04.03.20 · algebraic-geometry / cohomology

Hochschild homology and cohomology

shipped3 tiersLean: none

Anchor (Master): Hochschild *Ann. Math.* 46 (1945); Cartan-Eilenberg *Homological Algebra* (Princeton 1956) Ch. IX; Gerstenhaber *Ann. Math.* 78 (1963) and 79 (1964); Hochschild-Kostant-Rosenberg *Trans. AMS* 102 (1962); Loday *Cyclic Homology* (Springer Grundlehren 301, 1992; 2nd ed. 1998); Kontsevich *Lett. Math. Phys.* 66 (2003) — formality of $HH^*(A,A)$ and deformation quantisation

Intuition Beginner

An associative algebra over a field $k$ is a $k$ -vector space equipped with a multiplication that is associative but need not be commutative. The category of $k$ -algebras includes the commutative algebras (polynomial rings, function rings, coordinate rings of varieties) but also the noncommutative ones (matrix algebras, group algebras, universal enveloping algebras of Lie algebras, operator algebras). For commutative algebras, the standard cohomology theory is the Kähler differentials $Ω_{A / k}^{*}$ — a graded algebra of differential forms attached to the algebra. For a general associative algebra, no such differential-forms theory exists directly, and one needs a substitute.

Hochschild homology and cohomology are this substitute. Hochschild homology $H H_{*} (A, M)$ takes an associative algebra $A$ and an $A$ -bimodule $M$ and produces a sequence of $k$ -modules measuring how $M$ interacts with the multiplication of $A$ . Hochschild cohomology $H H^{*} (A, M)$ is the dual construction. In low degrees: $H H^{0}$ is the centre of $A$ , $H H^{1}$ is the space of outer derivations (symmetries of the multiplication modulo conjugations), and $H H^{2}$ records infinitesimal deformations of the multiplication. For a smooth commutative algebra, Hochschild homology recovers the Kähler differentials (the Hochschild-Kostant-Rosenberg theorem), bringing the classical differential-forms picture inside the noncommutative framework.

The everyday analogy is keeping accounting books for a small business. The ordinary multiplication is the cash transactions — what comes in and what goes out at each moment. The bimodule data is the inventory — how the cash flow interacts with the stock on hand. Hochschild homology and cohomology are the audit reports that record where the books fail to be perfectly consistent, what kinds of internal symmetries the accounting has, and where the multiplication can be perturbed (a small change in the rules) without breaking the whole structure. For a perfectly regular business (a smooth commutative algebra), the audit reports reduce to the standard differential-forms inventory, which is the HKR theorem.

Visual Beginner

A diagram showing the bar resolution of an associative algebra $A$ as a sequence of iterated tensor powers of $A$ mapping into $A$ , with the Hochschild complex assembled from the bimodule $M$ and the tensor powers of $A$ on one side, and the resulting homology and cohomology groups $H H_{n} (A, M)$ and $H H^{n} (A, M)$ extracted as cohomology of the complex. Arrows indicate that the cohomology in low degrees has direct interpretations: $H H^{0}$ is the centre, $H H^{1}$ is outer derivations, $H H^{2}$ is infinitesimal deformations.

The picture captures the structural shape: an algebra produces a canonical bar resolution, the bar resolution produces a Hochschild complex when tensored with a bimodule, and the cohomology of the Hochschild complex extracts both the homology and cohomology invariants of the algebra. A reader who internalises this picture will recognise the same template every time Hochschild theory appears — bar resolution then apply the bimodule then take cohomology.

Worked example Beginner

Compute the Hochschild homology and cohomology of the polynomial algebra $A = k [x]$ in one variable over a field $k$ of characteristic zero, with bimodule $M = A$ acting on both sides by multiplication.

Step 1. Identify the bar resolution. The bar resolution of $A = k [x]$ as an $A$ -bimodule is built from iterated tensor powers of $A$ over $k$ , with the multiplication map of $A$ providing the differential. For a polynomial algebra, this resolution simplifies considerably: $k [x]$ has a much shorter projective resolution as a bimodule over its enveloping algebra $A^{e}$ , namely a two-term complex whose two stages are both equal to the bimodule $A^{e}$ , with differential given by the difference between left and right multiplication by $x$ from the upper stage to the lower stage, and augmentation to $A = k [x]$ via the multiplication map of $A$ .

Step 2. Apply the bimodule. The Hochschild complex for $M = A = k [x]$ is obtained by tensoring the short resolution with $k [x]$ over the enveloping algebra. This produces a two-term complex with $k [x]$ in degree $1$ , $k [x]$ in degree $0$ , and the differential between them given by multiplication by $x - x = 0$ (since the difference between left and right multiplication by $x$ on a commutative algebra is zero). The differential is therefore the zero map.

Step 3. Compute the homology. Both differentials are zero (since the difference $x - x$ between left and right multiplication by $x$ acts as zero on the commutative algebra $k [x]$ ). So the Hochschild homology groups are $H H_{0} (k [x], k [x]) = k [x]$ and $H H_{1} (k [x], k [x]) = k [x]$ . All higher Hochschild homology groups vanish for length reasons.

Step 4. Match with Kähler differentials. The Kähler differentials of $k [x]$ are $Ω_{k [x] / k}^{0} = k [x]$ and $Ω_{k [x] / k}^{1} = k [x] \cdot d x$ (a free rank-one $k [x]$ -module generated by the symbol $d x$ ). The Hochschild-Kostant-Rosenberg map identifies $H H_{1} (k [x], k [x]) = k [x]$ with $k [x] \cdot d x$ by sending the generator to $1 \cdot d x$ , exhibiting the Hochschild homology as the Kähler differentials in low degrees.

What this tells us. For the polynomial algebra in one variable, Hochschild homology recovers exactly the Kähler differentials in degrees $0$ and $1$ , with all higher Hochschild groups vanishing. This matches the prediction of the Hochschild-Kostant-Rosenberg theorem: for a smooth commutative algebra, Hochschild homology is canonically the same as the Kähler differentials. The single Hochschild complex captures all the differential-forms information at once, and the recovery formula identifies $H H_{n} (A, A)$ with $Ω_{A / k}^{n}$ for each $n$ .

Check your understanding Beginner

Exercise (easy, multiple choice).

For an associative $k$ -algebra $A$ and an $A$ -bimodule $M$ , Hochschild homology $H H_{n} (A, M)$ is defined as:

A. $Tor_{n}^{A} (M, M)$ , the Tor over $A$ in both variables B. $Tor_{n}^{A^{e}} (A, M)$ , the Tor over the enveloping algebra $A^{e}$ (the tensor product of $A$ and its opposite algebra over $k$ ) C. The cohomology of $M$ itself, treated as a graded $k$ -module D. The exterior power $\land^{n} M$ of the bimodule

Hint

The enveloping algebra $A^{e}$ acts on $A$ in the natural left-right way (an element of the first factor multiplies from the left, an element of the second factor multiplies from the right), and a bimodule $M$ is precisely a module over $A^{e}$ . The Hochschild homology is built from a resolution of $A$ as an $A^{e}$ -module.

Answer

B. Hochschild homology of an associative $k$ -algebra $A$ with coefficients in an $A$ -bimodule $M$ is defined as $H H_{n} (A, M) = Tor_{n}^{A^{e}} (A, M)$ , where $A^{e}$ is the enveloping algebra (the tensor product over $k$ of $A$ with its opposite algebra). A bimodule is the same as a left $A^{e}$ -module, and the resolution of $A$ as an $A^{e}$ -module (via the bar resolution) computes Hochschild homology when paired with $M$ over $A^{e}$ . Option A confuses Tor over $A$ with Tor over $A^{e}$ — the bimodule structure requires the enveloping algebra. Option C ignores the algebra structure of $A$ . Option D is a different graded construction with no relationship to Hochschild.

Exercise (easy, true-false).

True or false: $H H^{0} (A, A)$ , the zeroth Hochschild cohomology of $A$ with coefficients in itself, is the centre $Z (A)$ of $A$ .

Hint

The zeroth cohomology of any cochain complex is the kernel of the first differential. For the Hochschild complex, the first differential applied to an element $a \in A = C^{0} (A, A)$ produces the map $b \mapsto ab - ba$ .

Answer

True. $H H^{0} (A, A) = Z (A)$ , the centre of $A$ . The Hochschild complex in cohomological grading has $C^{0} (A, A) = A$ and $C^{1} (A, A) = Hom_{k} (A, A)$ ; the first differential $δ : C^{0} \to C^{1}$ sends $a \mapsto (b \mapsto ab - ba)$ . The kernel of $δ$ is the set of $a \in A$ with $ab - ba = 0$ for all $b \in A$ , which is exactly the centre $Z (A)$ . This is the simplest of the low-degree interpretations of Hochschild cohomology: $H H^{0}$ records the elements that commute with everything in the algebra.

Exercise (easy, multiple choice).

For a smooth commutative $k$ -algebra $A$ in characteristic zero, the Hochschild-Kostant-Rosenberg theorem identifies $H H_{n} (A, A)$ with which object?

A. The $n$ -th exterior power $\land^{n} A$ B. The $n$ -th Kähler differential module $Ω_{A / k}^{n}$ C. The $n$ -th tensor power of $A$ as a $k$ -vector space D. The $n$ -th de Rham cohomology group $H_{dR}^{n} (A)$

Hint

For a smooth commutative algebra, Hochschild homology recovers the algebraic differential forms. The antisymmetrisation map $Ω_{A / k}^{n} \to C_{n} (A, A)$ is a quasi-isomorphism.

Answer

B. The Hochschild-Kostant-Rosenberg theorem (1962) identifies $H H_{n} (A, A) ≅ Ω_{A / k}^{n}$ , the $n$ -th module of Kähler differentials, for a smooth commutative $k$ -algebra $A$ in characteristic zero. The antisymmetrisation map $Ω_{A / k}^{n} \to C_{n} (A, A)$ in the Hochschild complex is a quasi-isomorphism. The result extends to smooth schemes via localisation. Option A is wrong because the exterior power is of the algebra rather than its differentials. Option C names the underived bar complex itself, not its cohomology. Option D names de Rham cohomology, which is the cohomology of the de Rham complex of forms — closely related to but distinct from Hochschild homology of the algebra.

Formal definition Intermediate+

Let $k$ be a field and let $A$ be an associative (not necessarily commutative) $k$ -algebra with multiplication $μ : A \otimes_{k} A \to A$ and unit $η : k \to A$ . Let $A^{op}$ denote the opposite algebra (same underlying $k$ -vector space, multiplication reversed), and let $A^{e} := A \otimes_{k} A^{op}$ be the enveloping algebra. An $A$ -bimodule is a $k$ -vector space $M$ equipped with commuting left and right $A$ -actions, equivalently a left module over $A^{e}$ via $(a \otimes b) \cdot m := amb$ .

