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A separability element contracts the normalized categorical Hochschild complex in positive degrees

Statement

Let $A$ be an algebra object in a $k$ -linear monoidal category and $B$ an $A$ -bimodule. Suppose $e : 1 \to A \otimes A$ satisfies $μ e = η, (μ \otimes i d) (i d \otimes e) = (i d \otimes μ) (e \otimes i d) .$ Set $C^{n} = Hom (A^{\otimes n}, B)$ , with the Hochschild differential given by the alternating sum of the left action, adjacent multiplications, and the right action. Its normalized subcomplex consists of maps vanishing whenever any input is the unit $η$ . For every $n \geq 1$ , define $h_{n} (f) = λ_{B} ({i d}_{A} \otimes f) (e \otimes {i d}_{A^{\otimes (n - 1)}}),$ where $λ_{B}$ denotes the left $A$ -action. Then $h$ preserves the normalized subcomplex and $d h + h d = i d$ in every positive degree.

Proof

Given: The two separability identities and the commuting left and right $A$ -actions on $B$ . We suppress coherent associators. The formal notation $e = \sum_{r} x_{r} \otimes y_{r}$ denotes a morphism composite; it does not assume that objects have elements.

1.1givenalgebra

Write the differential explicitly as $(d f) (a_{1}, \dots, a_{n + 1}) = a_{1} f (a_{2}, \dots, a_{n + 1}) + \sum_{j = 1}^{n} (- 1)^{j} f (a_{1}, \dots, a_{j} a_{j + 1}, \dots, a_{n + 1}) + (- 1)^{n + 1} f (a_{1}, \dots, a_{n}) a_{n + 1} .$ Each pair of successive faces occurs twice with opposite signs in $d^{2}$ , by associativity of multiplication and the bimodule axioms; hence $d^{2} = 0$ . At a unit input, the two faces adjoining that input cancel, and all other faces vanish for a normalized cochain. Thus normalization defines a subcomplex. The formula $h_{n} f = \sum_{r} x_{r} f (y_{r}, -)$ preserves normalization because every original input is still an input of $f$ .

2.1givenstep 1.1algebra∎

For $n \geq 1$ , expansion gives $(h_{n + 1} d f) (a_{1}, \dots, a_{n}) = f (a_{1}, \dots, a_{n}) - \sum_{r} x_{r} f (y_{r} a_{1}, a_{2}, \dots, a_{n}) + \sum_{j = 1}^{n - 1} (- 1)^{j + 1} \sum_{r} x_{r} f (y_{r}, a_{1}, \dots, a_{j} a_{j + 1}, \dots, a_{n}) + (- 1)^{n + 1} \sum_{r} x_{r} f (y_{r}, a_{1}, \dots, a_{n - 1}) a_{n} .$ The first term uses $\sum_{r} x_{r} y_{r} = 1$ . The other expansion is $(d h_{n} f) (a_{1}, \dots, a_{n}) = \sum_{r} a_{1} x_{r} f (y_{r}, a_{2}, \dots, a_{n}) + \sum_{j = 1}^{n - 1} (- 1)^{j} \sum_{r} x_{r} f (y_{r}, a_{1}, \dots, a_{j} a_{j + 1}, \dots, a_{n}) + (- 1)^{n} \sum_{r} x_{r} f (y_{r}, a_{1}, \dots, a_{n - 1}) a_{n} .$ Centrality of $e$ identifies the first term of the second expansion with $\sum_{r} x_{r} f (y_{r} a_{1}, a_{2}, \dots, a_{n})$ . All remaining terms cancel in pairs, proving $h_{n + 1} d + d h_{n} = {i d}_{C^{n}}$ , including $n = 1$ where the interior sums are empty. This identity restricts to the normalized subcomplex by the preceding step.

In degree zero the corresponding identity is $h_{1} d = i d - P$ , where $P (b) = \sum_{r} x_{r} b y_{r}$ projects onto the bimodule-invariant part. No degree-zero vanishing is asserted.

Depends on

Separable algebra objects and bimodules in a fusion category

Used by

Categorical separability contracts the module deformation complex Lemma

Dependency tree · two levels

2 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

Etingof, Gelaki, Nikshych, and Ostrik, Tensor Categories, Proposition 9.1.1 and formula (9.2) (standard reference, not scraped)