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Hochschild homology is Tor over the enveloping algebra

Statement

Assume the Axiom of Choice (AC). Let $k$ be a field, let $A$ be a unital associative $k$ -algebra, and let $M$ be a $k$ -central $A$ -bimodule. Regard $A$ as a right $A^{e} = A \otimes_{k} A^{o p}$ -module by $a (c \otimes d^{o p}) = d a c$ , and regard $M$ as a left $A^{e}$ -module by $(c \otimes d^{o p}) m = c m d$ . For every $n \geq 0$ , there is a canonical isomorphism

$H H_{n} (A, M) ≅ {Tor}_{n}^{A^{e}} (A, M),$

natural in the coefficient bimodule $M$ . The Axiom of Choice is assumed; no $A^{e}$ -projectivity of $M$ is assumed.

Facts & Assumptions

Given: AC, a field $k$ , a unital associative $k$ -algebra $A$ , and a $k$ -central $A$ -bimodule $M$ .

[F1]

A $k$ -central $A$ -bimodule $M$ is a left $A^{e}$ -module by $(c \otimes d^{o p}) m = c m d$ (Enveloping algebra and the bimodule–module dictionary).

[F2]

The regular bimodule $A$ is a right $A^{e}$ -module by $a (c \otimes d^{o p}) = d a c$ (Enveloping algebra and the bimodule–module dictionary).

[F3]

The augmented two-sided bar complex has terms ${Bar}_{n} (A) = A \otimes_{k} A^{\otimes_{k} n} \otimes_{k} A$ and the specified alternating adjacent-multiplication differential, including the empty middle tensor in degree zero (The augmented two-sided bar complex).

[F4]

Under AC, ${Bar}_{∙} (A) \to A$ is a projective resolution of the regular right $A^{e}$ -module $A$ (The two-sided bar complex is a projective $A^{e}$ -resolution).

[F5]

The Hochschild complex has $C_{0} (A, M) = M$ , $C_{n} (A, M) = M \otimes_{k} A^{\otimes_{k} n}$ for $n \geq 1$ , and $H H_{n} (A, M) = H_{n} (C_{∙} (A, M))$ (Hochschild chains and Hochschild homology with coefficients).

[F6]

The maps $(a_{0} \otimes \dots \otimes a_{n + 1}) \otimes m \mapsto (a_{n + 1} m a_{0}) \otimes a_{1} \otimes \dots \otimes a_{n}$ give a chain isomorphism ${Bar}_{∙} (A) \otimes_{A^{e}} M ≅ C_{∙} (A, M)$ , natural in $M$ , with no projectivity assumption on $M$ (Hochschild chains are bar tensor chains).

[F7]

If $Q_{∙} ↠ N$ is a specified projective resolution of a right $R$ -module, the right-resolution construction is ${Tor}_{n}^{R, Q} (N, L) : = H_{n} (Q_{∙} \otimes_{R} L)$ for a left $R$ -module $L$ (Tor from a projective resolution of the right module).

[F8]

Under DC and with projective resolutions supplied for both modules, balanced ${Tor}_{n}^{R} (N, L)$ is identified from either resolution; the identifications are canonical under change of resolution and define a covariant bifunctor up to those canonical isomorphisms (The balanced Tor bifunctor).

[F9]

Under AC, every left module over a unital ring admits a projective resolution (Under the Axiom of Choice, every module admits a projective resolution).

[F10]

AC means every family of nonempty sets has a choice function (The Axiom of Choice).

[F11]

In ZF, AC implies DC (AC implies DC implies countable choice).

Proof

technique · direct

1.1F1F2given

By [F1] and [F2], the right module in the Tor expression is the regular right $A^{e}$ -module $A$ , and the coefficient bimodule is a left $A^{e}$ -module. Thus the tensor products and resolution statements below use the stated sides of the enveloping algebra.

1.2F3F4given

Put $P_{∙} = {Bar}_{∙} (A)$ . By [F4], under the assumed AC this is a projective resolution of the right $A^{e}$ -module $A$ . The augmentation endpoint is ${Bar}_{0} (A) \to A$ by multiplication.

2.1F7step 1.2

Applying the right-resolution definition [F7] to this specified resolution gives ${Tor}_{n}^{A^{e}, P} (A, M) = H_{n} (P_{∙} \otimes_{A^{e}} M)$ . This construction does not require $M$ itself to be projective.

2.2F4F8F9step 1.2

Applying [F9] to the unital ring $A^{e}$ supplies a left projective resolution of $M$ . Together with the right projective resolution $P_{∙}$ of $A$ from step 1.2, this supplies both resolutions required in [F8]. The corollary does not assert that $M$ is itself projective.

3.1F5F6step 2.1

The chain isomorphism [F6], followed by the homology definition [F5], gives for every $n \geq 0$ the identity $H H_{n} (A, M) = H_{n} (C_{∙} (A, M)) ≅ H_{n} (P_{∙} \otimes_{A^{e}} M) = {Tor}_{n}^{A^{e}, P} (A, M)$ . In degree zero, [F6] maps $(a_{0} \otimes a_{1}) \otimes m$ to $a_{1} m a_{0}$ ; in degree one its chain-map identity uses $b_{1} (m \otimes a) = m a - a m$ , so the first nonzero boundary is included.

4.1F4F5F6F8F10F11step 3.1step 2.2

By [F10] AC is the assumed choice principle, and [F11] gives the DC hypothesis of [F8]. Hence the right-resolution group in step 3.1 is canonically identified with the balanced ${Tor}_{n}^{A^{e}} (A, M)$ , independently of the chosen projective resolutions. For a bimodule map $f : M \to M^{'}$ , the map $1_{P_{∙}} \otimes f$ induces the right-resolution homology map; [F5] commutes with $f$ , and [F8] makes the balanced Tor identifications natural. Thus the isomorphism in the statement is natural in $M$ . AC is used in [F4] to choose a $k$ -basis of $A$ and make the resulting free bar terms projective, in [F9] to make the canonical free resolution of $M$ projective, and through AC $\Rightarrow$ DC for balanced Tor comparison and naturality; no choice is used in the chain isomorphism itself.

5.1F3F4F6F7F8step 3.1∎

If $M = 0$ , both chain complexes in step 3.1 vanish and both sides are zero. If $A = k$ , every bar term identifies with $k$ and every adjacent-multiplication face identifies with $1_{k}$ , so $d_{n} = \sum_{r = 0}^{n} (- 1)^{r} 1_{k}$ : it is the identity for even $n$ and zero for odd $n$ . After tensoring with $M$ , this augmented complex has homology $M$ in degree zero and zero in positive degrees; the length-zero resolution of the projective right $k$ -module $k$ gives the same Tor groups. The degree-zero and degree-one endpoints are those already checked in step 3.1.

Source comparison

Weibel, An Introduction to Homological Algebra, §9.1.3, Lemma 9.1.3, printed pp. 302–303 (PDF pp. 2–3), identifies Hochschild homology with relative Tor over $k \to A^{e}$ and gives the bar-tensor chain isomorphism. Section 9.1.4 and Corollary 9.1.5, printed p. 303 (PDF p. 3), explain that when $A$ is projective over $k$ , the bar terms are projective over $A^{e}$ and the relative Tor computation agrees with absolute Tor. These passages corroborate the relative/absolute distinction; the proof above obtains the absolute Tor resolution directly from the AC-qualified projectivity theorem [F3].

Depends on

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Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9, §§9.1.3–9.1.5 (standard reference, not scraped)