Definition (bar resolution). The bar resolution of $A$ as an $A^{e}$ -module is the simplicial $A^{e}$ -module $B_{∙} (A)$ with $B_{n} (A) := A \otimes_{k} A^{\otimes n} \otimes_{k} A$ in degree $n$ , with face maps $$ d_i(a_0 \otimes a_1 \otimes \cdots \otimes a_n \otimes a_{n+1}) := a_0 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_{n+1} $$ for $0 \leq i \leq n$ , and degeneracy maps inserting the unit $1 \in A$ in the appropriate position. The associated chain complex $B_{∙} (A)$ with differential $\sum_{i = 0}^{n} (- 1)^{i} d_{i}$ is exact in positive degrees, with augmentation $B_{0} (A) = A \otimes_{k} A \to A$ given by multiplication, so $B_{∙} (A) \to A$ is a projective $A^{e}$ -resolution of $A$ .

Definition (Hochschild homology). For an $A$ -bimodule $M$ , the Hochschild homology is $$ HH_n(A, M) := \mathrm{Tor}n^{A^e}(A, M). $$ Computed via the bar resolution: $HH_n(A, M) = H_n(B\bullet(A) \otimes_{A^e} M) $. A f t er t h e i d e n t i f i c a t i o n$ B_n(A) \otimes_{A^e} M \cong M \otimes_k A^{\otimes n} $, t hi s i s t h e h o m o l o g y o f t h e * * H oc h sc hi l d co m pl e x * *$ C_*(A, M) $w i t h$ C_n(A, M) = M \otimes_k A^{\otimes n}$ and differential $$ b(m \otimes a_1 \otimes \cdots \otimes a_n) := m a_1 \otimes a_2 \otimes \cdots \otimes a_n + \sum_{i=1}^{n-1} (-1)^i m \otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n + (-1)^n a_n m \otimes a_1 \otimes \cdots \otimes a_{n-1}. $$ The last term records the bimodule action of $a_{n}$ on $m$ from the right; the cyclic appearance of $a_{n}$ at the beginning of the output records that the Hochschild complex sees the bimodule structure on both sides.

Definition (Hochschild cohomology). For an $A$ -bimodule $M$ , the Hochschild cohomology is $$ HH^n(A, M) := \mathrm{Ext}^n_{A^e}(A, M). $$ Computed dually via the bar resolution: $H H^{n} (A, M) = H^{n} (Hom_{A^{e}} (B_{∙} (A), M))$ . After the identification $Hom_{A^{e}} (B_{n} (A), M) ≅ Hom_{k} (A^{\otimes n}, M)$ , this is the cohomology of the Hochschild cochain complex $C^{*} (A, M)$ with $C^{n} (A, M) = Hom_{k} (A^{\otimes n}, M)$ and coboundary $$ (\delta f)(a_1 \otimes \cdots \otimes a_{n+1}) := a_1 f(a_2 \otimes \cdots \otimes a_{n+1}) + \sum_{i=1}^{n} (-1)^i f(a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_{n+1}) + (-1)^{n+1} f(a_1 \otimes \cdots \otimes a_n) a_{n+1}. $$

Definition (low-degree identifications, $M = A$ ). For $M = A$ as a bimodule (left and right multiplication of $A$ on itself): $$ HH^0(A, A) = Z(A), \qquad HH_0(A, A) = A / [A, A], $$ where $Z (A) = {a \in A : ab = ba for all b \in A}$ is the centre and $[A, A]$ is the $k$ -vector subspace spanned by commutators $ab - ba$ . The cocenter $A / [A, A]$ is the universal $k$ -vector space on which both left and right multiplication agree, often called the trace space of $A$ .

H H^{1} (A, A) = Der_{k} (A, A) / Inn (A),

the space of $k$ -linear derivations $D : A \to A$ (satisfying the Leibniz rule $D (ab) = D (a) b + a D (b)$ ) modulo the inner derivations $Inn (A) = {[a, -] : a \in A}$ . This is the space of outer derivations.

Definition ( $H H^{2}$ as deformations; Gerstenhaber 1963). For $M = A$ , an element $η \in H H^{2} (A, A)$ represents an infinitesimal deformation of the multiplication of $A$ : a $k [ϵ] / (ϵ^{2})$ -bilinear multiplication $μ_{ϵ} : A_{ϵ} \otimes A_{ϵ} \to A_{ϵ}$ on $A_{ϵ} := A \otimes_{k} k [ϵ] / (ϵ^{2})$ extending $μ$ in the sense that $μ_{ϵ} (a, b) = ab + ϵ \cdot η (a, b)$ to first order in $ϵ$ , with associativity to first order in $ϵ$ equivalent to the Hochschild cocycle condition $δ η = 0$ , and two such deformations equivalent iff they differ by a coboundary $η - η^{'} = δ f$ for $f \in C^{1} (A, A) = Hom_{k} (A, A)$ .

Definition ( $H H^{3}$ as obstructions; Gerstenhaber 1964). The obstruction to extending an infinitesimal deformation $η \in H H^{2} (A, A)$ to a second-order deformation modulo $ϵ^{3}$ lies in $H H^{3} (A, A)$ . The obstruction class is the Gerstenhaber bracket $[η, η] \in H H^{3}$ ; vanishing of this class is necessary (and, under suitable hypotheses, sufficient) for the existence of a second-order extension. Higher obstructions to extending to all orders also live in $H H^{3}$ .

Definition (cup product). The Hochschild cohomology $H H^{*} (A, A)$ carries a cup product $$ \smile : HH^p(A, A) \otimes HH^q(A, A) \to HH^{p+q}(A, A), \qquad (f \smile g)(a_1 \otimes \cdots \otimes a_{p+q}) := f(a_1 \otimes \cdots \otimes a_p) \cdot g(a_{p+1} \otimes \cdots \otimes a_{p+q}), $$ making $H H^{*} (A, A)$ into a graded-commutative $k$ -algebra (Gerstenhaber 1963; graded commutativity holds at the level of cohomology, not the cochain level).

Definition (Gerstenhaber bracket). The Hochschild cohomology $H H^{*} (A, A)$ also carries a graded Lie bracket $$ [-, -] : HH^p(A, A) \otimes HH^q(A, A) \to HH^{p + q - 1}(A, A), $$ the Gerstenhaber bracket, defined at the cochain level by the formula $[f, g] = f \overset{ˉ}{\circ} g - (- 1)^{(p - 1) (q - 1)} g \overset{ˉ}{\circ} f$ , where $f \overset{ˉ}{\circ} g = \sum_{i = 1}^{p} (- 1)^{(q - 1) (i - 1)} f \circ_{i} g$ and $f \circ_{i} g$ is the insertion of $g$ into the $i$ -th slot of $f$ . The bracket shifts degree by $- 1$ and makes $H H^{*+ 1} (A, A)$ a graded Lie algebra; together with the cup product, this is the Gerstenhaber algebra structure on $H H^{*} (A, A)$ (Gerstenhaber 1963).

Definition (Hochschild-Kostant-Rosenberg theorem; HKR 1962). Let $A$ be a smooth commutative $k$ -algebra in characteristic zero. The antisymmetrisation map $$ \Omega^n_{A/k} \to C_n(A, A), \qquad a_0 , da_1 \wedge \cdots \wedge da_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) , a_0 \otimes a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)} $$ is a quasi-isomorphism, inducing canonical isomorphisms $$ HH_n(A, A) \cong \Omega^n_{A/k}, \qquad HH^n(A, A) \cong \wedge^n_A \mathrm{Der}_k(A, A) $$ for all $n \geq 0$ . The HKR theorem identifies Hochschild homology of a smooth commutative algebra with Kähler differentials, recovering the classical differential-forms picture from the Hochschild framework.

Counterexamples to common slips

The enveloping algebra $A^{e} := A \otimes_{k} A^{op}$ uses $A^{op}$ in the second factor; missing the opposite-algebra structure converts a bimodule into a $k$ -module over $A \otimes A$ rather than an $A$ -bimodule, and the resulting Tor groups compute a different invariant.
The bar resolution differential involves $μ$ (multiplication of $A$ ) applied to adjacent factors, with alternating signs. Mistaking the signs produces a chain complex that fails to compute $Tor_{*}^{A^{e}}$ ; the alternating-sum structure is what encodes the simplicial nature of the resolution.
The HKR theorem requires smoothness (formally smooth as a $k$ -algebra) and characteristic zero (or at least characteristic large enough relative to the dimensions involved). Without smoothness, the antisymmetrisation map is no longer a quasi-isomorphism; in positive characteristic, the antisymmetrisation involves division by $n!$ , which fails when the characteristic divides $n!$ .
The Gerstenhaber bracket on $H H^{*+ 1} (A, A)$ is not the commutator of cup products (those would be zero by graded commutativity of $⌣$ ); it is a separate operation defined by the insertion formula at the cochain level. The compatibility of the bracket and the cup product (the Poisson-like compatibility) is the Gerstenhaber algebra axiom.
$H H^{2} (A, A)$ does not classify all deformations of $A$ ; it classifies first-order infinitesimal deformations modulo equivalence. Higher-order deformations require checking obstructions in $H H^{3} (A, A)$ and may not exist even when the infinitesimal deformation class is nonzero.

Key theorem with proof Intermediate+

Theorem (Hochschild-Kostant-Rosenberg; HKR 1962 Trans. AMS 102). Let $k$ be a field of characteristic zero and let $A$ be a smooth commutative $k$ -algebra of finite type. The antisymmetrisation map $ε_{n} : Ω_{A / k}^{n} \to C_{n} (A, A)$ defined by $$ \varepsilon_n(a_0 , da_1 \wedge \cdots \wedge da_n) := \frac{1}{n!} \sum_{\sigma \in S_n} \mathrm{sgn}(\sigma) , a_0 \otimes a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)} $$ is a quasi-isomorphism of chain complexes $\Omega^{A/k} \to C(A, A) $(w h er e$ \Omega^{A/k}$ has zero differential), inducing canonical isomorphisms* $$ HH_n(A, A) \xrightarrow{\sim} \Omega^n{A/k}, \qquad HH^n(A, A) \xrightarrow{\sim} \wedge^n_A \mathrm{Der}_k(A, A) $$ for all $n \geq 0$ .

Proof. The argument has four steps. First, verify that the antisymmetrisation map is a well-defined chain map. Second, treat the localisation case to reduce to a polynomial algebra. Third, prove the theorem for $A = k [x_{1}, \dots, x_{d}]$ by direct computation. Fourth, extend to general smooth $A$ via the étale-local structure.

Step 1: $ε_{*}$ is a chain map. The Kähler differentials $Ω_{A / k}^{n}$ have zero differential (regarded as a graded $A$ -module). The Hochschild differential $b$ on $C_{*} (A, A)$ has the alternating-sum-of-multiplications form. Direct computation: $b (ε_{n} (a_{0} d a_{1} \land \dots \land d a_{n}))$ produces terms of the form $a_{0} a_{σ (1)} \otimes \dots$ and $a_{0} \otimes a_{σ (1)} \otimes \dots$ paired with the multiplications. After collecting and using the antisymmetry over $S_{n}$ , the signed multiplications cancel pairwise (because each adjacent multiplication $a_{i} a_{i + 1}$ produces two contributions with opposite signs in the antisymmetric average, except at the endpoints where the bimodule cyclicity introduces the term $a_{n} m = a_{n} a_{0}$ that recombines after using commutativity of $A$ ). Hence $b (ε_{n} (ω)) = 0$ for all $ω \in Ω_{A / k}^{n}$ , and $ε$ is a chain map from $(Ω_{A / k}^{*}, 0)$ to $(C_{*} (A, A), b)$ .

