The shifted Hochschild cochains form a differential graded Lie algebra

Write $\varphi {\circ }_i\psi$ for insertion of a $q$-cochain into slot $i$ of a $p$-cochain, with slots numbered from $0$, so $\varphi \circ \psi ={\sum }_{i=0}^{p-1}(-1{)}^{i(q-1)}\varphi {\circ }_i\psi$. In $(\varphi \circ \psi )\circ \chi -\varphi \circ (\psi \circ \chi )$, terms where $\chi$ is inserted into the block occupied by $\psi$ cancel. If that block starts at slot $i$ and $\chi$ is inserted into its local slot $s$, the first expression has coefficient $(-1{)}^{i(q-1)+(i+s)(r-1)}$, while the second has coefficient $(-1{)}^{i(q+r-2)+s(r-1)}$; these exponents agree. Every remaining term is a pair of insertions into distinct original slots of $\varphi$. For original slots $i<j$, the coefficient with $\psi$ inserted first and $\chi$ second is $(-1{)}^{i(q-1)+(j+q-1)(r-1)}$; with the insertion order swapped it is $(-1{)}^{j(r-1)+i(q-1)}$. Their ratio is $(-1{)}^{(q-1)(r-1)}$. The same comparison for $j<i$ puts the earlier insertion first and gives that same ratio. Thus, with $\mathrm{\mid }\psi \mathrm{\mid }=q-1$ and $\mathrm{\mid }\chi \mathrm{\mid }=r-1$, $\mathrm{as}(\varphi ,\psi ,\chi )=(-1{)}^{\mathrm{\mid }\psi \mathrm{\mid }\mathrm{\mid }\chi \mathrm{\mid }}\mathrm{as}(\varphi ,\chi ,\psi ),\mathrm{as}(u,v,w):=(u\circ v)\circ w-u\circ (v\circ w)\mathrm{.}$

Since $m$ has shifted degree $1$, $[m,m{]}_G(a,b,c)=2((ab)c-a(bc))=0$ by associativity of $A$. Thus $\mathrm{\partial }=[m,-{]}_G$ has degree $+1$.

The bracket is graded skew by its defining formula. Put $e_{uv}=(-1{)}^{\mathrm{\mid }u\mathrm{\mid }\mathrm{\mid }v\mathrm{\mid }}$. Expanding the graded Jacobi identity, the difference between its right and left sides is $\begin{array}{rl}{\displaystyle }& {\displaystyle \mathrm{as}(\varphi ,\psi ,\chi )-e_{\varphi \psi }\mathrm{as}(\psi ,\varphi ,\chi )+e_{\varphi \psi }{e}_{\varphi \chi }\mathrm{as}(\psi ,\chi ,\varphi )-e_{\varphi \psi }{e}_{\varphi \chi }{e}_{\psi \chi }\mathrm{as}(\chi ,\psi ,\varphi )}\\ {\displaystyle }& {\displaystyle +e_{\varphi \chi }{e}_{\psi \chi }\mathrm{as}(\chi ,\varphi ,\psi )-e_{\psi \chi }\mathrm{as}(\varphi ,\chi ,\psi )\mathrm{.}}\end{array}$ Step 1.1 gives $\mathrm{as}(\varphi ,\psi ,\chi )=e_{\psi \chi }\mathrm{as}(\varphi ,\chi ,\psi )$, $\mathrm{as}(\psi ,\varphi ,\chi )=e_{\varphi \chi }\mathrm{as}(\psi ,\chi ,\varphi )$, and $\mathrm{as}(\chi ,\psi ,\varphi )=e_{\varphi \psi }\mathrm{as}(\chi ,\varphi ,\psi )$. Substitution cancels the six terms in three pairs, proving graded Jacobi.

Apply the graded Jacobi identity from step 2.1 with the first two entries equal to $m$. It gives $2{\mathrm{\partial }}^2\varphi =[[m,m{]}_G,\varphi {]}_G=0$, so ${\mathrm{\partial }}^2=0$ over $\mathbb{R}$. Applying the same identity with entries $m,\varphi ,\psi$ gives $\mathrm{\partial }[\varphi ,\psi {]}_G=[\mathrm{\partial }\varphi ,\psi {]}_G+(-1{)}^{\mathrm{\mid }\varphi \mathrm{\mid }}[\varphi ,\mathrm{\partial }\psi {]}_G$. Hence these operations form a differential graded Lie algebra.

Substitute $m$ and $\varphi$ into the insertion formula in [F1]. The two outer insertions give $\varphi (a_1,\dots ,a_p)a_{p+1}$ and $(-1{)}^{p-1}{a}_1\varphi (a_2,\dots ,a_{p+1})$. The internal insertions give $-(-1{)}^{p-1}{\sum }_{i=1}^p(-1{)}^{i-1}\varphi (a_1,\dots ,a_{i-1},a_i{a}_{i+1},a_{i+2},\dots ,a_{p+1})$. Comparing this with the displayed formula for $\delta$ in [F1] gives $\mathrm{\partial }\varphi =(-1{)}^{p-1}\delta \varphi$. Since ${\mathrm{\partial }}^2=0$, this also proves ${\delta }^2=0$.

If a normalized cochain is evaluated on a tuple containing $1$, every insertion term vanishes: the unit either enters the inner cochain, which is normalized, or remains an input of the outer one. In $\delta \varphi$, the only potentially nonzero terms when $a_j=1$ are the two neighboring products (or the corresponding endpoint product and boundary term); they have identical cochain arguments and opposite signs, so cancel. Finally, finite compositions of local finite-order differential operators remain finite-order differential operators by the product and chain rules. Thus the normalized multidifferential cochains are closed under the differential and the bracket.