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Termwise and total Hochschild theories have different grading outputs

Example

Assume the Axiom of Choice (used only in step 1.1, through the polynomial diagonal Koszul theorem). Let $R = Q [x]$ with $\deg x = 2$ and let $F$ be the two-term complex of $R$ -bimodules with $F^{0} = F^{1} = R, d_{F}^{0} = 0,$ concentrated in cohomological degrees $0$ and $1$ . Since $R$ is a polynomial ring in one variable, its diagonal Koszul complex has $H H_{0} (R, R) = R, H H_{1} (R, R) = R {2}, H H_{h} (R, R) = 0 (h \geq 2),$ where the generator of $H H_{1}$ is the Koszul symbol of internal degree $2$ . The termwise Hochschild complex of The termwise Hochschild complex of a Rouquier complex and the groups HHH therefore has $E_{1}^{c, - h} = H H_{h} (R, F^{c}) = H H_{h} (R, R)$ for $(c, h) \in {0, 1}^{2}$ and zero otherwise, all differentials $H H_{h} (R, d_{F}^{c})$ vanish because $d_{F} = 0$ , and the page $E_{2} = E_{\infty}$ carries four labeled free $R$ -generators $(c, h, p) \in {(0, 0, 0), (1, 0, 0), (0, 1, 2), (1, 1, 2)},$ each generating its displayed copy of $R$ or $R {2}$ : the termwise theory retains the full $(c, h)$ bigrading inside the trigrading, with generator degree $p = 2 h$ and further homogeneous classes in degrees $p = 2 h + 2 d$ , $d \geq 0$ , from multiplication by $x^{d}$ . The total Hochschild hyperhomology of Hochschild hyperhomology of a bounded bimodule complex has total degree $n$ term $T^{n} = ⨁_{i - j = n} C_{j} (R, F^{i})$ ; because $d_{F} = 0$ the total complex is the direct sum of the two shifted Hochschild complexes of the columns, and $H H^{h y p e r, n} ≅ {\begin{cases} R {2}, & n = - 1, \\ R \oplus R {2}, & n = 0, \\ R, & n = 1, \\ 0, & otherwise . \end{cases}$ The abutment retains only the total degree $n = c - h$ and the internal degree, together with the induced filtration: the four termwise labels regroup as $(c, h) = (0, 0), (1, 1)$ in total degree $0$ , $(0, 1)$ in degree $- 1$ and $(1, 0)$ in degree $1$ , In this zero-outer-differential example the column decomposition actually gives a canonical splitting $R \oplus R {2}$ in degree $0$ ; the filtration is therefore split here. The declared hyperhomology grading records only $(n, p)$ , although this particular model also retains the column labels through its canonical direct sum. The spectral sequence of Termwise Hochschild spectral sequence of a bounded bimodule complex collapses at $E_{1} = E_{2}$ in this example, and no nonzero higher differential is asserted.

Facts & Assumptions

Given: the field $Q$ , the ring $R = Q [x]$ with $\deg x = 2$ , the complex $F$ with $F^{0} = F^{1} = R$ and zero differential, and AC.

[L1]

The one-variable diagonal Koszul bimodule complex for $R$ is $K (u; R^{e})$ with $u = x^{L} - x^{R}$ , degree-one term $R^{e} {2}$ and degree-zero term $R^{e}$ ; the symbol has internal degree $2$ and the differential preserves internal degree (The polynomial diagonal Koszul bimodule complex).

[L2]

Under AC, $H H_{h} (R, M) ≅ H_{h}$ of the diagonal Koszul complex of The polynomial diagonal Koszul bimodule complex tensored with $M$ , naturally in the $k$ -central $R$ -bimodule $M$ and preserving internal degree (Polynomial Hochschild homology from the diagonal Koszul complex).

[L3]

$F$ is a bounded complex of $Q$ -central graded $R$ -bimodules with differentials of internal degree zero; the maps induced on Hochschild homology by its (zero) differentials make $(H H_{h} (R, F^{∙}), H H_{h} (R, d_{F}^{∙}))$ a cochain complex, whose cohomology is the iterated (termwise) Hochschild homology (Termwise Hochschild homology and iterated homology, The termwise Hochschild complex of a Rouquier complex and the groups HHH).

[L4]

The hyperhomology total complex has $T^{n} = ⨁_{i - j = n, j \geq 0} C_{j} (R, F^{i})$ with differential $D = d_{F} + (- 1)^{i} b$ on the $(i, j)$ summand, and $H H^{h y p e r, n} = H^{n} (T^{∙})$ ; the Hochschild complex $C_{∙} (R, M) = M \otimes_{Q} R^{\otimes_{Q} ∙}$ has the alternating boundary $b$ (Hochschild hyperhomology of a bounded bimodule complex, Hochschild chains and Hochschild homology with coefficients).

[L5]

The spectral sequence of the filtration $F^{p} T^{n} = ⨁_{i \geq p, i - j = n} C_{j} (R, F^{i})$ has $E_{1}^{i, - j} = H H_{j} (R, F^{i})$ , $E_{2}^{i, - j} = H^{i} (H H_{j} (R, F^{∙}))$ , differentials $d_{r} ⁣ : E_{r}^{i, - j} \to E_{r}^{i + r, - j - r + 1}$ , and abuts to the image filtration with $E_{\infty}^{i, - j} ≅ {gr}^{i} H H^{h y p e r, i - j}$ (Termwise Hochschild spectral sequence of a bounded bimodule complex).

