Total and separate Hochschild degrees for a two-term complex

The grading puts $R_d=k{x}^{d\mathrm{/}2}$ for even $d\ge 0$ and $R_d=0$ for odd $d$, so $R$ is graded with $1_R$ of internal degree $0$; the regular bimodule is $k$-central, and by [F2] the shift $R\{4\}$ is again a graded $k$-central $R$-bimodule, with $(R\{4\}{)}_d=R_{d-4}$ and $1$ of internal degree $4$. Hence $F$, with $F^0=R$ in cochain degree $0$, $F^1=R\{4\}$ in cochain degree $1$ and all other terms zero, is a bounded cochain complex of graded $k$-central $R$-bimodules whose differential $d_F=0$ is a degree-zero bimodule map.

By [F3] the termwise groups are $H{H}_j(R,F^i)=H_j(C_{\bullet }(R,F^i))$ with $C_j(R,F^i)=F^i{\otimes }_k{R}^{{\otimes }_{k}j}$, and the induced map $H{H}_j(R,d_F^0)$ is induced by the zero chain map, hence is zero; so by [F4] the termwise complex is the two-term complex $H{H}_j(R,F^0)\stackrel{0}{\to }H{H}_j(R,F^1)$ concentrated in cochain degrees $0$ and $1$, with $H^0=H{H}_j(R,F^0)$, $H^1=H{H}_j(R,F^1)$ and all other cohomology zero.

By [F5] the total complex is $T^n(R,F)={⨁}_{i-j=n,j\ge 0}{C}_j(R,F^i)$ with $D=d_F+(-1{)}^{i}b$; since $d_F=0$ the differential acts on $C_j(R,F^i)$ by $(-1{)}^{i}b$ alone, so $T^{\bullet }(R,F)$ is the direct sum of the two column complexes $(C_{\bullet }(R,F^0),b)$ and $(C_{\bullet }(R,F^1),-b)$, which have the same cycles and the same boundaries, and hence the same homology, as $(C_{\bullet }(R,F^0),b)$ and $(C_{\bullet }(R,F^1),b)$. Therefore $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},n}(R,F)\cong {⨁}_{i-j=n,i\in \{0,1\},j\ge 0}H{H}_j(R,F^i)$ as graded $k$-modules.

Applying [F1] with $n=1$ gives $H{H}_0(R,R)\cong R^{\left(\genfrac{}{}{0px}{}{1}{0}\right)}\{0\}=R$ and $H{H}_1(R,R)\cong R^{\left(\genfrac{}{}{0px}{}{1}{1}\right)}\{2\}=R\{2\}$, while $H{H}_j(R,R)=0$ for $j>1$.

By [F3] and [F7] the chain module $C_j(R,R\{4\})=(R\{4\}){\otimes }_k{R}^{{\otimes }_{k}j}$ is identified with $C_j(R,R)\{4\}=(R{\otimes }_k{R}^{{\otimes }_{k}j})\{4\}$ by the identity on elementary tensors, a degree-zero $k$-linear isomorphism; the faces of [F3] use only the left action, the right action and the multiplication of $R$, none of which the shift changes, so the identification intertwines the Hochschild boundaries and induces $H{H}_j(R,R\{4\})\cong H{H}_j(R,R)\{4\}$ for every $j$ as graded $k$-modules.

Combining steps 1.4 and 1.5 with $F^0=R$ and $F^1=R\{4\}$: $H{H}_0(R,F^0)\cong R$ and $H{H}_1(R,F^0)\cong R\{2\}$, while $H{H}_0(R,F^1)\cong H{H}_0(R,R)\{4\}\cong R\{4\}$ and $H{H}_1(R,F^1)\cong H{H}_1(R,R)\{4\}\cong R\{2\}\{4\}=R\{6\}$ by [F2]; moreover $H{H}_j(R,F^i)=0$ for $j>1$ and $i=0,1$, so these four groups are the only nonzero termwise groups.

The four nonzero termwise groups sit in bidegree $(i,j)$ with their free $R$-generators in internal degrees $0,2,4,6$: $(0,0)$ has $R$, $(0,1)$ has $R\{2\}$, $(1,0)$ has $R\{4\}$, and $(1,1)$ has $R\{6\}$. Since the total index of [F5] is $n=i-j$, they project to total cochain degrees $0$, $-1$, $1$ and $0$ respectively, placing $R$ and $R\{6\}$ in total degree $0$, $R\{2\}$ in total degree $-1$ and $R\{4\}$ in total degree $1$.

By step 1.3 the hyperhomology is the direct sum of the termwise groups computed in step 3.1, so $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},0}(R,F)\cong R\oplus R\{6\}$, $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},-1}(R,F)\cong R\{2\}$ and $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},1}(R,F)\cong R\{4\}$, with all other total degrees zero; the two free generators in total degree $0$ have internal degrees $0$ and $6$.

By [F6] the filtration yields a spectral sequence with $E_1^{i,-j}=H{H}_j(R,F^i)$ and $E_2^{i,-j}=H^i(H{H}_j(R,F^{\bullet }))$, and step 1.2 identifies the second page as $E_2^{0,0}=R$, $E_2^{0,-1}=R\{2\}$, $E_2^{1,0}=R\{4\}$, $E_2^{1,-1}=R\{6\}$ and zero elsewhere; every differential $d_r$ with $r\ge 2$ has empty target because $E_2$ is supported only in the columns $i=0,1$, so $E_2=E_{\mathrm{\infty }}$. The abutment of [F6] then reads ${\mathrm{gr}}^{0}H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},0}=R$, ${\mathrm{gr}}^{1}H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},0}=R\{6\}$, ${\mathrm{gr}}^{0}H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},-1}=R\{2\}$ and ${\mathrm{gr}}^{1}H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},1}=R\{4\}$. The degree-zero extension is split because the zero differential makes the total complex the direct sum of the two Hochschild column complexes, as shown in 1.3; their free generators have internal degrees $0$ and $6$.

Collecting: the four nonzero termwise groups are $H{H}_0(R,F^0)\cong R$, $H{H}_1(R,F^0)\cong R\{2\}$, $H{H}_0(R,F^1)\cong R\{4\}$ and $H{H}_1(R,F^1)\cong R\{6\}$, with free-generator degrees $0,2,4,6$ at $(i,j)=(0,0),(0,1),(1,0),(1,1)$, projecting to total cochain degrees $0,-1,1,0$; the zero differential makes the total complex a direct sum of the two Hochschild complexes, giving $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},0}\cong R\oplus R\{6\}$, $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},-1}\cong R\{2\}$ and $H{H}^{\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r},1}\cong R\{4\}$ with all other total degrees zero, and the $E_2$ page equals the $E_{\mathrm{\infty }}$ page. In contrast to this split situation, for a general bounded $F$ the cochain-index pieces only filter the hyperhomology, as in [F6], where neither degeneration of the spectral sequence nor a splitting of the $(i,j)$ pieces is asserted.