$A$ is $k$-algebra, $A$-bimodule-$A$ structure on $k$

Question

Let $A$ is $k$-algebra ($k= \mathbb{C}$ e.g.), does it exists $A$-bimodule-$A$ structure on $k$? My question emerged when I reading this proof in which assumed that $k$ is $A \otimes A^{o}$ module.

Answer 1

No, if $A$ is a $k$-algebra, then $k$ usually doesn't even have an $A$-module structure. For instance, if $A$ is a proper field extension of $k$, then there is no homomorphism of $k$-algebras $A\to k$ (and if $A$ has greater cardinality than $k$, there is not even any ring-homomorphism $A\to k$).

In the answer you linked, though, $A$ is assumed to be much more than just a $k$-algebra: it is assumed to be a graded connected $k$-algebra. This means $A=\bigoplus_{n\in\mathbb{N}}A_n$ is a graded $k$-algebra such that the algebra structure map $k\to A$ is an isomorphism between $k$ and $A_0$. There is then a natural quotient map of algebras $A\to A_0\cong k$ which sends $A_n$ to $0$ for all $n>0$. Via this map, $k$ can be considered as an $A$-bimodule. Explicitly, given $a\in A$, it acts on $k$ on either side by multiplication by the degree $0$ part of $a$ (which is just an element of $A_0\cong k$).