[L6]

AC is the choice-function principle (The Axiom of Choice), used only through [L2].

Verification

technique · direct

1.1L1L2L6givenalgebra

Compute $H H_{∙} (R, R)$ . By [L1] the diagonal Koszul complex of $R$ over $R$ has terms $R {2}$ in degree one and $R$ in degree zero, and the differential is multiplication by $u = x^{L} - x^{R}$ , which acts on the regular bimodule $R$ as $x m - m x = 0$ . Hence the complex is $0 \to R {2} \overset{0}{\to} R \to 0$ and [L2] gives $H H_{0} (R, R) = R$ with generator in internal degree $0$ , $H H_{1} (R, R) = R {2}$ with generator in internal degree $2$ (the Koszul symbol contributes the degree), and $H H_{h} (R, R) = 0$ for $h \geq 2$ , since the complex has no terms above degree one. AC is used only here, in [L2].

2.1L3step 1.1algebra

The termwise complex has four free $R$ -generators. By [L3] the termwise complex in Hochschild degree $h$ has terms $H H_{h} (R, F^{c})$ for $c = 0, 1$ ; by step 1.1 these are nonzero exactly for $h \in {0, 1}$ , where they are copies of $R {2 h}$ , with generator degree $2 h$ , and the induced differentials vanish because $d_{F} = 0$ . The page $E_{1}$ therefore has the four displayed free generators, and every higher differential $d_{r} ⁣ : E_{r}^{c, - h} \to E_{r}^{c + r, - h - r + 1}$ vanishes for bidegree reasons: for $r \geq 2$ the target column $c + r \geq 2$ is zero since $F$ is concentrated in degrees $0, 1$ . Hence $E_{2} = E_{\infty} = E_{1}$ and the termwise output is the trigraded object with the four labels $(c, h, p)$ displayed in the Example.

2.2L3L4step 1.1algebra

The total hyperhomology. By [L4] the total complex has $T^{n} = C_{- n} (R, F^{0}) \oplus C_{1 - n} (R, F^{1})$ for $n \leq 1$ and $T^{n} = 0$ for $n \geq 2$ , with $D = + b$ on the $F^{0}$ -summand and $D = - b$ on the $F^{1}$ -summand because $d_{F} = 0$ ; hence $T^{∙}$ is the direct sum of the two column complexes $C_{∙} (R, F^{0})$ and $C_{∙} (R, F^{1}) [- 1]$ in which the differential is $\pm b$ up to the index. Taking homology and using $H H_{j} (R, R) = 0$ for $j \geq 2$ from step 1.1 gives $H^{- 1} = H H_{1} (R, F^{0}) = R {2}$ , $H^{0} = H H_{0} (R, F^{0}) \oplus H H_{1} (R, F^{1}) = R \oplus R {2}$ , $H^{1} = H H_{0} (R, F^{1}) = R$ , and $H^{n} = H H_{- n} (R, F^{0}) \oplus H H_{1 - n} (R, F^{1}) = 0$ for $n \leq - 2$ and $n \geq 2$ ; this is the displayed computation of the Example.

3.1L5step 2.1step 2.2algebra∎

Compare the two outputs and identify the filtration. By [L5] the page $E_{\infty}^{i, - j}$ contributes to $H H^{h y p e r, i - j}$ as ${gr}^{i}$ , so the four classes of step 2.1 regroup by the total degree $n = c - h$ as follows: $(c, h) = (0, 1)$ gives ${gr}^{0}$ of $H H^{h y p e r, - 1}$ , the pairs $(0, 0)$ and $(1, 1)$ give ${gr}^{0}$ and ${gr}^{1}$ of $H H^{h y p e r, 0}$ , and $(1, 0)$ gives ${gr}^{1}$ of $H H^{h y p e r, 1}$ . This matches step 2.2, where the two summands of $H H^{h y p e r, 0} = R \oplus R {2}$ are exactly the two filtered pieces $H H_{0} (R, F^{0})$ and $H H_{1} (R, F^{1})$ : the declared hyperhomology grading uses $(n, p)$ , while in this particular example the vanishing outer differential supplies a canonical additional column decomposition. The termwise theory of step 2.1 keeps $c$ and $h$ separately inside the trigrading $(c, h, p)$ , while the total theory of step 2.2 is declared graded by $n = c - h$ and $p$ , with the indicated split filtration in this example; this is the asserted difference of grading outputs, and no nonzero higher differential occurs by step 2.1.

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Sources

Anna Beliakova, Krzysztof K. Putyra and Stephan M. Wehrli, Quantum link homology via trace functor I, arXiv:1605.03523v2 (85 printed pages); section 3.8.6, equation (3.44), printed p. 39 (standard reference, not scraped)
Mikhail Khovanov, Triply graded link homology and Hochschild homology of Soergel bimodules, arXiv:math/0510265v3; printed pp. 6-7 (standard reference, not scraped)