I don't think so. If A is commutative and of finite global dimension, say n, finitely generated as k-algebra, then by HKR you have $HH_n(A)\cong\Omega^n(A)$, and you want it to ve isomorphic to $A$ as an $A$-module.This is more or less the same as saying "every manifold is orientable". However, I would like to see an example

I don't think so. If A is commutative and of finite global dimension, say n, finitely generated as k-algebra, then by HKR you have $HH_n(A)\cong\Omega^n(A)$, and you want it to be isomorphic to $A$ as an $A$-module. This is more or less the same as saying "every manifold is orientable". However, I would like to see an example

I don't think so. If A is commutative and of finite global dimension, say n, finitely generated as k-algebra, then by HKR you have $HH_n(A)\cong\Omega^n(A)$, and you want it to ve isomorphic to $A$ as an $A$-module.This is more or less the same as saying "every manifold is orientable"

I don't think so. If A is commutative and of finite global dimension, say n, finitely generated as k-algebra, then by HKR you have $HH_n(A)\cong\Omega^n(A)$, and you want it to ve isomorphic to $A$ as an $A$-module.This is more or less the same as saying "every manifold is orientable". However, I would like to see an example