Step 2: Localisation reduces to a polynomial algebra. Both Hochschild homology and Kähler differentials commute with localisation at a multiplicative set $S \subset A$ (and more generally with étale extensions): $H H_{n} (S^{- 1} A, S^{- 1} A) = S^{- 1} H H_{n} (A, A)$ and $Ω_{S^{- 1} A / k}^{n} = S^{- 1} Ω_{A / k}^{n}$ . Hence the HKR map is an isomorphism iff it is so locally, and we may localise to an étale neighbourhood of any closed point of $Spec (A)$ . Smoothness gives an étale map to an affine space $A_{k}^{d} = Spec (k [x_{1}, \dots, x_{d}])$ at every closed point, reducing the problem to the polynomial case.

Step 3: Polynomial case. For $A = k [x_{1}, \dots, x_{d}]$ , the Kähler differentials are the free $A$ -module $Ω_{A / k}^{n} = \land_{A}^{n} (A \cdot d x_{1} \oplus \dots \oplus A \cdot d x_{d})$ , of rank $(n d)$ as an $A$ -module. The Hochschild homology is computed via the Koszul-style resolution of $A$ as an $A^{e} = A \otimes_{k} A$ -module: the complex with $A \otimes_{k} \land^{n} (⨁_{i} A \cdot e_{i}) \otimes_{k} A$ in degree $n$ , with differential $e_{i} \mapsto x_{i} \otimes 1 - 1 \otimes x_{i}$ extended as a Koszul derivation. Tensoring this Koszul resolution with $A$ over $A^{e}$ gives the complex with $A \otimes_{k} \land^{n} (⨁_{i} k \cdot e_{i})$ in degree $n$ and zero differential (because the Koszul differential $x_{i} \otimes 1 - 1 \otimes x_{i}$ acts as zero after the identification $A \otimes_{A^{e}} (A \otimes A) = A$ ). Hence $H H_{n} (A, A) = A \otimes_{k} \land^{n} (k^{d}) = Ω_{A / k}^{n}$ as an $A$ -module, with the antisymmetrisation map realising the isomorphism explicitly.

Step 4: General smooth $A$ via étale descent. For a smooth $k$ -algebra $A$ of finite type, every closed point $m \subset A$ has an étale neighbourhood realising $Spec (A)_{locat m}$ as an étale extension of an affine space $A_{k}^{d}$ . By étale descent of both $H H_{n}$ and $Ω^{n}$ (which follows from the universal property of Kähler differentials and from étale base change for Tor over the enveloping algebra), the HKR isomorphism for the polynomial algebra extends locally to $A$ . Gluing over a Zariski cover of $Spec (A)$ produces the global isomorphism $H H_{n} (A, A) ≅ Ω_{A / k}^{n}$ .

The cohomology statement $H H^{n} (A, A) ≅ \land_{A}^{n} Der_{k} (A, A)$ follows by duality: $Der_{k} (A, A) = Hom_{A} (Ω_{A / k}^{1}, A)$ for smooth $A$ , and the Hochschild cohomology is the cohomology of the dual Hochschild complex. Applying the same antisymmetrisation argument dually identifies $H H^{n} (A, A)$ with $\land_{A}^{n} Hom_{A} (Ω_{A / k}^{1}, A) = \land_{A}^{n} Der_{k} (A, A)$ . $□$

Bridge. The HKR theorem builds the bridge from the noncommutative Hochschild framework back to the commutative-algebra differential-forms picture, and the foundational reason the construction works is that smoothness gives the algebra $A$ a Koszul-style resolution of $A$ as an $A \otimes A$ -module by free modules, and this Koszul structure is what makes the antisymmetrisation map a quasi-isomorphism. The bridge is the antisymmetrisation map $ε : Ω_{A / k}^{*} \to C_{*} (A, A)$ , which identifies the classical Kähler differentials as a subcomplex of the Hochschild complex on which the differential vanishes; the smoothness hypothesis ensures that this subcomplex absorbs all the cohomology. Putting these together, the HKR theorem identifies the Hochschild homology of a smooth commutative algebra with its Kähler differentials, recovering the entire commutative-algebra differential-forms theory as a special case of the more general Hochschild framework.

This pattern appears again in 04.03.17 (the derived tensor product $\otimes^{L}$ and Tor), where the Hochschild homology is the special case $H H_{*} (A, M) = Tor_{*}^{A^{e}} (A, M)$ of derived Tor over the enveloping algebra, and in 04.03.12 (derived functors), where the bar resolution is the canonical projective resolution computing the derived functor $Tor^{A^{e}}$ on the abelian category of $A$ -bimodules. The central insight is that Hochschild homology is the universal noncommutative replacement for differential forms — for smooth commutative algebras it reduces to the classical theory, and for noncommutative algebras it provides the natural extension that detects deformations, derivations, and the noncommutative geometric content of the algebra.

Exercises Intermediate+

Exercise 2 (easy, short-answer).

Compute $H H_{0} (A, A)$ for $A = M_{n} (k)$ , the algebra of $n \times n$ matrices over a field $k$ . Identify the answer with the trace space (cocenter) of the matrix algebra.

Hint

$H H_{0} (A, A) = A / [A, A]$ is the trace space, generated by traces of matrices.

Answer

$H H_{0} (A, A) = A / [A, A] = M_{n} (k) / [M_{n} (k), M_{n} (k)]$ . The subspace $[M_{n} (k), M_{n} (k)]$ is the set of $k$ -linear combinations of commutators $A B - B A$ of $n \times n$ matrices; standard fact: this subspace equals the trace-zero matrices $sl_{n} (k) \subset M_{n} (k)$ (every trace-zero matrix is a commutator, and conversely commutators have trace zero). Hence $H H_{0} (M_{n} (k), M_{n} (k)) = M_{n} (k) / sl_{n} (k) ≅ k$ via the trace map $tr : M_{n} (k) \to k$ . This recovers the classical trace as the universal $k$ -linear map on which $A B$ and $B A$ agree. Rubric: full credit for the commutator-subspace identification, the trace isomorphism, and the universal-trace interpretation.

Exercise 3 (medium, short-answer).

Show that for $A = k$ (the base field as a $k$ -algebra), all Hochschild homology and cohomology groups vanish in positive degrees, with $H H_{0} (k, k) = H H^{0} (k, k) = k$ .

Hint

The bar resolution of $k$ as a $k \otimes k = k$ -module is the length-zero resolution $k \to k$ . Tensoring with any bimodule gives a complex concentrated in degree zero.

Answer

For $A = k$ , the enveloping algebra is $A^{e} = k \otimes_{k} k = k$ . The module $A = k$ is already a free (rank-one) $A^{e} = k$ -module, so the bar resolution simplifies to $k \to k$ in degree zero with zero in all higher degrees. Tensoring with any bimodule $M$ gives the complex $M$ in degree zero and zero elsewhere; cohomology gives $H H_{n} (k, M) = 0$ for $n \geq 1$ and $H H_{0} (k, M) = M$ . Dually $H H^{n} (k, M) = 0$ for $n \geq 1$ and $H H^{0} (k, M) = M$ . For $M = A = k$ , this gives $H H_{0} (k, k) = H H^{0} (k, k) = k$ and all higher groups zero. Rubric: full credit for the length-zero resolution, the cohomology calculation, and the recovery of the base-field case.

Exercise 4 (medium, short-answer).

Show that $H H^{1} (A, A)$ is canonically isomorphic to the space of outer derivations $Der_{k} (A, A) / Inn (A)$ , where $Inn (A) = {[a, -] : a \in A}$ is the space of inner derivations.

Hint

A $1$ -cocycle in the Hochschild cochain complex is a $k$ -linear map $f : A \to A$ satisfying $a f (b) - f (ab) + f (a) b = 0$ , equivalently $f (ab) = a f (b) + f (a) b$ — the Leibniz rule. A $1$ -coboundary is the differential of an element $a \in A$ regarded as a $0$ -cochain, giving the inner derivation $[a, -]$ .

Answer

The Hochschild cochain complex has $C^{1} (A, A) = Hom_{k} (A, A)$ and the coboundary $δ : C^{0} \to C^{1}$ sends $a \mapsto [a, -] : b \mapsto ab - ba$ . The next coboundary $δ : C^{1} \to C^{2}$ sends $f : A \to A$ to $(δ f) (a, b) = a f (b) - f (ab) + f (a) b$ . The kernel of this second coboundary is the set of $f$ with $f (ab) = a f (b) + f (a) b$ , which is exactly the Leibniz condition defining a $k$ -linear derivation $f \in Der_{k} (A, A)$ . The image of the first coboundary is the set of inner derivations ${[a, -] : a \in A} = Inn (A) \subseteq Der_{k} (A, A)$ .

Hence $H H^{1} (A, A) = ker δ / im δ = Der_{k} (A, A) / Inn (A)$ , the space of outer derivations. This is the universal symmetry group of the multiplication of $A$ : outer derivations are the infinitesimal automorphisms of $A$ modulo the inner ones generated by conjugation by elements of $A$ .

Rubric: full credit for the cocycle identification with derivations, the coboundary identification with inner derivations, and the outer-derivation conclusion.

Exercise 5 (medium, short-answer).

Verify the worked-example computation $H H_{*} (k [x], k [x])$ over a field $k$ of characteristic zero: $H H_{0} = k [x]$ , $H H_{1} = k [x]$ , $H H_{n} = 0$ for $n \geq 2$ .

Hint

Use the short Koszul-style resolution of $k [x]$ as a $k [x] \otimes k [x]$ -module: $0 \to k [x] \otimes k [x] x \otimes 1 - 1 \otimes x k [x] \otimes k [x] \to k [x] \to 0$ . Tensor with $k [x]$ over $k [x] \otimes k [x]$ .

Answer

The short Koszul resolution of $k [x]$ as a $k [x]^{e} = k [x] \otimes k [x]$ -module is the two-term complex $0 \to k [x] \otimes k [x] x \otimes 1 - 1 \otimes x k [x] \otimes k [x] \to k [x] \to 0$ , with augmentation by multiplication. Tensoring with $k [x]$ over $k [x]^{e}$ produces the complex $$ 0 \to k[x] \xrightarrow{\cdot 0} k[x] \to 0, $$ because the differential $x \otimes 1 - 1 \otimes x$ acts on $k [x] = k [x] \otimes_{k [x]^{e}} (k [x] \otimes k [x])$ as multiplication by $x - x = 0$ after the identification.

Reading off cohomology: $H H_{0} (k [x], k [x]) = k [x]$ (the cokernel of zero) and $H H_{1} (k [x], k [x]) = k [x]$ (the kernel of zero). All higher Hochschild homology groups vanish since the resolution has length one.

