The Hochschild Differential as an Operator

Introduction

A cochain complex is a graded module with an operator of degree one whose square is zero, and the first cochain complex that an algebra produces from itself is the Hochschild complex. Its cochains are the multilinear maps on the algebra with values in a bimodule, and its operator is the Hochschild differential $b$, the alternating sum of the ways of multiplying the arguments together and acting on the value. The whole content of the complex is two properties of this single operator: it is $k$-linear of degree one, and its square is zero.

This article studies the differential as an operator. It fixes the cochain module, defines $b$ on it, proves $b^2 = 0$ by a cancellation of the twelve terms, and reads the consequence — the cocycles, the coboundaries and the cohomology — as the kernel, the image and the quotient of the operator. The cohomology itself, its identification with the derived functors of the invariants, its low-dimensional interpretations and its products are the subject of Hochschild Homology, and the general theory of a cochain complex, its differential and its cohomology belongs to Homological Algebra; the present article is the operator layer of the first and cites the second for the ambient theory.

Throughout, $k$ is a commutative ring with identity $1 \neq 0$, $A$ is a unital associative $k$-algebra, not necessarily commutative, and $M$ is an $A$-bimodule. The tensor powers are over $k$, and $\operatorname{Hom}$ without a subscript means $\operatorname{Hom}_k$; the enveloping algebra is $A^{\mathrm{e}} = A \otimes_k A^{\mathrm{op}}$, so that $A$-bimodules are the left $A^{\mathrm{e}}$-modules. No norm, distance or completion occurs.

The Cochain Module

Definition

Definition. For $n \geq 0$ the module of $n$-cochains of $A$ with coefficients in $M$ is

$$ C^n(A, M) = \operatorname{Hom}_k(A^{\otimes n}, M), $$

the $k$-module of $k$-multilinear maps $f : A^n \to M$. For $n = 0$ the empty tensor power is $A^{\otimes 0} = k$, so that $C^0(A,M) = \operatorname{Hom}_k(k, M) \cong M$; the element of $M$ corresponding to a $0$-cochain is written $m$, and the map is $k \mapsto km$.

The Hochschild cochain module is the graded module

$$ C^{\bullet}(A, M) = \bigoplus_{n \geq 0} C^n(A, M). $$

An element of $C^\bullet(A,M)$ is a finite sequence $f = (f_0, f_1, f_2, \dots)$ with $f_n \in C^n(A,M)$ and $f_n = 0$ for all but finitely many $n$; the module is graded, and a single $n$-cochain is homogeneous of degree $n$.

The Bimodule Structure

The cochain module carries more than its $k$-module structure, and the extra structure is what makes the differential a map of the right kind.

Proposition. For $a \in A$ and $f \in C^n(A,M)$ the assignments

$$ (af)(a_1,\dots,a_n) = a\,f(a_1,\dots,a_n), \qquad (fa)(a_1,\dots,a_n) = f(a_1,\dots,a_n)\,a $$

turn $C^n(A,M)$ into an $A$-bimodule, and the same formulas turn $C^{\bullet}(A,M)$ into a graded $A$-bimodule: the two actions commute, each is $k$-linear in $f$, and $C^{\bullet}(A,M)$ is a left module over the enveloping algebra $A^{\mathrm{e}}$.

Proof. The module $M$ is an $A$-bimodule, so its left and right actions commute; applying them pointwise to the values of $f$ gives two actions on $C^n(A,M)$ that are $k$-linear in $f$ and commute. A left $A$-module that is also a right $A$-module with commuting actions is a left $A^{\mathrm{e}}$-module by $(a \otimes a') \cdot f = a f a'$, and the grading is preserved because the actions do not change the number of arguments.

Remark. The cochain module is therefore not merely a graded $k$-module but a graded $A^{\mathrm{e}}$-module, and a differential of the theory is required to commute with this action. The Hochschild differential below does; this is the operator form of the statement that the Hochschild complex is a complex of $A^{\mathrm{e}}$-modules and not only of $k$-modules.

