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Degree-zero Hochschild homology is bimodule coinvariants

Statement

Let $k$ be a field, $A$ a unital associative $k$ -algebra, and $M$ a $k$ -central $A$ -bimodule. Then there is a canonical $k$ -module isomorphism

$H H_{0} (A, M) ≅ M / D (A, M), D (A, M) : = {span}_{k} {a m - m a : a \in A, m \in M} .$

The denominator $D (A, M)$ is a $k$ -subspace of $M$ . It need not be a two-sided ideal or a sub-bimodule: this failure occurs for the regular bimodule of $M_{2} (k)$ .

Facts & Assumptions

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

[F1]

The Hochschild chain definition sets $C_{0} (A, M) = M$ (Hochschild chains and Hochschild homology with coefficients).

[F2]

The Hochschild chain definition sets $b_{0} = 0$ and $b_{1} (m \otimes a) = m a - a m$ (Hochschild chains and Hochschild homology with coefficients).

[F3]

The Hochschild homology definition sets $H H_{n} (A, M) = H_{n} (C_{∙} (A, M))$ (Hochschild chains and Hochschild homology with coefficients).

[F4]

The degree- $n$ cycle and boundary subobjects are respectively $\ker (d_{n})$ and $im (d_{n + 1})$ (Cycle and boundary subobjects of a complex).

[F5]

Homology is the cokernel of the boundary-to-cycle map, equivalently the quotient of cycles by boundaries (Homology object of a chain complex).

[F6]

Every element of $M \otimes_{k} A$ is a finite sum of elementary tensors (The tensor product $M \otimes_{R} N$ from the additive group underlying the free $Z$ -module on $M \times N$ , elementary tensors, and finite tensor sums).

[F7]

The span of a subset of a vector space is a linear subspace and consists of its finite linear combinations, including the empty sum (Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , $span (S)$ is exactly the set of linear combinations of finite lists of elements of $S$ , and $span (\emptyset) = {0_{V}}$ ).

[F8]

The cokernel of a module homomorphism is the quotient by its image (Module homomorphism and isomorphism, kernel, image and cokernel).

[F9]

The image of a module homomorphism is a submodule (Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel).

[F10]

A quotient by a submodule has the induced module structure (Quotient module $M / N$ with scalar multiplication on additive cosets).

[F11]

The quotient action is well-defined and satisfies the module laws (The quotient action is well defined and makes $M / N$ a module).

[F12]

For every ring $R$ , the category of left $R$ -modules is abelian (Modules over a ring form an abelian category).

[F13]

$M_{n} (k)$ is the vector space of $n$ by $n$ matrices with entrywise addition and scalar multiplication (The vector space $M_{m \times n} (F) : = F^{m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations).

[F14]

$M_{n} (k)$ is a unital ring with entrywise addition and matrix multiplication ( $M_{n} (F)$ is a ring under entrywise addition and matrix multiplication, including the zero ring $M_{0} (F)$ ).

[F15]

Matrix multiplication is associative and unital, distributes over addition, and is compatible with scalar multiplication (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

[F16]

A $k$ -algebra is a unital ring with a unital ring map from $k$ whose image is central (Algebras over a commutative ring, central structure maps, and algebra homomorphisms).

[F17]

Matrix products are defined by row-by-column sums and $I_{n}$ is the identity matrix (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F18]

The matrix unit $E_{i j}$ has a single $1$ in entry $(i, j)$ and zeros elsewhere (Matrix units $E_{i j}$ and the Kronecker delta).

[F19]

Matrix units multiply by $E_{i j} E_{r s} = δ_{j r} E_{i s}$ ( $E_{i j} E_{k ℓ} = δ_{j k} E_{i ℓ}$ ).

[F20]

The trace of a square matrix is the sum of its diagonal entries (The trace $tr (A)$ as the sum of the diagonal entries).

[F21]

For square matrices $X, Y$ over $k$ , $tr (X Y) = tr (Y X)$ (For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ ).

[F22]

Trace is $k$ -linear (Trace is a linear functional on $M_{n} (F)$ ).

[F23]

The regular bimodule has left and right actions given by multiplication (Enveloping algebra and the bimodule–module dictionary).

Source notes

Weibel, An Introduction to Homological Algebra, §9.1.1, printed p.300/PDF p.0, lines 17–19, identifies the image of the degree-one face difference with the commutator submodule and gives the degree-zero quotient. Khovanov, “Triply-graded link homology and Hochschild homology of Soergel bimodules,” “Hochschild homology,” PDF p.1, lines 9–13, defines the coinvariant quotient by the span of commutators and identifies it with $R \otimes_{R^{e}} M$ . These passages confirm the convention up to the harmless sign reversal in our $b_{1}$ ; the equality $im b_{1} = D (A, M)$ is proved in steps 2.1–2.2.

