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HHH is isomorphic to reduced Khovanov-Rozansky homology

Statement

Assume the Axiom of Choice (inherited through the diagonal Koszul comparison of A closed MOY resolution's Koszul complex computes Hochschild homology of its Soergel bimodule). Let $σ$ be a braid word on $m \geq 1$ strands with closure $\hat{σ}$ , let $H H H (σ)$ be the termwise Hochschild theory of The termwise Hochschild complex of a Rouquier complex and the groups HHH, and let $\overline{H} (\hat{σ})$ be the reduced Khovanov-Rozansky homology of The reduced Khovanov-Rozansky homology evaluated on the corresponding braid diagram. Then there is an isomorphism of trigraded $Q$ -vector spaces $H H H^{c, h, p} (σ) ≅ {\overline{H}}^{c, - h - 1, p - h + 1} (\hat{σ})$ under the trigrading correspondence of The Koszul-Hochschild comparison respects crossing differentials and trigradings: writing $k_{c o r r} = k + 1$ and $l_{c o r r} = l - 1$ for the corrected Khovanov-Rozansky grading, $k_{c o r r} = - h$ , $l_{c o r r} = p - h$ , $j = c$ ; equivalently $a = - h$ , $q = p - h$ , $t = c$ in the marked corrected variables $(a, q, t) = (k_{c o r r}, l_{c o r r}, j)$ . The unreduced Khovanov-Rozansky theory is recovered from the reduced one by adjoining the trivial polynomial factor, as recorded in The reduced Khovanov-Rozansky homology.

Caveats: only the reduced theory is compared; the isomorphism is trigrading-preserving only after the global correction, and no absolute normalization beyond it is claimed; the compatibility has to hold over all resolutions, including the corrected negative crossing and the signs of the Koszul total differentials; the Axiom of Choice enters only through the comparison of the bar and diagonal Koszul resolutions inside the cited identification, not through the construction of $H H H$ .

Facts & Assumptions

Given: a braid word $σ$ with $N$ crossings and closure $\hat{σ}$ , the cube of resolutions of the braid diagram, the termwise Hochschild complex of The termwise Hochschild complex of a Rouquier complex and the groups HHH and the reduced Khovanov-Rozansky complex of The reduced Khovanov-Rozansky homology, and AC.

[F1]

For the generator complex $F (σ)$ of Khovanov's generator complexes for the HHH construction the groups $H H H^{c, h, p} (σ) = H^{c} (H H_{h} (R, F (σ)^{∙}))_{p}$ are the cohomology of a bounded complex whose terms are the Hochschild homologies of the resolution terms $F (σ)^{ν} = ⨂_{ν} (local term)$ over the crossings, with differentials induced by the maps $r b_{s}, b r_{s}$ (The termwise Hochschild complex of a Rouquier complex and the groups HHH).

[F2]

The reduced Khovanov-Rozansky theory is built over the ring of differences with the coefficient $a$ retained; for a fixed resolution $D_{ν}$ the reduced summand of the Koszul homology is $H H_{∙} (R, B (D_{ν}))$ with $B (D_{ν}) = ⨂_{s} B_{s}$ over the wide edges, and the unreduced theory differs by adjoining the trivial polynomial factor (The reduced Khovanov-Rozansky homology, A closed MOY resolution's Koszul complex computes Hochschild homology of its Soergel bimodule).

[F3]

The $a = 0$ specialization of the Khovanov-Rozansky complex is the folded Koszul complex of the resolution sequence, and the crossing complexes are the cones of $χ_{0}$ and $χ_{1}$ with the shifts ${0, 2}$ and ${0, - 2}$ (The Khovanov-Rozansky complex and trigraded braid homology, The positive and negative Khovanov-Rozansky crossing complexes).