The Differential

Definition

Definition. The Hochschild differential is the family of $k$-linear maps

$$ b = b_n : C^n(A, M) \longrightarrow C^{n+1}(A, M), \qquad n \geq 0, $$

given on $f \in C^n(A,M)$ and $a_0, \dots, a_n \in A$ by

$$ (b f)(a_0, \dots, a_n) = a_0 f(a_1, \dots, a_n) + \sum_{i=1}^{n} (-1)^i f(a_0, \dots, a_{i-1}a_i, \dots, a_n) + (-1)^{n+1} f(a_0, \dots, a_{n-1})\, a_n . $$

The definition is $k$-multilinear in the arguments, so the value on $f$ is an $(n+1)$-cochain; the formula is the alternating sum of the $n+2$ ways of inserting the product of two adjacent arguments, with the two end terms acting on the value in $M$ instead of multiplying two arguments together. The family $b = (b_n)$ is also written $\delta$ when the cochain reading is to be distinguished from the chain boundary of Hochschild Homology.

Low Degrees

The formula is best read in low degrees, where the operator acquires its meaning.

Degree zero. Here $n = 0$ and $f = m \in M$; the sum is empty and only the two end terms survive:

$$ (b m)(a_0) = a_0 m - m a_0 . $$

So $b : M \to \operatorname{Hom}_k(A, M)$ sends an element of the bimodule to the operator $a \mapsto am - ma$, the inner multiplication of the bimodule. Its kernel is the invariant submodule $M^A$.

Degree one. Here $f \in \operatorname{Hom}_k(A, M)$ and

$$ (b f)(a_0, a_1) = a_0 f(a_1) - f(a_0 a_1) + f(a_0) a_1 . $$

This is the operator form of the Leibniz rule: $b f = 0$ is exactly the statement that $f$ is a derivation $A \to M$.

Degree two. Here $g \in \operatorname{Hom}_k(A^{\otimes 2}, M)$ and

$$ (b g)(a_0, a_1, a_2) = a_0 g(a_1, a_2) - g(a_0 a_1, a_2) + g(a_0, a_1 a_2) - g(a_0, a_1) a_2 . $$

The alternating pattern of signs is visible: $-, +, -$ on the inner terms and $+$ then $-$ on the two end terms.

The Square-Zero Property

Theorem. The Hochschild differential satisfies

$$ b \circ b = 0, \qquad \text{that is} \qquad b_{n+1} b_n = 0 \quad \text{for every } n \geq 0 . $$

Proof. Fix $f \in C^n(A,M)$ and arguments $a_0, \dots, a_{n+1}$. The cochain $b f$ is a sum of $n+2$ terms, each of which is the value of $f$ at an $(n+1)$-tuple together with a product of two adjacent arguments contracted or an end action applied; the differential $b(bf)$ is the alternating sum over the $n+3$ positions of $(b f)$ of the same operation. Expand: the double sum ranges over the pairs $(i,j)$ with $0 \leq i < j \leq n+1$, where $i$ is the position contracted in the inner application and $j$ that contracted in the outer one; each pair $(i,j)$ arises in exactly two ways, once by contracting $a_i a_{i+1}$ first and then the resulting adjacent product with $a_{i+2}$, and once by contracting $a_{i+1}a_{i+2}$ first and then $a_i$ with the result. The two contributions are equal as values of $f$ — both amount to the single value $f(\dots, a_i a_{i+1} a_{i+2}, \dots)$ — and their coefficients in the alternating sum are $(-1)^i(-1)^{j-1}$ and $(-1)^{i+1}(-1)^j$, which are opposite. The two end terms are paired with the contractions at the ends in the same way, and every remaining term cancels; hence $b^2 f = 0$.

Corollary. The pair $(C^{\bullet}(A,M), b)$ is a cochain complex: the image of $b_n$ is contained in the kernel of $b_{n+1}$,

$$ \operatorname{im} b_n \;\subseteq\; \ker b_{n+1} . $$

The corollary is the definition of a cochain complex in Homological Algebra, and the two properties that make the pair one — $b$ of degree one and $b^2 = 0$ — are exactly the two properties proved above.