Matching with HKR: $Ω_{k [x] / k}^{0} = k [x]$ and $Ω_{k [x] / k}^{1} = k [x] \cdot d x$ , both free rank-one $k [x]$ -modules. The HKR isomorphism sends $1 \in H H_{1} = k [x]$ to $1 \cdot d x \in Ω_{k [x] / k}^{1}$ , recovering the classical differential. Rubric: full credit for the Koszul resolution, the cohomology computation, and the HKR identification.

Exercise 6 (medium, short-answer).

Compute $H H^{*} (k [x], k [x])$ over a field $k$ of characteristic zero and identify the cup product structure on $H H^{*}$ .

Hint

Apply HKR dually: $H H^{n} (A, A) ≅ \land_{A}^{n} Der_{k} (A, A)$ . For $A = k [x]$ , $Der_{k} (A, A) = k [x] \cdot \partial_{x}$ is a free rank-one module.

Answer

By HKR, $H H^{n} (k [x], k [x]) = \land_{k [x]}^{n} Der_{k} (k [x], k [x])$ . Compute $Der_{k} (k [x], k [x])$ : a $k$ -linear derivation of $k [x]$ is determined by its value on the generator $x$ , since $\partial (x^{n}) = n x^{n - 1} \partial (x)$ by the Leibniz rule. Hence $Der_{k} (k [x], k [x])$ is free of rank one over $k [x]$ , generated by $\partial_{x}$ (the standard derivation $x \mapsto 1$ ).

Therefore $H H^{0} (k [x], k [x]) = k [x]$ (the centre, which is all of $k [x]$ since it is commutative), $H H^{1} (k [x], k [x]) = k [x] \cdot \partial_{x}$ (all derivations are outer since inner derivations vanish on a commutative algebra), and $H H^{n} (k [x], k [x]) = 0$ for $n \geq 2$ (the wedge powers vanish for $n > rank Der = 1$ ).

Cup product structure: $H H^{0} \otimes H H^{0} \to H H^{0}$ is the ordinary multiplication of $k [x]$ ; $H H^{0} \otimes H H^{1} \to H H^{1}$ is the $k [x]$ -action on $Der$ ; $H H^{1} \otimes H H^{1} \to H H^{2} = 0$ is the zero map (the cup product of two derivations vanishes by the wedge-power dimension count). The resulting cup-product algebra is the polynomial algebra $k [x] \oplus k [x] \cdot \partial_{x}$ truncated at degree $1$ , isomorphic to $k [x] [\partial_{x}] / (\partial_{x}^{2})$ as a graded $k$ -algebra. Rubric: full credit for the HKR computation, the derivation identification, and the cup-product structure.

Exercise 7 (hard, short-answer).

Show that for a commutative algebra $A$ , the Gerstenhaber bracket on $H H^{*} (A, A)$ vanishes identically. Use this and HKR to identify the Gerstenhaber algebra $H H^{*} (A, A)$ for $A$ smooth commutative in characteristic zero with the algebra of polyvector fields under the Schouten-Nijenhuis bracket.

Hint

For $A$ commutative, every inner derivation $[a, -] = 0$ , so the bracket on $H H^{1} (A, A) = Der_{k} (A, A)$ reduces to the commutator of derivations. The Schouten-Nijenhuis bracket on polyvector fields generalises the commutator of vector fields to higher-degree polyvectors.

Answer

For commutative $A$ , the Gerstenhaber bracket on $H H^{1} \otimes H H^{1} \to H H^{1}$ reduces to the commutator of $k$ -linear derivations: $[D, D^{'}] = D D^{'} - D^{'} D$ . The general Gerstenhaber bracket on $H H^{*} (A, A)$ at the cochain level still vanishes when restricted to the antisymmetrised subcomplex $Ω_{A / k}^{*}$ (with zero differential) at the cohomology level, but the bracket on the antisymmetrised subspace lifts to the Schouten-Nijenhuis bracket on the polyvector-field side.

By HKR, $H H^{n} (A, A) ≅ \land_{A}^{n} Der_{k} (A, A) = Λ^{n} T_{A / k}$ (the $n$ -th polyvector module, with $T_{A / k} = Der_{k} (A, A)$ the tangent module). The Gerstenhaber bracket $[-, -] : H H^{p} \otimes H H^{q} \to H H^{p + q - 1}$ transfers under HKR to a bracket on polyvector fields, which is the Schouten-Nijenhuis bracket $[-, -]_{S N} : Λ^{p} T \otimes Λ^{q} T \to Λ^{p + q - 1} T$ . The Schouten-Nijenhuis bracket is the unique extension of the Lie bracket of vector fields to a graded Lie bracket on polyvector fields satisfying the graded Leibniz rule with respect to the wedge product.

Hence for smooth commutative $A$ in characteristic zero, $H H^{*} (A, A)$ with its cup product and Gerstenhaber bracket is isomorphic, as a Gerstenhaber algebra, to $Λ^{*} T_{A / k}$ with the wedge product and Schouten-Nijenhuis bracket. This is the algebraic shadow of the Schouten algebra on polyvector fields of a smooth variety. The deformation-theoretic content: a degree- $2$ class $η \in H H^{2} (A, A) = Λ^{2} T_{A / k}$ is a bivector field, and the obstruction $[η, η]_{S N} = 0 \in H H^{3} (A, A) = Λ^{3} T_{A / k}$ to extending $η$ to a deformation is exactly the Poisson condition $[π, π]_{S N} = 0$ for a Poisson bivector $π$ . Rubric: full credit for the commutator identification, the HKR transfer, the Schouten-Nijenhuis identification, and the deformation-theoretic interpretation.

Exercise 8 (hard, short-answer).

State the Connes SBI long exact sequence relating Hochschild homology $H H_{*}$ and cyclic homology $H C_{*}$ , and use it to compute the cyclic homology $H C_{*} (k [x])$ for $k$ a field of characteristic zero.

Hint

The SBI sequence reads $\dots \to H C_{n - 1} (A) S H C_{n + 1} (A) B H H_{n + 1} (A) I H C_{n} (A) \to \dots$ . For $A = k [x]$ with $H H_{0} = k [x]$ , $H H_{1} = k [x]$ , $H H_{n} = 0$ for $n \geq 2$ , the long exact sequence collapses.

Answer

The Connes SBI long exact sequence for an associative $k$ -algebra $A$ reads $$ \cdots \to HC_{n-1}(A) \xrightarrow{S} HC_{n+1}(A) \xrightarrow{B} HH_{n+1}(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n+2}(A) \to \cdots $$ where $S$ is the periodicity operator (shifting degree up by $2$ ), $B$ is the Connes operator (the boundary map from cyclic to Hochschild), and $I$ is the inclusion of Hochschild cycles into cyclic cycles. The sequence is the long exact sequence of the short exact sequence of bicomplexes encoding the cyclic-Hochschild relationship via Connes' cyclic bicomplex.

For $A = k [x]$ with $H H_{0} = k [x]$ , $H H_{1} = k [x]$ , $H H_{n} = 0$ for $n \geq 2$ : substituting into the SBI sequence and using the periodicity $S$ , the long exact sequence reduces to a tower of short exact sequences. The base case $n = 0$ reads $H C_{- 1} = 0 \to H C_{1} \to H H_{1} = k [x] \to H C_{0} \to H C_{2} \to 0$ , with $H C_{0} (k [x]) = H H_{0} (k [x]) = k [x]$ (the cocenter of a commutative algebra is itself). Working through the recursive structure of the SBI sequence: $$ HC_0(k[x]) = k[x], \quad HC_1(k[x]) = k[x] / dk[x] \cong k \cdot 1, \quad HC_n(k[x]) = k \text{ for } n \ge 1. $$ More precisely, for $A = k [x]$ in characteristic zero, $H C_{n} (k [x]) ≅ Ω_{k [x] / k}^{n} / d Ω_{k [x] / k}^{n - 1}$ for $n$ odd and $H C_{n} (k [x]) ≅ ker (d : Ω_{k [x] / k}^{n} \to Ω_{k [x] / k}^{n + 1})$ for $n$ even. With $Ω^{0} = k [x]$ , $Ω^{1} = k [x] \cdot d x$ , and $Ω^{n} = 0$ for $n \geq 2$ , this collapses to $H C_{0} = k [x]$ , $H C_{1} = k [x] \cdot d x / d (k [x]) = k \cdot 1$ (since the image of $d : k [x] \to k [x] \cdot d x$ has codimension $1$ , generated by the constants modulo $d x$ exact forms — equivalently $k [x] \cdot d x / d (k [x]) ≅ k$ via the residue $\oint$ ), and $H C_{n} = k$ for all $n \geq 1$ (alternating between $ker d$ in even degrees and $Ω/ d Ω$ in odd degrees).

The full HC computation: $H C_{2 n} (k [x]) = k$ for $n \geq 0$ (with $H C_{0} = k [x]$ recovering the algebra in degree $0$ via the trace) and $H C_{2 n + 1} (k [x]) = k$ for $n \geq 0$ (the residues of forms modulo exact). The pattern is the periodic cyclic homology $H P_{*} (k [x]) = k$ in degrees $\equiv 0$ and $\equiv 1 (mod 2)$ , matching the periodic de Rham homology of $A_{k}^{1}$ — illustrating Connes' philosophy that cyclic homology is the noncommutative replacement for de Rham. Rubric: full credit for the SBI sequence statement, the recursive HC computation, and the periodic-cyclic identification with de Rham. (Defer the detailed double-complex computation to the dedicated cyclic-homology unit at 04.03.22.)

Lean formalization Intermediate+

lean_status: none — Mathlib has the underived tensor product TensorProduct R M N and a partial RingTheory.HochschildCohomology skeleton with the bar complex statement, but the named Hochschild homology and cohomology packages with their universal-property characterisation via $Tor_{*}^{A^{e}} (A, M)$ and $Ext_{A^{e}}^{*} (A, M)$ are not yet assembled. The intended formalisation reads schematically:

import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.RingTheory.TensorProduct
import Mathlib.Algebra.Homology.HomologicalComplex

namespace Codex.HomologicalAlgebra.Hochschild

variable (k : Type*) [Field k]
variable (A : Type*) [Ring A] [Algebra k A]
variable (M : Type*) [AddCommGroup M] [Module A M] [Module Aᵐᵒᵖ M]

/-- The enveloping algebra A^e := A ⊗_k A^op. -/
abbrev envelopingAlgebra : Type* := TensorProduct k A Aᵐᵒᵖ

/-- The bar resolution of A as an A^e-module. -/
noncomputable def barResolution :
    ChainComplex (ModuleCat (envelopingAlgebra k A)) ℕ := sorry

/-- Hochschild homology HH_n(A, M) = Tor_n^{A^e}(A, M). -/
noncomputable def HH (n : ℕ) : Type* := sorry

/-- Hochschild cohomology HH^n(A, M) = Ext^n_{A^e}(A, M). -/
noncomputable def HHCoh (n : ℕ) : Type* := sorry

/-- HH^0(A, A) = Z(A), the centre of A. -/
theorem HH_zero_centre :
    HHCoh k A A 0 ≃ Subring.center A := sorry

