is a coboundary in \(C^3(A,A)\), i.e., if and only if \(\frac{1}{2}[[\mu_1, \mu_1]_G] = 0\) in \(HH^3(A,A)\).

Definition (Obstruction class). The class \(\mathrm{obs}(\mu_1) := \frac{1}{2}[[\mu_1, \mu_1]_G] \in HH^3(A,A)\) is the obstruction class to extending the first-order deformation \(\mu_1\) to second order.

Order \(t^n\): \(\delta\mu_n + \frac{1}{2}\sum_{i=1}^{n-1}[\mu_i, \mu_{n-i}]_G = 0\). The obstruction at order \(n\) is the class \([\sum_{i=1}^{n-1}[\mu_i, \mu_{n-i}]_G] \in HH^3(A,A)\).

7.3 Rigidity and Formality

Definition (Rigidity). An algebra \(A\) is rigid if \(HH^2(A,A) = 0\). For rigid \(A\), every formal deformation is trivial (isomorphic to the undeformed algebra \(A[t](/notes/t/)\)).

Definition (Formality). A dg-algebra \((V, d)\) is formal if there exists a zigzag of dg-algebra quasi-isomorphisms connecting \((V, d)\) to \((H^\bullet(V), 0)\) (its cohomology with zero differential).

Theorem (Kontsevich formality theorem, 1997). For any smooth manifold \(X\) (or smooth affine variety over \(\mathbb{Q}\)), the Hochschild cochain complex \(C^\bullet(\mathcal{O}(X), \mathcal{O}(X))\) is formal as an \(L_\infty\)-algebra: there is an \(L_\infty\)-quasi-isomorphism

\[C^\bullet(\mathcal{O}(X), \mathcal{O}(X)) \xrightarrow{\sim} HH^\bullet(\mathcal{O}(X), \mathcal{O}(X)) \cong \bigwedge^\bullet_{\mathcal{O}(X)} \mathrm{Der}(\mathcal{O}(X))\]

(by HKR). This implies that deformations of \(\mathcal{O}(X)\) as an associative algebra are classified by Maurer-Cartan elements in the Schouten algebra of polyvector fields — i.e., by Poisson structures on \(X\). Every Poisson manifold admits a formal deformation quantization.

8. Operadic Cohomology as the General Framework 🔑

This section closes the loop by situating everything in the operadic framework from Koszul Duality and the Bar Construction §8.

8.1 The Two-Sided Bar Resolution for Ass

Recall from Algebras and Modules that the enveloping algebra of an \(\mathrm{Ass}\)-algebra \(A\) is \(U_{\mathrm{Ass}}(A) = A \otimes_k A^{\mathrm{op}}\), and an \(A\)-bimodule is a left \(U_{\mathrm{Ass}}(A)\)-module. The operadic cohomology is computed by a projective resolution of \(A\) as a \(U_{\mathrm{Ass}}(A)\)-module.

Definition (Bar resolution). The two-sided bar resolution (or standard bar resolution) of \(A\) over \(A \otimes A^{\mathrm{op}}\) is the chain complex

\[B(A) : \cdots \to A \otimes \bar{A}^{\otimes 2} \otimes A \xrightarrow{d} A \otimes \bar{A} \otimes A \xrightarrow{d} A \otimes A \xrightarrow{m} A \to 0,\]

where \(\bar{A} = A/k\cdot 1\) (the augmentation ideal) and the differential is

\[d(a_0 \otimes [a_1|\cdots|a_n] \otimes a_{n+1}) = \sum_{i=0}^n (-1)^i a_0 \otimes [a_1|\cdots|a_i a_{i+1}|\cdots|a_n] \otimes a_{n+1}\]

(with \(a_0 a_1\) replacing \(a_0 \otimes [a_1|\cdots]\) at \(i=0\) and \(a_n a_{n+1}\) at \(i=n\)). This is a free resolution of \(A\) as an \(A \otimes A^{\mathrm{op}}\)-module.