Proof

technique · direct

1.1F1F2F3F4F5F12given

By [F1], [F2], and [F4], $Z_{0} (C) = \ker b_{0} = M$ and $B_{0} (C) = im b_{1}$ . Thus [F5] identifies $H H_{0} (A, M)$ with the cokernel of the inclusion $im b_{1} ↪ M$ .

1.2F2F6F7given

Let $z \in C_{1} = M \otimes_{k} A$ . By [F6], write $z = \sum_{i < r} m_{i} \otimes a_{i}$ . Then [F2] gives $b_{1} (z) = \sum_{i < r} (m_{i} a_{i} - a_{i} m_{i}) = - \sum_{i < r} (a_{i} m_{i} - m_{i} a_{i}) \in D (A, M) .$ So $im b_{1} \subseteq D (A, M)$ .

1.3F13F14F15F16F17F23given

To verify the asserted failure of ideal and sub-bimodule closure, take $A = M = M_{2} (k)$ with the regular bimodule. By [F13] and [F14] this is a vector space and a unital ring. Define $η : k \to M_{2} (k)$ by $η (λ) = λ I_{2}$ . Entrywise operations and [F15], [F17] give $η (1) = I_{2}, η (λ + μ) = η (λ) + η (μ), η (λ μ) = η (λ) η (μ),$ and for every $X \in M_{2} (k)$ , $η (λ) X = λ X = X η (λ) .$ Thus [F16] makes $A$ a unital associative $k$ -algebra. The regular left and right actions in [F23] commute by associativity, and the displayed centrality shows they agree on $k$ , so $M$ is $k$ -central.

1.4F7F21F22given

For any $X, Y \in M_{2} (k)$ , [F21] and [F22] imply $tr (X Y - Y X) = 0$ . Since every element of $D (A, A)$ is a finite $k$ -linear combination of commutators by [F7], [F22] implies $D (A, A) \subseteq \ker (tr)$ .

1.5F1F2F3F7given

If $M = 0$ , then $C_{0} = 0$ , $b_{1} = 0$ , $D (A, M) = 0$ , and both sides of the isomorphism are zero. If $A = k$ , $k$ -centrality gives $a m = m a$ for every $a \in k, m \in M$ , so $D (k, M) = 0$ and [F3] gives $H H_{0} (k, M) = M$ . In both cases the canonical quotient map is the asserted isomorphism. No basis, projectivity, or choice is used; in particular, AC is neither assumed nor invoked.

2.1F2F6F7step 1.2given

Conversely, by [F7] an arbitrary $d \in D (A, M)$ has the form $d = \sum_{i < r} λ_{i} (a_{i} m_{i} - m_{i} a_{i})$ . The tensor $w = - \sum_{i < r} λ_{i} m_{i} \otimes a_{i}$ satisfies $b_{1} (w) = - \sum_{i < r} λ_{i} (m_{i} a_{i} - a_{i} m_{i}) = d$ by linearity of $b_{1}$ and [F2]. This also covers the empty sum $r = 0$ , for which $d = w = 0$ . Hence $D (A, M) \subseteq im b_{1}$ , so $im b_{1} = D (A, M)$ .

2.2F18F19F20step 1.4given

By [F18] and [F19], $E_{01} E_{11} = E_{01}, E_{11} E_{01} = 0,$ so $E_{01} = E_{01} E_{11} - E_{11} E_{01} \in D (A, A)$ . But $E_{01} E_{10} = E_{00}, tr (E_{00}) = 1_{k} \neq 0_{k}$ by [F19], [F20], and the field axiom $1_{k} \neq 0_{k}$ . Therefore $E_{00} \notin D (A, A)$ by step 1.4.

3.1F7F8F9F10F11step 1.1step 2.1given

By [F7], $D (A, M)$ is a $k$ -subspace. Steps 1.1 and 2.1 therefore identify the homology cokernel with the quotient module $M / D (A, M)$ by [F8], [F9], [F10], and [F11]. The isomorphism is induced by the identity on $M$ , so it is canonical.

3.2step 2.2F23

Since $E_{01} \in D (A, A)$ while $E_{01} E_{10} \notin D (A, A)$ , this subspace is not closed under the right regular action. Hence it is neither a two-sided ideal nor an $A$ -sub-bimodule, proving the stated qualification.

∎

Depends on

Used by

Hochschild homology of the ground field Example

Dependency tree · two levels

74 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 9: Hochschild and Cyclic Homology, §9.1.1 (standard reference, not scraped)
Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, Hochschild homology section (standard reference, not scraped)