I'm trying to compute/find simple examples the relative Hochschild cohomology of an extension of algebras. Let $$B \subseteq A$$ be an extension of algebras. Firstly there is the relative bar complex which Hochschild defined in his original article: $$ \dotsb \rightarrow A^{\otimes_{B}n + 2} \rightarrow \dotsb \rightarrow A^{\otimes_B 2} \rightarrow 0 $$ where we tensor over $B$ instead of $k$ as in the non-relative case. Now to get relative Hochschild cohomology we can apply $\mathrm{Hom}(-, A)$ to this complex. As in the non-relative case I get the feeling that trying to compute using the bar complex is quite difficult/not feasible. Secondly one can also define what we mean by a relative projective resolution of $A$ as follows. We say that an exact sequence of $A$-$A$-bimodules $$\dotsb \rightarrow C_3 \rightarrow C_2 \rightarrow C_1 \rightarrow C_0 \rightarrow A \rightarrow 0$$ is a $A^e|B^e$-relative projective resolution if
- Relative exact: $\ker(d)$ is a direct summand of $C_i$ as a $B$-$B$-bimodule.
- Relative projective: We say that an $A$-$A$-bimodule $M$ is $A^e|B^e$-relative projective if for any module $N$ and any map $M \rightarrow N$ this map factor via any relative exact map $$L \rightarrow N \rightarrow 0$$ This is equivalent to M being isomorphic to a summand of $A^e \otimes_{B^e} C$ for some $B$-$B$-bimodule $C$.
Then the relative Hochschild cohomology is relative $\mathrm{Ext}_{(A^e|B^e)}(A,A)$. Does anybody know any easy(-ish) examples to just get a hang on the relative theory? It doesn't necessarily have to be Hochschild, cyclic (co)homology also works. I haven't found any concrete computation in any of the articles/books I read.
Thank you very much for any help!
