The termwise Hochschild complex of a Rouquier complex and the groups HHH

Each differential of $F(\sigma )$ commutes with the Hochschild faces, hence induces a chain map of the Hochschild complexes. By [L1] $d^j$ is a degree-zero map of $R$-bimodules; by [L2] the Hochschild faces are built from the bimodule actions and multiplication in $R$, so $d^j\circ {\delta }_i={\delta }_i\circ d^j$ for every face, and therefore $d^j$ commutes with the boundary $b$ and defines a chain map $C_{\bullet }(R,F(\sigma {)}^j)\to C_{\bullet }(R,F(\sigma {)}^{j+1})$.

The induced maps make a cochain complex in each Hochschild degree. Since $d^{j+1}{d}^j=0$ as bimodule maps, the composite chain map is induced by the zero map, hence is zero and induces the zero map on homology; identities induce identities and composition is respected because the construction is functorial in the coefficient bimodule. By [L3] the family $(H{H}_h(R,F(\sigma {)}^{\bullet }),H{H}_h(R,d^{\bullet }))$ is therefore a cochain complex of graded $\mathbb{Q}$-vector spaces for every $h$, with cohomology $H^j$ as in the definition.

The trigrading is well defined and finite in each degree. The internal grading is preserved by [L1], so all induced Hochschild maps have degree zero. For fixed $h$ and internal degree $d$, the chain group $F(\sigma {)}^j{\otimes }_{\mathbb{Q}}{R}^{\otimes h}$ has finite-dimensional degree-$d$ part: $F(\sigma {)}^j$ is a finite direct sum of shifts of the positive-degree polynomial ring $R$, and the other $h$ factors are the same polynomial ring, so only finitely many monomials of the required total degree occur. Its subquotient $H{H}_h$ and the subsequent bounded cochain homology therefore have finite-dimensional graded pieces. Thus $HH{H}^{j,h,d}$ is defined for every $j\in \mathbb{Z}$, $h\ge 0$, $d\in \mathbb{Z}$; the possible $j$ are bounded by the finite word complex.

Distinguish termwise from total. By [L4] the total hyperhomology complex combines $b$ and $d_F$ into one differential, its homology is the abutment, and the iterated homology of the termwise complex of step 2.1 is the second page of the spectral sequence; consequently the termwise groups carry the separate $(j,h,d)$ trigrading whereas the hyperhomology retains the total degree and the internal degree together with a finite filtration. The definition therefore records the second-page groups, as in the source, and asserts nothing about degeneration of the spectral sequence.