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Hochschild chains and Hochschild homology with coefficients

Definition

Let $k$ be a field, $A$ a unital associative $k$ -algebra, and $M$ a $k$ -central $A$ -bimodule (Enveloping algebra and the bimodule–module dictionary). For $n \geq 1$ put

$C_{n} (A, M) : = M \otimes_{k} A^{\otimes_{k} n},$

and put $C_{0} (A, M) : = M$ , using the canonical tensor-unit identification $M \otimes_{k} k ≅ M$ . For $n \geq 1$ and $0 \leq i \leq n$ , define the faces on elementary tensors by

$δ_{i}^{(n)} (m \otimes a_{1} \otimes \dots \otimes a_{n}) = {\begin{cases} (m a_{1}) \otimes a_{2} \otimes \dots \otimes a_{n}, & i = 0, \\ m \otimes a_{1} \otimes \dots \otimes (a_{i} a_{i + 1}) \otimes \dots \otimes a_{n}, & 0 < i < n, \\ (a_{n} m) \otimes a_{1} \otimes \dots \otimes a_{n - 1}, & i = n . \end{cases}$

These are well-defined $k$ -linear maps by the multilinear universal property of the finite tensor product. Set $C_{- 1} (A, M) = 0$ and $b_{0} = 0$ . The Hochschild boundary is

$b_{n} : = \sum_{i = 0}^{n} (- 1)^{i} δ_{i}^{(n)} : C_{n} (A, M) ⟶ C_{n - 1} (A, M) (n \geq 1) .$

In particular, $b_{1} (m \otimes a) = m a - a m$ . For $n \geq 2$ the faces satisfy

$δ_{i}^{(n - 1)} δ_{j}^{(n)} = δ_{j - 1}^{(n - 1)} δ_{i}^{(n)} (0 \leq i < j \leq n),$

so the alternating-sum boundaries satisfy $b_{n - 1} b_{n} = 0$ for $n \geq 2$ ; $b_{0} b_{1} = 0$ because $b_{0} = 0$ . Thus $C_{∙} (A, M)$ is a chain complex of $k$ -modules. Its $n$ th homology object is the Hochschild homology with coefficients in $M$ ,

$H H_{n} (A, M) : = H_{n} (C_{∙} (A, M)) .$

Facts & Assumptions

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

[F1]

The left and right $A$ -actions on $M$ commute, and their scalar actions agree because $M$ is $k$ -central (Enveloping algebra and the bimodule–module dictionary).

[F2]

A finite tensor product over a commutative ring represents multilinear maps into a module, so a multilinear face formula induces a unique linear map on the tensor product (Finite iterated tensor products represent multilinear maps independently of parenthesization).

[F3]

The canonical map $M \otimes_{k} k \to M$ , $m \otimes λ \mapsto m λ$ , is an isomorphism with inverse $m \mapsto m \otimes 1$ (The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[F4]

Disjoint adjacent multiplication faces satisfy the reindexed face identity (The bar boundary squares to zero and is augmented).

[F5]

Overlapping adjacent multiplication faces satisfy the face identity by associativity (The bar boundary squares to zero and is augmented).

[F6]

The homology object $H_{n} (C)$ is defined when $C_{∙}$ is a chain complex (Homology object of a chain complex).

Proof

technique · direct

1.1F2F3givenalgebra

Each endpoint formula is multilinear because the bimodule actions are $k$ -bilinear, and each interior formula is multilinear because multiplication in $A$ is $k$ -bilinear; hence [F2] induces the displayed $k$ -linear face maps and every face sends a zero input to zero. The tensor-unit isomorphism [F3] identifies the degree-zero term with $M$ .

1.2givenalgebra

The degree-zero boundary is zero by definition, while in degree one the two faces are $δ_{0}^{(1)} (m \otimes a) = m a$ and $δ_{1}^{(1)} (m \otimes a) = a m$ , so $b_{1} (m \otimes a) = m a - a m$ and $b_{0} b_{1} = 0$ .

1.3F4F5givenalgebra

If $1 \leq i < j \leq n - 1$ , both faces are internal adjacent multiplications among the $A$ -slots. For disjoint slots their operations commute and reindex as in [F4]; for overlapping slots the two composites multiply the same triple, and associativity gives [F5]. Thus $δ_{i}^{(n - 1)} δ_{j}^{(n)} = δ_{j - 1}^{(n - 1)} δ_{i}^{(n)}$ for all such internal pairs.

1.4F1givenalgebra

For the adjacent first pair $i = 0, j = 1$ , the two composites have first coefficients $(m a_{1}) a_{2}$ and $m (a_{1} a_{2})$ , which agree by the right module law. For $i = n - 1, j = n$ , they have first coefficients $a_{n - 1} (a_{n} m)$ and $(a_{n - 1} a_{n}) m$ , which agree by the left module law. For the pair $i = 0, j = n$ , they have first coefficients $(a_{n} m) a_{1}$ and $a_{n} (m a_{1})$ , which agree because the left and right actions commute by [F1]. The untouched slots agree in their original order in all three cases.

1.5givenalgebra

The remaining pairs with one endpoint face and one internal face act on disjoint data: for $i = 0$ and $2 \leq j < n$ , the first face acts on $m, a_{1}$ while the other multiplies $a_{j}, a_{j + 1}$ ; for $1 \leq i < n - 1$ and $j = n$ , the internal face multiplies $a_{i}, a_{i + 1}$ while the last face acts by $a_{n}$ on $m$ . In either order the same multiplication/action is applied to each of these disjoint slots, with the same remaining tensor factors and order, proving the face identity. Together with 1.3 and 1.4 this covers every $0 \leq i < j \leq n$ .

1.6F2F3givenalgebra

In the degenerate case $A = k$ , the canonical tensor-unit identifications turn each face into the identity on $M$ ; hence $b_{n} = \sum_{i = 0}^{n} (- 1)^{i} {id}_{M}$ , which is the identity for even $n$ and zero for odd $n$ . Consecutive composites therefore vanish in this case too.

2.1step 1.3step 1.4step 1.5algebra

Expand $b_{n - 1} b_{n}$ as the sum of terms $(- 1)^{i + j} δ_{i}^{(n - 1)} δ_{j}^{(n)}$ . For every $i < j$ , the face identity from 1.3--1.5 pairs this term with the term indexed by $(j - 1, i)$ ; their composites agree and their signs are opposite because $(- 1)^{i + j} = - (- 1)^{(j - 1) + i}$ . Every term with first index $r \geq s$ is uniquely such a partner, obtained from $i = s, j = r + 1$ ; thus every term occurs in exactly one pair. Hence $b_{n - 1} b_{n} = 0$ for $n \geq 2$ ; the case $n = 1$ was checked in 1.2.

3.1step 1.1step 1.2step 2.1F6∎

The maps are $k$ -linear by 1.1 and their consecutive composites vanish by 1.2 and 2.1, so they form a chain complex. The definition of homology applies by [F6], giving $H H_{n} (A, M) = H_{n} (C_{∙} (A, M))$ .

Depends on

Used by

Diagonal Hochschild homology of a polynomial ring Corollary
Hochschild homology of the ground field Example
Hochschild chains are bar tensor chains Lemma
Degree-zero Hochschild homology is bimodule coinvariants Proposition
Functoriality and coefficient long exact sequences for Hochschild homology Theorem
Hochschild homology is Tor over the enveloping algebra Theorem
Polynomial Hochschild homology from the diagonal Koszul complex Theorem

Dependency tree · two levels

17 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)