Naturality and the Bimodule Action

Proposition. The differential $b$ is natural in $A$ and in $M$, and it commutes with the $A^{\mathrm{e}}$-action on the cochains: for $a, a' \in A$ and $f \in C^n(A,M)$,

$$ b(a f a') = a\, (bf)\, a' . $$

Proof. Naturality in $M$ is the observation that every term of the formula uses only the bimodule operations and evaluates $f$, so a bimodule map $M \to M'$ carries $bf$ to $b(f')$ where $f'$ is the composite. For the action, multiply the defining formula on the left by $a$ and on the right by $a'$; the two end terms acquire $a a_0 f(\dots) a'$ and $a f(\dots) a_n a'$, which are the end terms of $b(a f a')$, and the inner terms acquire $a f(\dots) a'$ at each contracted position, which are the corresponding inner terms.

Corollary. The Hochschild complex is a complex of $A^{\mathrm{e}}$-modules: the differential is $A^{\mathrm{e}}$-linear, and so the kernels, the images and the cohomology $H^{\bullet}(A,M)$ carry $A^{\mathrm{e}}$-module structures.

Cocycles, Coboundaries and Cohomology

The Definitions

Definition. Let $Z^n(A,M) = \ker b_n$ be the module of $n$-cocycles and let $B^n(A,M) = \operatorname{im} b_{n-1}$ be the module of $n$-coboundaries, with the convention $B^0(A,M) = 0$. Because $b^2 = 0$, one has $B^n \subseteq Z^n$, and the Hochschild cohomology is the graded quotient

$$ H^n(A, M) = Z^n(A, M) / B^n(A, M), \qquad n \geq 0 . $$

The definitions are those of a cochain complex, and they are the reason the square-zero property is worth proving: without it the quotient would not be defined.

The Low-Dimensional Reading

Proposition. The low-dimensional modules of the Hochschild complex are

$$ H^0(A,M) = M^A, \qquad H^1(A,M) = \operatorname{Der}_k(A,M) / \operatorname{InnDer}_k(A,M), \qquad H^2(A,M) = \text{the abelian extensions up to equivalence}, $$

where $M^A = \{m : am = ma\}$ is the invariant submodule, $\operatorname{Der}_k(A,M)$ is the module of $k$-linear maps satisfying the Leibniz rule, and $\operatorname{InnDer}_k(A,M)$ is the submodule of inner derivations $a \mapsto am - ma$.

Proof. The degree-zero statement is the computation $b m = 0 \iff am = ma$ for all $a$. The degree-one statement is the identification of the kernel of $b_1$ with the derivations, by the degree-one formula, and of the image of $b_0$ with the inner derivations. The degree-two statement is the interpretation of the second cohomology of the Hochschild complex as the group of abelian extensions of $A$ by $M$; it is the classical interpretation of the Hochschild theory and belongs, with its proof, to Hochschild Homology.

Remark. The identifications with the derived functors — $H^n(A,M) \cong \operatorname{Ext}^n_{A^{\mathrm{e}}}(A,M)$ — and the products on the cohomology are not developed here. The present article owns only the operator: the differential, its degree, its square-zero property, and the complex and the cohomology that these define. The reader who needs the derived-functor theory, the Hochschild–Kostant–Rosenberg theorem, the cup product or the Gerstenhaber bracket is directed to Hochschild Homology.

The Differential as an Operator

Degree and Square

The differential is an operator on a graded module, and its two defining properties are statements about that grading.

Proposition. With respect to the grading $C^{\bullet}(A,M) = \bigoplus_n C^n(A,M)$ the differential has degree one, $b(C^n) \subseteq C^{n+1}$, and its square is zero, $b^2 = 0$. Conversely a $k$-linear map of degree one with square zero on a graded module is exactly a differential of a cochain complex.

Proof. The first statement is the definition: $b$ raises the number of arguments by one. The second is the theorem. The converse is the definition of a cochain complex in Homological Algebra.

