I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and there's something I'm trying to prove for what I may need some help.
More precisely, is this statement from Example 1.2:
"Dually, $A$ is a $K[G]$-comodule algebra if and only if $A$ is a $G$-graded algebra, that is, $A= \bigoplus_{g\in G} A_g$, where the $A_g$ are $K$-subspaces of $A$ and $A_gA_h\subseteq A_{gh}$, for all $g,h\in G$. To see this, note that the $K[G]$-comodule algebra structure is given by $$\rho: A\to A\otimes_K K[G] \qquad \text{via} \qquad a\mapsto a\otimes g,$$ for each homogeneus element $a\in A_g$. Moreover the subalgebra of coinvariants is $A^{\text{co}K[G]}= A_1$, the identity component."
I am trying to prove $A$ has $K[G]$-comodule algebra structure when $A=\bigoplus_{g\in G} A_g$. Given $H$ a Hopf algebra and $A$ an algebra with $1$ over the field $K$, the definition this paper gives for comodule algebra is the following:
$A$ is a right $H$-comodule algebra if $A$ is a counital right $H$-comodule, via $\delta_A(a)\mapsto \sum_aa_0\otimes a_1\in A\otimes_KH$, such that $\delta$ is an algebra map.
My first problem is that I'm not sure if I get what this definition means. Does "algebra map" mean "algebra homomorphism"? What is exactly $\delta$? The structure map of the $H$-comodule structure? At first I thought it was, and I managed to prove $A$ is $K[G]$-comodule with the given $\rho$, doing the following:
$$(id\otimes \Delta)\circ \rho \left(\sum_{g\in G} a_g\right) = (id\otimes \Delta) \left(\sum_{g\in G} a_g\otimes g\right) = \sum_{g\in G} a_g\otimes g\otimes g$$ $$(\rho \otimes id)\circ \rho \left(\sum_{g\in G} a_g\right) = (\rho \otimes id)\left(\sum_{g\in G} a_g\otimes g \right) = \sum_{g\in G} a_g\otimes g\otimes g$$
$$(id\otimes \varepsilon)\circ \rho \left(\sum_{g\in G} a_g\right) = (id\otimes \varepsilon)\left(\sum_{g\in G} a_g\otimes g \right) = \sum_{g\in G} a_g\otimes 1 \cong \sum_{g\in G} a_g$$
And it's trivial (I think) that $\rho$ is $K$-linear, so then $A$ is $H$-comodule via that $\rho$. How can I end the proof? What do I have to check to prove it's $H$-comodule algebra? What is that $\delta$? Is it $\rho$? Any help or hint will be appreciated, thanks in advance.