/-- HH_0(A, A) = A / [A, A], the cocenter. -/
theorem HH_zero_cocentre :
    HH k A A 0 ≃ A ⧸ (Submodule.span A {x : A | ∃ a b : A, x = a * b - b * a}) := sorry

/-- HH^1(A, A) = Der_k(A, A) / Inn(A), the outer derivations. -/
theorem HH_one_outer_derivations :
    HHCoh k A A 1 ≃ (Derivation k A A) ⧸ (innerDerivationsSubmodule k A) := sorry

/-- Hochschild-Kostant-Rosenberg theorem: for a smooth commutative
    algebra A in characteristic zero, HH_n(A, A) ≅ Ω^n_{A/k}. -/
theorem HKR [CommRing A] [Algebra k A] [Smooth k A] [CharZero k] (n : ℕ) :
    HH k A A n ≃ KahlerDifferential.exteriorPower k A n := sorry

/-- Cup product on Hochschild cohomology makes HH^*(A, A) a
    graded-commutative algebra. -/
noncomputable def cupProduct (p q : ℕ) :
    HHCoh k A A p →ₗ[k] HHCoh k A A q →ₗ[k] HHCoh k A A (p + q) := sorry

/-- Gerstenhaber bracket on Hochschild cohomology gives HH^{*+1} a
    graded Lie algebra structure. -/
noncomputable def gerstenhaberBracket (p q : ℕ) :
    HHCoh k A A p →ₗ[k] HHCoh k A A q →ₗ[k] HHCoh k A A (p + q - 1) := sorry

end Codex.HomologicalAlgebra.Hochschild

The proof gap is substantive. The construction of the bar resolution as a chain complex of $A^{e}$ -modules requires the simplicial-to-chain-complex passage (Dold-Kan style) at the level of $A^{e}$ -modules, which depends on the general derived-category machinery currently being assembled in Mathlib. The identification $H H^{0} (A, A) ≅ Z (A)$ is direct once the cochain complex is in place, but the higher cohomology identifications require the explicit cocycle/coboundary calculations against a known group-theoretic structure (derivations, deformations). The HKR theorem requires both the Kähler differentials package (partially in Mathlib via KahlerDifferential) and the smoothness/characteristic-zero hypothesis, plus the étale-descent argument that reduces to the polynomial case. The cup product and Gerstenhaber bracket require the operadic structure on the Hochschild cochain complex; the Deligne conjecture (Tamarkin 1998, McClure-Smith 2002, Kontsevich-Soibelman 2009) lifting the Gerstenhaber structure to an $E_{2}$ -algebra structure is far beyond the current Mathlib API. Each component is formalisable in principle but requires substantial coordinated infrastructure that the Mathlib homological-algebra and derived-category projects have not yet completed as of 2026.

Advanced results Master

Theorem (Hochschild-Kostant-Rosenberg; HKR 1962 Trans. AMS 102). For a smooth commutative $k$ -algebra $A$ of finite type with $k$ a field of characteristic zero, the antisymmetrisation map $\varepsilon : \Omega^{A/k} \to C(A, A) $i s a q u a s i - i so m or p hi s m, in d u c in g c an o ni c a l i so m or p hi s m s$ HH_n(A, A) \cong \Omega^n_{A/k} $an d$ HH^n(A, A) \cong \wedge^n_A \mathrm{Der}_k(A, A) $f or a l l$ n \ge 0$.

The HKR theorem is the central identification of Hochschild homology with the classical Kähler differentials in the smooth commutative case. It identifies $H H_{*} (A, A)$ for a smooth commutative algebra with the algebra of differential forms, recovering the classical commutative-algebra picture from the more general noncommutative Hochschild framework. The theorem extends to smooth schemes via localisation and globalises to the statement that $H H_{*} (X, O_{X}) ≅ H^{*} (X, Ω_{X / k}^{*})$ for $X$ a smooth $k$ -scheme, where the right side is the hypercohomology of the de Rham complex. In positive characteristic the theorem fails (the antisymmetrisation involves $1/ n!$ which is zero in characteristic $p \leq n$ ), and the correct replacement involves the derived de Rham complex of Illusie 1971/72.

Theorem (Gerstenhaber algebra structure on $H H^{*}$ ; Gerstenhaber 1963 Ann. Math. 78). Hochschild cohomology $HH^(A, A) $c a r r i es t h es t r u c t u r eo f a G er s t e nhab er a l g e b r a : a g r a d e d - co mm u t a t i v ec u pp r o d u c t$ \smile : HH^p \otimes HH^q \to HH^{p+q} $, a g r a d e d L i e b r a c k e t$ [-,-] : HH^p \otimes HH^q \to HH^{p+q-1} $o f d e g r ee$ -1 $, an d a co m p a t ibi l i t y (t h e b r a c k e t i s a g r a d e dd er i v a t i o n o f t h ec u pp r o d u c t) . T h e G er s t e nhab er b r a c k e t i sco n s t r u c t e d a tt h ecoc hain l e v e l v ia t h e in ser t i o n o p er a t i o n s$ f \circ_i g $t ha t co m p ose a$ p $- coc hainan d a$ q $- coc hainin t o a$ (p + q - 1)$-cochain.*

The Gerstenhaber algebra structure on $H H^{*} (A, A)$ encodes the noncommutative geometric structure of the algebra $A$ : the cup product is the algebraic product on Hochschild cohomology, the bracket is the deformation-theoretic Lie structure, and the compatibility is the Poisson-like compatibility between products and brackets that arises naturally for deformation problems. The bracket of two $1$ -cochains (derivations) is the commutator $[D, D^{'}] = D D^{'} - D^{'} D$ , recovering the standard Lie algebra of derivations; the bracket of a $2$ -cochain $η$ with itself $[η, η] \in H H^{3}$ is the obstruction to extending the deformation classified by $η$ to higher order (Gerstenhaber 1964 Ann. Math. 79).

Theorem (Kontsevich formality; Kontsevich 2003 Lett. Math. Phys. 66). For a smooth commutative $k$ -algebra $A$ in characteristic zero (or more generally a smooth Poisson manifold), the Gerstenhaber algebra $HH^(A, A) $i s * * f or ma l * * : t h er e i s an$ L_\infty $- q u a s i - i so m or p hi s mb e tw ee n t h eH oc h sc hi l d coc hain co m pl e x$ C^(A, A) $(a s a d i f f er e n t ia l g r a d e d L i e a l g e b r a u n d er t h e G er s t e nhab er b r a c k e t) an d t h e p o l y v ec t or f i e l d a l g e b r a$ \Lambda^ T_{A/k} $(w i t h t h e S c h o u t e n - N ij e nh u i s b r a c k e t an d z er o d i f f er e n t ia l) . T h e f or ma l i t y t h eor e mim pl i es t ha t e v er y P o i sso n s t r u c t u r eo n$ A $(a d e g r ee -$ 2 $G er s t e nhab er - b r a c k e t - c l ose d e l e m e n t o f$ HH^2(A, A) = \Lambda^2 T_{A/k} $) a d mi t s a d e f or ma t i o n q u an t i s a t i o n : ana ssoc ia t i v es t a r p r o d u c t$ \star : A[[\hbar]] \otimes A[[\hbar]] \to A[[\hbar]]$ deforming the original multiplication.*

Kontsevich's formality theorem is one of the deepest results in derived algebraic geometry and a landmark of mathematical physics: it solves the deformation quantisation problem for arbitrary Poisson manifolds, generalising the Moyal product on a symplectic vector space and the earlier results of Gerstenhaber-Schack (1986) and Tamarkin (1998). The proof uses the explicit Kontsevich formality morphism, given by an integral over compactified configurations of points in the upper half plane, with weights determined by graphs (the "Kontsevich graphs"). The formality theorem is the algebraic shadow of the topological formality of the little disks operad $E_{2}$ , and the modern proofs go through the Deligne conjecture on the $E_{2}$ -action on $H H^{*}$ .

Theorem (Deligne conjecture, Tamarkin 1998 / McClure-Smith 2002 / Kontsevich-Soibelman 2009). The Hochschild cochain complex $C^(A, A) $o f ana ssoc ia t i v e$ k $- a l g e b r a$ A $in c ha r a c t er i s t i cz er oc a r r i es ana t u r a l a c t i o n o f t h e l i ttl e d i s k so p er a d$ E_2 $, l i f t in g t h e G er s t e nhab er a l g e b r a s t r u c t u r eo n co h o m o l o g y . E q u i v a l e n tl y,$ C^(A, A) $i s an$ E_2 $- a l g e b r a, an d t h e in d u ce d s t r u c t u r eo n$ HH^(A, A)$ is the Gerstenhaber algebra structure of the previous theorem.*

The Deligne conjecture, proved independently by Tamarkin (1998), McClure-Smith (2002), and Kontsevich-Soibelman (2009) using different techniques (operadic, simplicial-set-theoretic, and graph-theoretic respectively), promotes the Gerstenhaber algebra structure on $H H^{*} (A, A)$ to an $E_{2}$ -algebra structure on the chain-level Hochschild cochain complex. The $E_{2}$ -structure is the algebraic shadow of the action of the framed little disks operad on the chains of based loops in a manifold, and is the foundational structure of the topological field-theoretic interpretation of Hochschild cohomology in the cobordism-hypothesis framework of Lurie. The Deligne conjecture is the input to Kontsevich's formality theorem and to the modern $\infty$ -categorical reformulation of deformation quantisation in derived algebraic geometry.

Theorem (Connes SBI long exact sequence; Connes 1985 Publ. Math. IHES 62; Loday-Quillen 1984; Tsygan 1983). For an associative $k$ -algebra $A$ in characteristic zero, Hochschild homology $HH_(A) $an d cy c l i c h o m o l o g y$ HC_*(A)$ fit into a long exact sequence* $$ \cdots \to HC_{n-1}(A) \xrightarrow{S} HC_{n+1}(A) \xrightarrow{B} HH_{n+1}(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n+2}(A) \to \cdots, $$ the Connes SBI sequence, where $S$ is the periodicity operator, $B$ is the Connes coboundary, and $I$ is the inclusion of Hochschild cycles. The sequence packages the cyclic-vs-Hochschild relationship into a single tower of exact sequences and identifies cyclic homology as the "periodic" version of Hochschild homology.

The Connes SBI sequence is the bridge from Hochschild to cyclic homology, and the cyclic-homology framework that it organises is the noncommutative differential geometry analogue of de Rham cohomology. For a smooth commutative $k$ -algebra $A$ in characteristic zero, the HKR identification of $H H_{*} (A, A)$ with Kähler differentials lifts via SBI to an identification of cyclic homology with the de Rham cohomology of the affine scheme, illustrating Connes' philosophy that cyclic homology is the noncommutative version of de Rham. The periodicity operator $S$ in SBI corresponds to the integration map in de Rham; the Connes operator $B$ corresponds to the de Rham differential. Forward-link to 04.03.22 for the full cyclic-homology unit.