The Shift

The differential is a morphism of graded modules from the complex to its shift.

Proposition. Let $C^{\bullet}[1]$ be the graded module with $C^{\bullet}[1]^n = C^{n+1}$. Then $b$ is a morphism of graded $A^{\mathrm{e}}$-modules $C^{\bullet} \to C^{\bullet}[1]$, and the condition $b^2 = 0$ is the statement that the composite

$$ C^{\bullet} \xrightarrow{\;b\;} C^{\bullet}[1] \xrightarrow{\;b\;} C^{\bullet}[2] $$

is zero.

Proof. The first statement restates the degree of $b$ and its $A^{\mathrm{e}}$-linearity. For the second, $b^2$ as a map $C^n \to C^{n+2}$ is $b_{n+1}b_n$, zero by the theorem.

The Transpose Relation

The differential of the cochain complex is the transpose of the boundary of the chain complex, and the two readings of the algebra are dual.

Proposition. Let $B_\bullet(A)$ be the bar resolution of Hochschild Homology, with boundary $b^{\mathrm{chain}} : B_n(A) \to B_{n-1}(A)$. Then the Hochschild differential on the cochains is the transpose of the chain boundary under the pairing $\operatorname{Hom}_k(B_n(A), M) \times B_n(A) \to M$,

$$ \langle b f, x \rangle = \langle f, b^{\mathrm{chain}} x \rangle . $$

In particular the cochain differential is determined by the chain boundary and vice versa, and the cochain complex is the dual of the chain complex.

Proof. The bar resolution has $B_n(A) = A^{\otimes(n+2)}$ and the cochains are the $A^{\mathrm{e}}$-linear maps $B_n(A) \to M$, whose underlying $k$-linear maps are $\operatorname{Hom}_k(A^{\otimes n}, M)$; the transpose of the boundary is computed by evaluating the defining formula of Hochschild Homology against a cochain, and it gives the formula of the present article. The identification of the two sign conventions is the standard one for a chain complex and its dual.

Remark. The corollary is a statement about operators and not a new construction: the cochain differential and the chain boundary are one operator read on a module and on its dual. The reader should not take the two symbols $b$ of the two articles as two operators; they are transposes.

The Algebra as Coefficients

The Case $M = A$

The most used case is $M = A$, with the bimodule structure given by the product.

Proposition. For $M = A$ one has $H^0(A,A) = Z(A)$, the centre; $H^1(A,A) = \operatorname{Der}_k(A)/\operatorname{InnDer}_k(A)$, the space of outer derivations; and the degree-one cocycles are exactly the derivations of $A$.

Proof. The invariant submodule of $A$ under the bimodule structure $ama'$ is the set of $z$ with $az = za$ for all $a$, which is the centre. The degree-one statement is the low-dimensional proposition with $M = A$, and the inner derivations are the maps $a \mapsto am - ma$ with $m \in A$, which are the inner derivations $\mathrm{ad}_m$ of The Commutator Operator.

Corollary. The differential $b_0 : A \to \operatorname{Hom}_k(A,A)$ is the operator $m \mapsto \mathrm{ad}_m$, whose image is the space of inner derivations of Automorphisms and Derivations of Algebras and whose kernel is the centre. Hence $H^1(A,A)$ vanishes exactly when every derivation of $A$ is inner; in particular $H^1(M_n(k), M_n(k)) = 0$ over a field, because every derivation of the full matrix algebra is inner.

Proof. The identification $b_0(m) = \mathrm{ad}_m$ is the degree-zero computation. The vanishing for $M_n(k)$ is the statement of Automorphisms and Derivations of Algebras that every derivation of a central simple algebra is inner.

The Examples

A Commutative Algebra

Let $A$ be commutative and let $M = A$ with the product bimodule structure. Then the invariant submodule $M^A$ is all of $A$, so $H^0(A,A) = A$; the inner derivations vanish, so $H^1(A,A) = \operatorname{Der}_k(A)$ is the whole module of derivations; and the differential $b_0$ is the zero operator. The centre is the whole algebra, and the operator $b_0$ is identically zero, which is the operator form of the commutativity of $A$.

