I am trying to compute $HH^i(kG, kG)$ where $G=C_2$ using that $HH^i(kG, kG) = Ext_i^{(kG)^e} (kG, kG)$, where chark=2.
The first step is to construct an injective resolution of $kG$ as a $kG \otimes_k kG^{op}$ module. Note that $kG^{op} \simeq kG$.
I am a bit stuck after this. Since chark=2, can we simplify $kG \otimes_k kG$? If we take $C_2$ to be generated by $e$ and $g$, then the generators of $kG \otimes_k kG$ would be $e \otimes e$, $e \otimes g$ , $g \otimes e$, and $g \otimes g$. Can we simplify this at all using the characteristic of the field?
Also, could you give me a hint on constructing the injective resolution? I am guessing that we want to use the agumentation map $k C_2 \to k$ sending $g$ and $e$ to $1$.