By [1]}, the category
\(\operatorname{mf}({\mathbb {A}}^{n+2},\Gamma _{{\bf w}},{\bf w})\)  is the \(n\) -Calabi-Yau
completion \(\Pi _{n}(\operatorname{mf}({\mathbb {A}}^{n+1},\Gamma _{{\bf w}},{\bf w}))\)  (defined in
[2]}; see also [1]}). By Conjecture REF , this is quasi-equivalent
to \(\Pi _{n}(\operatorname{perf}(A))\simeq \operatorname{perf}(\Pi _{n}(A))\) . Thus
\(\operatorname{HH}^*(\operatorname{mf}({\mathbb {A}}^{n+2},\Gamma _{{\bf w}},{\bf w}))\cong \operatorname{HH}^*(\Pi _{n}(A)).\) 

По [1], категория \(\operatorname{mf}({\mathbb {A}}^{n+2},\Gamma _{{\bf w}},{\bf w})\) является \(n\)-Calabi-Yau

завершением \(\Pi _{n}(\operatorname{mf}({\mathbb {A}}^{n+1},\Gamma _{{\bf w}},{\bf w}))\) (определена в

[2]; см. также [1]). По Гипотезе REF, она квазиэквивалентна

\(\Pi _{n}(\operatorname{perf}(A))\simeq \operatorname{perf}(\Pi _{n}(A))\). Таким образом,

\(\operatorname{HH}^*(\operatorname{mf}({\mathbb {A}}^{n+2},\Gamma _{{\bf w}},{\bf w}))\cong \operatorname{HH}^*(\Pi _{n}(A)).\)