Theorem. Applying \(\mathrm{Hom}_{A \otimes A^{\mathrm{op}}}(-, M)\) to the bar resolution \(B(A)\) yields the Hochschild cochain complex \(C^\bullet(A,M)\). Hence

\[HH^n(A,M) = \mathrm{Ext}^n_{A \otimes A^{\mathrm{op}}}(A, M) = H^n_{\mathrm{Ass}}(A, M).\]

This is the identification of Hochschild cohomology as operadic cohomology for \(\mathcal{O} = \mathrm{Ass}\). The two-sided bar complex \(B(\mathrm{Ass}^!, A, M)\) of the general operadic framework (with \(\mathrm{Ass}^! = \mathrm{Ass}\) as a cooperad, reflecting the Koszul self-duality of \(\mathrm{Ass}\)) reduces to this bar resolution.

8.2 Analogues for Com and Lie

The operadic framework unifies Hochschild cohomology with other classical cohomology theories:

The key point: each cohomology theory is operadic cohomology for the relevant operad. The Koszul dual pairs govern the form of the bar complex: - \(\mathrm{Ass}^! = \mathrm{Ass}\): the bar complex of \(\mathrm{Ass}\) is \(\mathrm{Ass}\)-shaped (sequences of associative multiplications). - \(\mathrm{Com}^! = \mathrm{Lie}\): Harrison cohomology uses the Lie-type bar complex (antisymmetric tensors); this is why \(\mathrm{Harr}^\bullet(A,M)\) is a summand of \(HH^\bullet(A,M)\) for commutative \(A\). - \(\mathrm{Lie}^! = \mathrm{Com}\): the CE complex uses the exterior algebra (com-type structure).

8.3 Closing the Loop: the Deformation Complex of Ass

The deformation theory of the operad \(\mathrm{Ass}\) itself (as an operad, not just as an algebra) produces a complex whose cohomology encodes the space of all deformations of associative algebra structures. This is the Gerstenhaber–Schack complex or the operadic deformation complex \(\mathrm{Def}(\mathrm{Ass})\).

Theorem. The deformation complex of \(\mathrm{Ass}\) (the dg-Lie algebra governing deformations of \(\mathrm{Ass}\) as an operad) is quasi-isomorphic to the full Hochschild cochain complex \(C^\bullet(A,A)\) for any \(\mathrm{Ass}\)-algebra \(A\). The Gerstenhaber bracket on \(C^\bullet(A,A)\) is the Lie bracket of the deformation complex. The Maurer-Cartan elements of \(\mathrm{Def}(\mathrm{Ass})\) are exactly the formal deformations of \(\mathrm{Ass}\)-algebra structures.

This unification — Hochschild cohomology as deformation theory of the operad \(\mathrm{Ass}\) — is the deepest structural statement, and it specializes to all the low-degree interpretations (§2) simultaneously: \(HH^1\) = infinitesimal automorphisms of \(\mathrm{Ass}\)-structure, \(HH^2\) = infinitesimal deformations, \(HH^3\) = obstructions.

Algorithmic Applications

import numpy as np

# Basis elements: e0 = 1, e1 = x in A
# C^0(A,A) = A, spanned by m0=(1 component), m1=(x component)
# delta^0(m)(a) = a*m - m*a
# For commutative A (dual numbers is commutative!): delta^0 = 0
# So HH^0 = A = k^2

# C^1(A,A) = Hom(A, A), spanned by f_{ij} where f_{ij}(e_i) = e_j
# delta^1(f)(a,b) = a*f(b) - f(ab) + f(a)*b
# Build delta^1 as 4x4 matrix (dim C^2 = 4, dim C^1 = 4)

def mult(a, b):
    # a = (a0, a1) means a0*1 + a1*x, x^2=0
    return (a[0]*b[0], a[0]*b[1] + a[1]*b[0])

basis = [(1,0), (0,1)]
delta1 = np.zeros((4, 4))
for col, f_basis in enumerate([(i,j) for i in range(2) for j in range(2)]):
    def f(a, fi=f_basis):
        return (fi[1],) if a == basis[fi[0]] else (0, 0)
    # ... fill delta1 matrix