Theorem (Hochschild-Serre spectral sequence for group extensions). For a short exact sequence of groups $1 \to N \to G \to Q \to 1$ and a $Z [G]$ -module $M$ , there is a Hochschild-Serre spectral sequence $$ E_2^{p, q} = H^p(Q, H^q(N, M)) \Rightarrow H^{p + q}(G, M). $$ Specialising to the Hochschild-cohomology setting (algebras instead of groups), the Hochschild-Serre spectral sequence becomes the spectral sequence of an algebra extension $0 \to I \to A \to A / I \to 0$ computing $HH^(A, M) $f r o m$ HH^(I, M) $an d$ HH^(A/I, M)$.*

The Hochschild-Serre spectral sequence is the version of the Grothendieck spectral sequence (from 04.03.13) for the composition of derived functors that arise in extension-theoretic computations of Hochschild cohomology. It is the spectral-sequence translation of the Serre fibration $B N \to B G \to B Q$ in classifying-space topology, and it generalises directly to Hochschild cohomology via the algebra-extension formalism. The Hochschild-Serre formalism is the input to the standard computational machinery of group cohomology and its noncommutative analogues.

Theorem (topological Hochschild homology and the cyclotomic trace; Bökstedt 1985; Bökstedt-Hsiang-Madsen 1993). For an $E_{\infty}$ -ring spectrum $A$ , the topological Hochschild homology $THH (A)$ is the smash product $A \land_{A \land A^{op}}^{L} A$ in the symmetric monoidal $\infty$ -category of $A$ -bimodule spectra, generalising the Hochschild homology of an associative algebra to the world of spectra. The cyclotomic trace $K (A) \to THH (A) \to T C (A)$ (Bökstedt-Hsiang-Madsen 1993) maps algebraic $K$ -theory into topological Hochschild and topological cyclic homology, and is the foundational tool of trace-method computations of algebraic $K$ -theory.

Topological Hochschild homology $THH$ is the spectrum-level lift of Hochschild homology, taking into account the higher-coherence structure of $E_{\infty}$ -rings instead of merely associative algebras. The cyclotomic trace is the fundamental tool of modern $K$ -theory computations: it relates the difficult algebraic $K$ -theory $K (A)$ to the more computable $THH (A)$ via the trace map and to the topological cyclic homology $T C (A)$ via the cyclotomic structure. The Hesselholt-Madsen computation of $K (Z_{p})$ via cyclotomic trace and the Nikolaus-Scholze reformulation of topological cyclic homology (2018 Acta Math. 221) are foundational applications. The cyclotomic-trace framework is the input to chromatic homotopy theory's understanding of arithmetic, including the Bhatt-Morrow-Scholze theory of prismatic cohomology.

Theorem (Hochschild homology of schemes; Weibel 1996). For a quasi-compact, separated $k$ -scheme $X$ , the Hochschild homology of $X$ is defined as $$ HH_n(X) := \mathrm{Ext}^{-n}{D(X \times_k X)}(\Delta\mathcal{O}X, \Delta\mathcal{O}X), $$ where $Δ : X \to X \times_{k} X$ is the diagonal. For $X$ smooth of finite type in characteristic zero, the Hochschild-Kostant-Rosenberg theorem extends to the global statement $$ HH_n(X) \cong \bigoplus{p - q = n} H^p(X, \Omega^q_{X/k}), $$ identifying Hochschild homology with the Hodge cohomology of $X$ .

The scheme-theoretic Hochschild homology is the natural extension of the affine version to algebraic geometry, and the HKR identification with Hodge cohomology is one of the central calculations of derived algebraic geometry. The result implies that Hochschild homology of a smooth projective variety $X$ contains the Hodge numbers of $X$ , and the cup product on $H H^{*} (X)$ recovers the cohomological cup product on Hodge cohomology. The scheme-theoretic Hochschild homology is the input to derived noncommutative geometry (Kontsevich, Bondal-Orlov), where the Hochschild invariants of a triangulated category serve as the "Hodge invariants" of a noncommutative space; the Bondal-Orlov reconstruction theorem (from 04.03.18) is the geometric counterpart of this Hochschild-Hodge identification.

Synthesis. Hochschild homology and cohomology build toward a unified framework for the noncommutative invariants of an associative algebra, and the foundational reason the construction works is that the bar resolution provides a canonical projective resolution of $A$ as a module over its enveloping algebra $A^{e} = A \otimes_{k} A^{op}$ , so the Tor and Ext groups over $A^{e}$ produce well-defined invariants of $A$ and its bimodules. The package is structurally tight: HH^0 is the centre, HH^1 is the outer derivations, HH^2 classifies infinitesimal deformations, and HH^3 houses the obstructions, with the Gerstenhaber algebra structure on $H H^{*}$ packaging cup product and bracket into a single deformation-theoretic invariant. The HKR theorem reduces the noncommutative framework to the classical commutative differential-forms theory in the smooth case, and the Connes SBI sequence extends the framework forward into cyclic homology and noncommutative de Rham theory. The central insight is that Hochschild theory is the universal noncommutative replacement for differential forms, deformation theory, and de Rham cohomology — and that every classical commutative-algebra construction has a Hochschild-flavoured noncommutative analogue extracted from the bar resolution.

The framework appears again in 04.03.17 (the derived tensor product $\otimes^{L}$ and Tor), where Hochschild homology is the special case $H H_{*} (A, M) = Tor_{*}^{A^{e}} (A, M)$ of derived Tor over the enveloping algebra; in 04.03.12 (derived functors $R F$ and $L F$ ), where the bar resolution is the canonical projective resolution computing the derived functor $Tor^{A^{e}}$ on the abelian category of $A$ -bimodules; and forward in 04.03.22 (cyclic homology), where the Connes SBI long exact sequence ties Hochschild to its periodic refinement and identifies cyclic homology as the noncommutative analogue of de Rham. The recursion stabilises: every Hochschild computation produces a deformation-theoretic invariant, every smooth commutative algebra recovers its classical differential forms via HKR, every Gerstenhaber algebra carries a deformation-quantisation problem solved by Kontsevich formality, and the entire framework lifts to the $E_{2}$ -algebra structure on the cochain level via the Deligne conjecture.

Full proof set Master

Proposition (bar resolution is a projective resolution). Let $A$ be an associative $k$ -algebra and $A^{e} = A \otimes_{k} A^{op}$ its enveloping algebra. The bar complex $B_{∙} (A)$ with $B_{n} (A) = A \otimes_{k} A^{\otimes n} \otimes_{k} A$ and differential the alternating sum of multiplications is a projective resolution of $A$ as an $A^{e}$ -module.

Proof. Each $B_{n} (A)$ is a free $A^{e}$ -module via the action $(a^{'} \otimes b^{'}) \cdot (a_{0} \otimes a_{1} \otimes \dots \otimes a_{n} \otimes a_{n + 1}) = a^{'} a_{0} \otimes a_{1} \otimes \dots \otimes a_{n} \otimes a_{n + 1} b^{'}$ , with basis the elements $1 \otimes a_{1} \otimes \dots \otimes a_{n} \otimes 1$ for $a_{i} \in A$ ; hence each $B_{n} (A)$ is projective. The augmentation $B_{0} (A) = A \otimes_{k} A \to A$ is the multiplication map $μ : A \otimes A \to A$ , an $A^{e}$ -module surjection.

Exactness in positive degrees: the bar complex carries an explicit $k$ -linear contracting homotopy $s_{n} : B_{n} (A) \to B_{n + 1} (A)$ defined by $s_{n} (a_{0} \otimes \dots \otimes a_{n + 1}) := 1 \otimes a_{0} \otimes \dots \otimes a_{n + 1}$ , satisfying $d s + s d = id$ on $B_{n} (A)$ for $n \geq 0$ (with $s_{- 1} : A \to B_{0} (A)$ given by $a \mapsto 1 \otimes a$ as the unit map of the augmentation). The contracting homotopy is $k$ -linear but not $A^{e}$ -linear, so it does not split the resolution as $A^{e}$ -modules; however, it does establish that the bar complex is acyclic as a complex of $k$ -vector spaces, which is sufficient for the resolution property. Hence $B_{∙} (A) \to A$ is a projective $A^{e}$ -resolution of $A$ . $□$

Proposition (independence from resolution). The Tor groups $H H_{n} (A, M) = Tor_{n}^{A^{e}} (A, M)$ are independent of the choice of projective $A^{e}$ -resolution of $A$ used to compute them.

Proof. The general theory of derived functors (see 04.03.12) guarantees that $Tor_{n}^{A^{e}}$ is well-defined up to canonical isomorphism independent of the resolution. Concretely, any two projective resolutions $P_{∙} \to A$ and $P_{∙}^{'} \to A$ are related by a chain homotopy equivalence; tensoring with $M$ over $A^{e}$ yields chain-homotopic complexes whose homologies are canonically isomorphic. Hence the Hochschild homology $H H_{n} (A, M)$ defined via the bar resolution agrees with the Hochschild homology defined via any other projective $A^{e}$ -resolution. $□$

Proposition ( $H H^{0} (A, A) = Z (A)$ and $H H_{0} (A, A) = A / [A, A]$ ). For any associative $k$ -algebra $A$ and $M = A$ , the zeroth Hochschild cohomology is the centre and the zeroth Hochschild homology is the cocenter.

Proof. The Hochschild cochain complex in degree $0$ is $C^{0} (A, A) = Hom_{k} (k, A) = A$ , and the coboundary $δ : C^{0} \to C^{1} = Hom_{k} (A, A)$ sends $a \mapsto (b \mapsto ab - ba)$ . The kernel of $δ$ is ${a \in A : ab = ba for all b \in A} = Z (A)$ , the centre. Hence $H H^{0} (A, A) = Z (A)$ .

For homology: the Hochschild chain complex in degree $0$ is $C_{0} (A, A) = A$ (interpreting $A^{\otimes 0} = k$ ), and the differential $b : C_{1} = A \otimes A \to C_{0} = A$ sends $m \otimes a \mapsto ma - am$ . The image of $b$ is the $k$ -subspace $[A, A]$ spanned by commutators, and the cokernel is $A / [A, A]$ . Hence $H H_{0} (A, A) = A / [A, A]$ , the cocenter. $□$

Proposition ( $H H^{1} (A, A)$ is outer derivations). For any associative $k$ -algebra $A$ , $H H^{1} (A, A) = Der_{k} (A, A) / Inn (A)$ .

Proof. The Hochschild cochain complex in degree $1$ is $C^{1} (A, A) = Hom_{k} (A, A)$ . The coboundary $δ : C^{1} \to C^{2} = Hom_{k} (A \otimes A, A)$ sends $f : A \to A$ to $(δ f) (a, b) = a f (b) - f (ab) + f (a) b$ . The kernel of $δ$ consists of $f$ with $f (ab) = a f (b) + f (a) b$ — the Leibniz rule — which is precisely $Der_{k} (A, A)$ . The image of the previous coboundary $δ : C^{0} \to C^{1}$ consists of inner derivations $[a, -] = ab - ba$ . Hence $H H^{1} (A, A) = Der_{k} (A, A) / Inn (A)$ . $□$

Proposition ( $H H^{2} (A, A)$ classifies infinitesimal deformations). For any associative $k$ -algebra $A$ , $H H^{2} (A, A)$ classifies equivalence classes of infinitesimal deformations of the multiplication of $A$ .

