I asked the following question on mathstack but didn't receive any comments, so I thought I'd try my luck here.
Let $A$ be an associative algebra over a field $k$. One can define $HH^n (A,A)$ as $ Hom_{D(A^e)} ( A, A[n])$ where $A^e = A \otimes A^o$. So to compute what this is in practice one should take a resolution of $A$ but if we take the bar resolution with respect to $A$ as a left $A$ module it will also be a complex in $C(A^e)$ and so we recover the usual explicit description of the Hochschild cochain complex. But it seems to me that if I take the bar resolution with respect to $A^e$ I get a different complex that doesn't seem to compute the Hochschild cohomology correctly. I think I am making a silly mistake.