The Polynomial Algebra

Let $A = k[x]$ and $M = A$. A derivation is determined by its value $f = D(x) \in k[x]$, since the Leibniz rule gives $D(p) = p' f$ for the formal derivative $p'$; the module of derivations is free of rank one on $D_0 = \frac{d}{dx}$, and $H^1(k[x], k[x]) = k[x] \cdot \frac{d}{dx}$. The differential $b_1$ sends a derivation to $0$ and the inner derivations form the zero submodule, so the first cohomology is the whole derivation module. This is the smallest nonvanishing example, and the Hochschild–Kostant–Rosenberg theorem of Hochschild Homology is the general statement behind it.

The Algebra of Dual Numbers

Let $A = k[\varepsilon]/(\varepsilon^2)$ and $M = A$. The derivations are the maps with $D(\varepsilon) = c + d\varepsilon$ subject to $2\varepsilon D(\varepsilon) = 0$, so $D(\varepsilon) = c$ is a scalar and the derivation module has dimension one over $k$; the inner derivations are again the zero submodule, because $A$ is commutative, and $H^1(A,A)$ has dimension one. The degree-one differential $b_1$ therefore has a one-dimensional kernel and, since $b_2 b_1 = 0$, the image of $b_1$ lies in the degree-two cocycles; the operator picture is the short exact sequence of the complex in low degree.

Summary

The Hochschild cochains are the multilinear maps $C^n(A,M) = \operatorname{Hom}_k(A^{\otimes n},M)$, assembled into the graded $A^{\mathrm{e}}$-module $C^{\bullet}(A,M)$, and the Hochschild differential $b$ is the degree-one operator

$$ (b f)(a_0,\dots,a_n) = a_0 f(a_1,\dots,a_n) + \sum_{i=1}^{n}(-1)^i f(a_0,\dots,a_{i-1}a_i,\dots,a_n) + (-1)^{n+1}f(a_0,\dots,a_{n-1})a_n . $$

In low degrees it is $b m = (a \mapsto am - ma)$ on $M$, the Leibniz operator $b f(a_0,a_1) = a_0 f(a_1) - f(a_0a_1) + f(a_0)a_1$ on the linear maps, and the three-term alternating operator on the bilinear maps. The square-zero property $b^2 = 0$ holds, by the cancellation of the twelve terms of the double sum; it makes $(C^{\bullet}(A,M), b)$ a cochain complex, and it makes the quotient $H^n(A,M) = \ker b_n / \operatorname{im} b_{n-1}$ defined. The differential commutes with the $A^{\mathrm{e}}$-action, has degree one, is the transpose of the chain boundary of Hochschild Homology, and its kernel and image reproduce the low-dimensional theory: $H^0(A,M) = M^A$, $H^1(A,M)$ is the module of derivations modulo the inner ones, and $H^2(A,M)$ classifies the abelian extensions. For $M = A$ the degree-zero differential is $m \mapsto \mathrm{ad}_m$, with kernel the centre and image the inner derivations, so $H^1(A,A)$ is the space of outer derivations and vanishes for a central simple algebra. The cohomology's derived-functor description, its products and its theorems are the subject of Hochschild Homology; the complex and the differential as an operator are the subject of this article.

Summary of Notation

Further Reading

• Sarah J. Witherspoon, Hochschild Cohomology for Algebras (American Mathematical Society, 2019), for the differential, the complex and the whole cohomology theory.
• Murray Gerstenhaber, "The cohomology structure of an associative ring", Annals of Mathematics 78 (1963), 267–288, for the differential and the brackets it carries.
• Charles A. Weibel, An Introduction to Homological Algebra (Cambridge University Press, 1994), for the general theory of cochain complexes, differentials and their cohomology.
• Henri Cartan and Samuel Eilenberg, Homological Algebra (Princeton University Press, 1956), for the Hochschild complex as the standard complex of an algebra.