Proof. An infinitesimal deformation of $A$ is an associative multiplication $μ_{ϵ} : A_{ϵ} \otimes A_{ϵ} \to A_{ϵ}$ on $A_{ϵ} = A \otimes_{k} k [ϵ] / (ϵ^{2})$ extending the original $μ : A \otimes A \to A$ to first order in $ϵ$ . Write $μ_{ϵ} (a, b) = ab + ϵ \cdot η (a, b)$ for a $k$ -bilinear map $η : A \otimes A \to A$ . Associativity of $μ_{ϵ}$ modulo $ϵ^{2}$ reads $$ \mu_\epsilon(\mu_\epsilon(a, b), c) - \mu_\epsilon(a, \mu_\epsilon(b, c)) = 0 \pmod {\epsilon^2}. $$ Expanding and using associativity of $μ$ , the $O (ϵ)$ term is $$ a \eta(b, c) - \eta(ab, c) + \eta(a, bc) - \eta(a, b) c = (\delta \eta)(a, b, c), $$ which equals zero iff $η$ is a $2$ -cocycle in the Hochschild cochain complex, i.e., $δ η = 0$ .

Two deformations $μ_{ϵ}, μ_{ϵ}^{'}$ are equivalent if there is a $k [ϵ] / (ϵ^{2})$ -linear automorphism $ϕ_{ϵ} = id + ϵ \cdot f : A_{ϵ} \to A_{ϵ}$ (with $f : A \to A$ a $k$ -linear map) such that $ϕ_{ϵ}^{- 1} (μ_{ϵ} (ϕ_{ϵ} (a), ϕ_{ϵ} (b))) = μ_{ϵ}^{'} (a, b)$ . Expanding to $O (ϵ)$ , this gives $$ \eta(a, b) - \eta'(a, b) = a f(b) - f(ab) + f(a) b = (\delta f)(a, b), $$ which is exactly the condition that $η - η^{'}$ is a $2$ -coboundary. Hence equivalence classes of infinitesimal deformations correspond to elements of $ker (δ : C^{2} \to C^{3}) / im (δ : C^{1} \to C^{2}) = H H^{2} (A, A)$ . $□$

Proposition (HKR for the polynomial algebra). For $A = k [x_{1}, \dots, x_{d}]$ in characteristic zero, $H H_{n} (A, A) ≅ Ω_{A / k}^{n} = A \otimes_{k} \land^{n} (k \cdot d x_{1} \oplus \dots \oplus k \cdot d x_{d})$ .

Proof. Use the Koszul resolution of $A$ as an $A^{e}$ -module: take the algebra $A^{e} = A \otimes_{k} A$ as a $k$ -algebra in $2 d$ variables $x_{1}, \dots, x_{d}, y_{1}, \dots, y_{d}$ (where $y_{i} = 1 \otimes x_{i}$ and $x_{i} = x_{i} \otimes 1$ ). The kernel of $μ : A^{e} \to A$ is generated by the regular sequence $x_{i} - y_{i}$ for $i = 1, \dots, d$ . The Koszul resolution $$ K_\bullet := \wedge^*_{A^e} (A^e \cdot e_1 \oplus \cdots \oplus A^e \cdot e_d) \to A $$ with differential $\partial (e_{i}) = x_{i} - y_{i}$ extended as a Koszul derivation is a free $A^{e}$ -resolution of $A$ of length $d$ .

Tensoring $K_{∙}$ with $A$ over $A^{e}$ : under the identification $A \otimes_{A^{e}} A^{e} \cdot e_{i_{1}} \land \dots \land e_{i_{n}} = A \cdot e_{i_{1}} \land \dots \land e_{i_{n}}$ , the differential $\partial (e_{i}) = x_{i} - y_{i}$ acts as multiplication by $x_{i} - x_{i} = 0$ (after the identification $A \otimes_{A^{e}} A^{e} = A$ and the action of $y_{i}$ as right multiplication by $x_{i}$ , which equals left multiplication on the commutative algebra $A$ ). Hence $A \otimes_{A^{e}} K_{∙}$ is the complex with $A \otimes_{k} \land^{n} (k^{d})$ in degree $n$ and zero differential. Reading off homology: $H H_{n} (A, A) = A \otimes_{k} \land^{n} (k^{d}) = Ω_{A / k}^{n}$ , with the standard generators $d x_{i_{1}} \land \dots \land d x_{i_{n}}$ corresponding to the basis $e_{i_{1}} \land \dots \land e_{i_{n}}$ .

The antisymmetrisation map $ε : Ω_{A / k}^{n} \to C_{n} (A, A)$ realises this isomorphism explicitly: the symbol $a_{0} \cdot d a_{i_{1}} \land \dots \land d a_{i_{n}}$ maps to the antisymmetrised tensor $(1/ n!) \sum_{σ} sgn (σ) a_{0} \otimes a_{i_{σ (1)}} \otimes \dots \otimes a_{i_{σ (n)}}$ , and a direct calculation shows that this map agrees with the comparison map between the Koszul resolution and the bar resolution at the homology level. $□$

Proposition (Gerstenhaber bracket of derivations). For $f, g \in H H^{1} (A, A) = Der_{k} (A, A) / Inn (A)$ , the Gerstenhaber bracket $[f, g] \in H H^{1} (A, A)$ is the commutator $f g - g f$ of derivations.

Proof. The Gerstenhaber bracket at the cochain level is $[f, g] = f \overset{ˉ}{\circ} g - (- 1)^{(p - 1) (q - 1)} g \overset{ˉ}{\circ} f$ where $f \overset{ˉ}{\circ} g = \sum_{i = 1}^{p} (- 1)^{(q - 1) (i - 1)} f \circ_{i} g$ and $f \circ_{i} g$ inserts $g$ into the $i$ -th slot of $f$ . For $f, g$ both of degree $1$ (so $p = q = 1$ ), the bracket simplifies: $f \overset{ˉ}{\circ} g = f \circ_{1} g$ (only one insertion) and the sign $(- 1)^{(p - 1) (q - 1)} = 1$ for $p = q = 1$ , so $[f, g] = f \circ_{1} g - g \circ_{1} f$ . The insertion $f \circ_{1} g$ for $1$ -cochains is the composition $f \circ g : A \to A \to A$ , viewed as a $1$ -cochain. Hence $[f, g] = f \circ g - g \circ f = f g - g f$ , the commutator of $k$ -linear endomorphisms.

When $f$ and $g$ are derivations, the commutator $f g - g f$ is again a derivation (standard check: $(f g - g f) (ab) = f (g (a) b + a g (b)) - g (f (a) b + a f (b)) = \dots = (f g) (a) b + a (f g) (b) - (g f) (a) b - a (g f) (b) = [f, g] (a) b + a [f, g] (b)$ after collecting). Hence the bracket descends to a Lie bracket on $Der_{k} (A, A)$ , recovering the standard Lie algebra structure on derivations. $□$

Connections Master

Derived tensor product $\otimes^{L}$ and Tor 04.03.17. Hochschild homology is the special case of derived Tor over the enveloping algebra: $H H_{*} (A, M) = Tor_{*}^{A^{e}} (A, M)$ , computed via the bar resolution of $A$ as an $A^{e}$ -module. The recovery formula $H H_{n} = H_{n} (bar complex \otimes M)$ is the standard derived-tensor calculation specialised to the enveloping-algebra setting, and the entire derived-tensor-product framework specialises to give the Hochschild theory in this case.
Derived functors $R F$ and $L F$ via derived categories 04.03.12. The bar resolution $B_{∙} (A) \to A$ is the canonical projective resolution of $A$ as an $A^{e}$ -module, and the Hochschild homology $H H_{*}$ is the derived functor $Tor_{*}^{A^{e}}$ applied to $(A, M)$ . The general framework of left-derived functors and their independence from resolution choices applies directly, identifying Hochschild homology as a canonical noncommutative invariant of the algebra.
Tensor product of modules 01.02.10. The Hochschild complex $C_{*} (A, M) = M \otimes_{k} A^{\otimes*}$ is built from iterated tensor products of $A$ and $M$ over the base field $k$ . The bimodule structure on $M$ is the input that distinguishes Hochschild homology from ordinary Tor, and the alternating-sum differential involves the multiplication on $A$ together with the bimodule action on $M$ .
t-Structure on a triangulated category — heart and truncations 04.03.18. The Hochschild cohomology $H H^{*} (X)$ of a smooth projective variety $X$ is an invariant of the derived category $D^{b} (Coh (X))$ , and the Bondal-Orlov reconstruction theorem implies that the Hochschild cohomology of a derived category recovers a substantial portion of the geometric invariants of the underlying variety. The Hochschild theory of triangulated categories is the noncommutative-geometric input to derived algebraic geometry and Bridgeland stability.
Six-functor formalism 04.03.16. The Hochschild homology of a scheme $X$ is defined as $H H_{n} (X) = Ext_{D (X \times_{k} X)}^{- n} (Δ_{*} O_{X}, Δ_{*} O_{X})$ , using the diagonal $Δ : X \to X \times_{k} X$ and the six-functor operations on the derived category of $X \times_{k} X$ . The HKR identification with Hodge cohomology $⨁_{p - q = n} H^{p} (X, Ω_{X / k}^{q})$ packages the entire Hodge data of $X$ into the Hochschild invariant, demonstrating the bridge between derived noncommutative geometry and classical Hodge theory.
Cyclic homology and Connes' SBI sequence (forward to 04.03.22). Hochschild homology fits into the Connes SBI long exact sequence with cyclic homology: $\dots \to H H_{n} \to H C_{n} \to H C_{n - 2} \to H H_{n - 1} \to \dots$ . The cyclic homology $H C_{*} (A)$ is the noncommutative analogue of de Rham cohomology, and the SBI sequence makes the relationship explicit at the level of long exact sequences. The forthcoming unit 04.03.22 develops the full cyclic-homology framework with the bicomplex construction, the Connes periodicity operator $B$ , and the de Rham comparison for smooth commutative algebras.
Deformation theory and Kontsevich quantisation. The Gerstenhaber-algebra structure on $H H^{*} (A, A)$ encodes the deformation theory of the algebra $A$ : degree- $2$ classes are first-order deformations, degree- $3$ classes are obstructions, and the bracket structure controls higher-order extensions. Kontsevich's formality theorem (2003) for smooth Poisson manifolds states that the Hochschild cochain complex of the function algebra is $L_{\infty}$ -quasi-isomorphic to the polyvector field complex, implying that every Poisson structure admits a deformation quantisation as a star product on the algebra of functions. This is one of the central results of mathematical physics, providing a rigorous algebraic foundation for quantum mechanics on Poisson manifolds.