[F4]

The local maps induced by $χ_{0}$ and $χ_{1}$ on the reduced summands are, up to nonzero rational units, the maps $r b_{s}$ and $b r_{s}$ , the identification is compatible with the tensor products over the layers, and the trigradings correspond by $k = - h$ , $l = p - h$ , $j = c$ after the correction $(k, l) \mapsto (k + 1, l - 1)$ (The Koszul-Hochschild comparison respects crossing differentials and trigradings).

[F5]

AC is the choice-function principle (The Axiom of Choice), used only through the diagonal Koszul identification inside [F2].

Proof

technique · direct

1.1F1F2givenalgebra

Resolution-wise identification. Fix a resolution $D_{ν}$ of the braid diagram, obtained by replacing each crossing by an arc (0-resolution) or a wide edge (1-resolution). By [F2] the reduced Koszul homology of $D_{ν}$ is $H H_{∙} (R, B (D_{ν}))$ , and by [F1] the term of the termwise complex attached to the same choice of resolutions is $H H_{∙}$ of the corresponding tensor product of local terms, which is the same bimodule $B (D_{ν})$ when the reduced bimodules $B_{s}$ are assigned to the wide edges and $R$ to arcs; therefore the resolution-wise contributions coincide up to the fixed shifts, and the identification is natural for bimodule maps.

2.1F3F4step 1.1algebra

Intertwining the differentials. Two adjacent resolutions differ at one crossing, and the differential of the cube complex at that edge is $χ_{0}$ (positive crossing) or the corrected $χ_{1}$ (negative crossing) on the Khovanov-Rozansky side, and $r b_{s}$ or $b r_{s}$ on the termwise Hochschild side. By [F4] the resolution-wise identifications of step 1.1 carry one into the other up to nonzero rational units and the fixed shifts, and they are compatible with the tensor products over the layers and with the Koszul signs of the total differentials; hence they assemble over the $2^{N}$ resolutions into a chain isomorphism, up to the overall shift, between the complex computing $H H H (σ)$ and the corrected original reduced Khovanov-Rozansky resolution-homology complex. The complete $a = 0$ specialization has a second copy from its universal zero row; [F4] removes that row and retains the original theory's fixed ${- 1, 1}$ shift. The consistent vertex rescalings of [F4], not independent arbitrary edge scalars, give the stated chain isomorphism.

3.1F4step 2.1algebra

Cohomology and the trigrading. Taking cohomology of the chain isomorphism of step 2.1 gives a $Q$ -linear isomorphism $H H H^{c, h, p} (σ) ≅ {\overline{H}}^{j, k, l} (\hat{σ})$ ; by [F4] the grading classes correspond by $k = - h$ , $l = p - h$ , $j = c$ after the global correction, and the correction is the one that moves the one-strand class $(- 1, 1, 0)$ in the order $(k, l, j)$ of the Khovanov-Rozansky theory to $(0, 0, 0)$ , matching the one-dimensional class of $H H H$ in $(0, 0, 0)$ for the trivial one-strand braid.

4.1F2F5step 1.1step 3.1∎

The unreduced theory. The unreduced construction carries the coefficient variable $a$ and the trivial polynomial factor; by [F2] and The reduced Khovanov-Rozansky homology the unreduced groups are obtained from the reduced ones by adjoining that factor, so the isomorphism of step 3.1 is exactly the comparison stated in the source between $H H H$ and the reduced homology. The only use of AC is through [F2], in step 1.1.

Depends on

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Sources

Mikhail Khovanov, Triply-graded link homology and Hochschild homology of Soergel bimodules, arXiv:math/0510265v3 (19 printed pages); Theorem 1 and its proof, printed pp. 6-10 (standard reference, not scraped)
Mikhail Khovanov and Lev Rozansky, Matrix factorizations and link homology II, arXiv:math/0505056v2 (37 printed pages); end of section 1, printed pp. 11-12 (standard reference, not scraped)
Anna Beliakova, Krzysztof K. Putyra and Stephan M. Wehrli, Quantum link homology via trace functor I, arXiv:1605.03523v2; section 3.8.6, equation (3.44), printed p. 39 (standard reference, not scraped)