# HH^0 = ker(delta^0) = A (all of A, since A commutative)
# HH^1 = ker(delta^1)/im(delta^0) = Der(A)/0 = Der(A) = k*(x d/dx)
# HH^2 = ker(delta^2)/im(delta^1) -- periodic for dual numbers

import numpy as np

def pre_lie_product(f, g, structure_constants):
    """
    f, g: arrays of shape (dim, dim, dim) representing 2-cochains
          f[i,j,k] = coefficient of e_k in f(e_i, e_j)
    structure_constants: c[i,j,k] = coeff of e_k in e_i * e_j
    Returns: f circ g as array of shape (dim, dim, dim) (a 3-cochain)
    """
    dim = f.shape[0]
    result = np.zeros((dim, dim, dim, dim))  # 3-cochain: 3 inputs, 1 output
    c = structure_constants
    # (f circ g)(a1, a2, a3) = f(g(a1,a2), a3) - f(a1, g(a2,a3))
    # sign: (-1)^{(q-1)*i} with q=2, i=0: +1; i=1: -1
    for a1 in range(dim):
        for a2 in range(dim):
            for a3 in range(dim):
                for out in range(dim):
                    # Term i=0: f(g(a1,a2), a3)
                    for mid in range(dim):
                        result[a1,a2,a3,out] += g[a1,a2,mid] * f[mid,a3,out]
                    # Term i=1: -f(a1, g(a2,a3))
                    for mid in range(dim):
                        result[a1,a2,a3,out] -= g[a2,a3,mid] * f[a1,mid,out]
    return result

def gerstenhaber_bracket(f, g, c):
    """Both f, g are 2-cochains. Returns [f,g]_G = f circ g - g circ f."""
    return pre_lie_product(f, g, c) - pre_lie_product(g, f, c)
    # sign (-1)^{(2-1)(2-1)} = 1, so no extra sign

import numpy as np

dim = 2
unit = np.array([1, 0])  # e0 = 1 (unit element)

def B_operator(f, unit, dim):
    """
    f: array of shape (dim,)*n representing an n-cochain
       (i.e., (n+1)-linear form A^{n+1} -> k in the cyclic convention)
    Returns: Bf, an (n-1)-cochain
    """
    n = len(f.shape) - 1  # f is an n-cochain
    result = np.zeros((dim,) * n)
    # (Bf)(a0,...,a_{n-1}) = sum_{i=0}^{n-1} (-1)^{ni} f(1, a_i,...,a_{n-1},a_0,...,a_{i-1})
    for idx in np.ndindex((dim,) * (n - 1)):
        val = 0
        for i in range(n):
            # cyclic rotation: (1, a_i, ..., a_{n-1}, a_0, ..., a_{i-1})
            rotated = (slice(None),) + idx[i:] + idx[:i]
            sign = (-1) ** (n * i)
            # insert unit at position 0: contract with unit vector
            for u in range(dim):
                val += sign * unit[u] * f[(u,) + idx[i:] + idx[:i]]
        result[idx] = val
    return result

# Test B^2 = 0
f = np.random.randn(dim, dim, dim)  # random 2-cochain (trilinear form)
Bf = B_operator(f, unit, dim)       # 1-cochain
BBf = B_operator(Bf, unit, dim)     # 0-cochain
assert np.allclose(BBf, 0), f"B^2 != 0: {BBf}"
print("B^2 = 0 verified.")

import numpy as np
from scipy.linalg import lstsq

def extend_deformation(mu_1, delta_matrix, gerstenhaber_bracket, max_order=5):
    """
    mu_1: array (dim, dim, dim) -- the first-order deformation 2-cocycle
    delta_matrix: matrix of the Hochschild coboundary delta: C^2 -> C^3
    Returns: list of (mu_n, obstruction_n) up to max_order
    """
    deformation = [None, mu_1]  # mu_0 = 0, mu_1 given
    for n in range(2, max_order + 1):
        # Compute obstruction: sum_{i=1}^{n-1} [mu_i, mu_{n-i}]
        obs = sum(
            gerstenhaber_bracket(deformation[i], deformation[n - i])
            for i in range(1, n)
            if deformation[i] is not None and deformation[n - i] is not None
        ) / 2
        obs_flat = obs.flatten()
        # Solve delta * mu_n = -obs
        mu_n_flat, residual, rank, sv = lstsq(delta_matrix, -obs_flat)
        if residual > 1e-10:
            print(f"Obstruction at order {n} is non-zero: deformation blocked.")
            return deformation, obs
        deformation.append(mu_n_flat.reshape(mu_1.shape))
        print(f"Order {n}: mu_{n} found, obstruction class = 0.")
    return deformation, None

References