Historical & philosophical context Master

Hochschild homology and cohomology were introduced by Gerhard Hochschild in his 1945 paper "On the cohomology groups of an associative algebra" (Annals of Mathematics 46) ^{[source pending]}, where he defined the cochain complex $C^{n} (A, M) = Hom_{k} (A^{\otimes n}, M)$ with the alternating-sum coboundary and identified the low-degree cohomology with classical algebraic constructions: $H H^{1}$ with outer derivations and $H H^{2}$ with infinitesimal extensions of the algebra. The systematic treatment of Hochschild as a derived functor — $H H^{n} (A, M) = Ext_{A^{e}}^{n} (A, M)$ via the bar resolution — was developed by Henri Cartan and Samuel Eilenberg in their 1956 textbook Homological Algebra (Princeton) ^{[source pending]} Ch. IX, where the Hochschild groups appeared as the canonical example of derived Tor and Ext over an enveloping algebra. The Cartan-Eilenberg framework placed Hochschild theory firmly within the abelian-category derived-functor formalism and established the bar resolution as the standard computational device.

The deformation-theoretic interpretation of Hochschild cohomology was developed by Murray Gerstenhaber in two landmark papers: "The cohomology structure of an associative ring" (Annals of Mathematics 78, 1963) ^{[source pending]} introduced the cup product and the eponymous Gerstenhaber bracket, identifying $H H^{*} (A, A)$ as a Gerstenhaber algebra (a graded-commutative algebra with a graded Lie bracket of degree $- 1$ that is a derivation of the product), and "On the deformation of rings and algebras" (Annals of Mathematics 79, 1964) developed the deformation theory of associative algebras via Hochschild cohomology: $H H^{2} (A, A)$ classifies infinitesimal deformations, $H H^{3} (A, A)$ houses the obstructions, and the bracket $[η, η] \in H H^{3}$ controls when an infinitesimal deformation extends to higher order. Gerstenhaber's framework is the algebraic shadow of modern derived deformation theory and the input to Kontsevich's formality theorem (Kontsevich 2003 Letters in Mathematical Physics 66 ^{[source pending]}) which solves the deformation quantisation problem for arbitrary Poisson manifolds.

The Hochschild-Kostant-Rosenberg theorem identifying $H H_{n} (A, A) ≅ Ω_{A / k}^{n}$ for smooth commutative algebras in characteristic zero was proved by Gerhard Hochschild, Bertram Kostant, and Alex Rosenberg in 1962 ("Differential forms on regular affine algebras," Transactions of the American Mathematical Society 102) ^{[source pending]}, establishing the bridge from the noncommutative Hochschild framework back to the classical commutative-algebra differential-forms theory. The HKR theorem is the central calculation of Hochschild theory in the commutative case, and its globalisation to smooth schemes (Weibel 1996, then extensively developed in the derived algebraic geometry programme of Toën-Vezzosi and Lurie) identifies $H H_{*} (X) ≅ ⨁_{p, q} H^{p} (X, Ω_{X / k}^{q})$ as the Hodge cohomology of $X$ , packaging the entire Hodge data of a smooth variety into its Hochschild invariants.

The cyclic-homology framework extending Hochschild was developed independently by Alain Connes (1981 C.R. Acad. Sci. Paris 296 announcement, then 1985 Publ. Math. IHES 62) and Boris Tsygan (1983 Uspekhi Mat. Nauk 38), with the Connes SBI long exact sequence relating Hochschild and cyclic homology providing the algebraic infrastructure for what Connes called noncommutative differential geometry. Connes' philosophy that cyclic homology is the noncommutative replacement for de Rham cohomology has been the organising principle of an enormous body of subsequent work, including the Hesselholt-Madsen computation of algebraic $K$ -theory via topological Hochschild and cyclic homology (1997), the Bhatt-Morrow-Scholze theory of prismatic cohomology (2019 Publ. Math. IHES 129), and the Nikolaus-Scholze reformulation of topological cyclic homology (2018 Acta Math. 221). The forward-link from this unit is to 04.03.22 (cyclic homology), where the full Connes framework is developed.

The Deligne conjecture on the $E_{2}$ -action on the Hochschild cochain complex, proved independently by Dmitry Tamarkin (1998), James McClure and Jeffrey Smith (2002 Contemp. Math. 293), and Maxim Kontsevich and Yan Soibelman (2009) using different techniques, promoted the Gerstenhaber algebra structure on $H H^{*}$ to an $E_{2}$ -algebra structure on the cochain level, and is the foundational input to Kontsevich's formality theorem and to the modern $\infty$ -categorical reformulation of deformation quantisation. The Deligne conjecture is the algebraic shadow of the topological action of the framed little disks operad on based loops in a manifold, and the Lurie-Ayala-Francis cobordism-hypothesis programme identifies the Hochschild theory of an $\infty$ -category with its topological field theory invariants.

Bibliography Master

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  volume    = {46},
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  year      = {1945}
}

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}

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  year      = {1963}
}

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}

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}

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}

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}

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}

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}

@article{Tsygan1983,
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}

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}

@article{McClureSmith2002,
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}

@article{KontsevichSoibelman2009,
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  booktitle = {Homological Mirror Symmetry},
  series    = {Lecture Notes in Physics},
  volume    = {757},
  pages     = {153--219},
  year      = {2009},
  publisher = {Springer}
}

@article{Weibel1996,
  author    = {Weibel, Charles A.},
  title     = {Cyclic homology for schemes},
  journal   = {Proceedings of the American Mathematical Society},
  volume    = {124},
  number    = {6},
  pages     = {1655--1662},
  year      = {1996}
}

@book{GelfandManinAlgebraV,
  author    = {Gel'fand, Sergei I. and Manin, Yuri I.},
  title     = {Homological Algebra},
  publisher = {Springer-Verlag},
  series    = {Encyclopaedia of Mathematical Sciences},
  volume    = {38},
  year      = {1994},
  note      = {Algebra V; English translation of the 1989 VINITI Russian original}
}

@article{BokstedtHsiangMadsen1993,
  author    = {B{\"o}kstedt, Marcel and Hsiang, Wu-Chung and Madsen, Ib},
  title     = {The cyclotomic trace and algebraic $K$-theory of spaces},
  journal   = {Inventiones Mathematicae},
  volume    = {111},
  pages     = {465--539},
  year      = {1993}
}

@article{NikolausScholze2018,
  author    = {Nikolaus, Thomas and Scholze, Peter},
  title     = {On topological cyclic homology},
  journal   = {Acta Mathematica},
  volume    = {221},
  number    = {2},
  pages     = {203--409},
  year      = {2018}
}

Prerequisites

01.02.10
04.03.12
04.03.17

Tier anchors

beginner: Weibel *An Introduction to Homological Algebra* §9.1 informal — bar resolution and the Hochschild complex of an associative algebra; classical computation of $HH_*(k[x])$
intermediate: Weibel *An Introduction to Homological Algebra* Ch. 9; Loday *Cyclic Homology* Ch. 1; Gelfand-Manin *Homological Algebra* (Algebra V) Ch. V §§1-2
master: Hochschild *Ann. Math.* 46 (1945); Cartan-Eilenberg *Homological Algebra* (Princeton 1956) Ch. IX; Gerstenhaber *Ann. Math.* 78 (1963) and 79 (1964); Hochschild-Kostant-Rosenberg *Trans. AMS* 102 (1962); Loday *Cyclic Homology* (Springer Grundlehren 301, 1992; 2nd ed. 1998); Kontsevich *Lett. Math. Phys.* 66 (2003) — formality of $HH^*(A,A)$ and deformation quantisation

References

Hochschild — On the cohomology groups of an associative algebra · *Annals of Mathematics* 46 (1945), 58--67; the original definition of Hochschild cohomology $HH^n(A, M)$ for an associative algebra $A$ over a field $k$ and an $A$-bimodule $M$, defined as the cohomology of the cochain complex $C^n(A, M) = \mathrm{Hom}_k(A^{\otimes n}, M)$ with the alternating-sum coboundary; identification of $HH^1$ with outer derivations and $HH^2$ with infinitesimal extensions
Cartan-Eilenberg — Homological Algebra · Princeton University Press (1956), Princeton Mathematical Series 19; Ch. IX (Hochschild homology and cohomology as the derived functors $\mathrm{Tor}^{A^e}_*(A, M)$ and $\mathrm{Ext}_{A^e}^*(A, M)$ via the bar resolution of $A$ as an $A^e$-module, where $A^e = A \otimes_k A^{\mathrm{op}}$ is the enveloping algebra); Ch. X (interpretation of low-degree cohomology in terms of derivations and extensions)
Gerstenhaber — The cohomology structure of an associative ring · *Annals of Mathematics* 78 (1963), 267--288 (the cup product and Gerstenhaber bracket on $HH^*(A, A)$, making it into what is now called a Gerstenhaber algebra; the deformation-theoretic interpretation of $HH^2$ as infinitesimal deformations of the multiplication on $A$); and *Annals of Mathematics* 79 (1964), 59--103 (the obstruction theory: $HH^3$ houses the obstruction to extending an infinitesimal deformation to higher order)
Hochschild-Kostant-Rosenberg — Differential forms on regular affine algebras · *Transactions of the American Mathematical Society* 102 (1962), 383--408; the HKR theorem identifying $HH_n(A, A) \cong \Omega^n_{A/k}$ for a smooth commutative $k$-algebra $A$ in characteristic zero (Kähler differentials in degree $n$); the antisymmetrisation map $\Omega^n_{A/k} \to C_n(A, A)$ is a quasi-isomorphism in the smooth case
Loday — Cyclic Homology · Springer Grundlehren der mathematischen Wissenschaften 301 (1st ed. 1992; 2nd ed. 1998), $\approx 516$ pp.; Ch. 1 (the Hochschild complex of an associative algebra and its homology; bar resolution; identification with $\mathrm{Tor}^{A^e}_*$); Ch. 1.5 (cup product and Gerstenhaber bracket on $HH^*(A, A)$); Ch. 3 (HKR theorem and the smooth case); Ch. 4 (cyclic homology and the Connes SBI long exact sequence)
Weibel — An Introduction to Homological Algebra · Cambridge Studies in Advanced Mathematics 38 (1994); Ch. 9 (Hochschild and cyclic homology of associative algebras; bar resolution; identification of low-degree cohomology with derivations and extensions; HKR theorem in the smooth case; cup product and Gerstenhaber bracket; relation to cyclic homology via the Connes long exact sequence)
Gelfand-Manin — Homological Algebra (Algebra V) · Encyclopaedia of Mathematical Sciences Vol. 38 (Springer 1994; English translation of the 1989 VINITI Russian original); Ch. V (Hochschild homology and cohomology; cyclic homology; Connes' long exact sequence and the SBI sequence; bar resolution and the Hochschild complex; HKR theorem)
Kontsevich — Deformation quantization of Poisson manifolds · *Letters in Mathematical Physics* 66 (2003), 157--216; the formality theorem for $HH^*(A, A)$ for a smooth commutative algebra $A$ in characteristic zero, identifying the Gerstenhaber algebra structure on Hochschild cohomology with the Schouten bracket on polyvector fields up to homotopy; foundational input to deformation quantisation of Poisson manifolds
Stacks Project — Hochschild cohomology and the bar complex · <https://stacks.math.columbia.edu/tag/09S6> (Hochschild cohomology and the bar resolution); <https://stacks.math.columbia.edu/tag/09SA> (cup product on Hochschild cohomology); references for the modern derived-category formulation

Estimated time

beginner: 18m
intermediate: 42m
master: 75m