Let $A$ be an $\mathbb{N}$-graded $\Bbbk$-algebra, where $\Bbbk$ is a field, and where $\dim_\Bbbk A_n < \infty$ for all $n \in \mathbb{N}$. I can't see anything preventing me from constructing a ring $\widehat{A}$ whose elements are formal infinite sums $\sum_{n=0}^\infty a_n$, where $a_n \in A_n$, and with the obvious notions of addition and multiplication. An obvious special case is $\widehat{A} = \Bbbk[[x]]$ when $A = \Bbbk[x]$. However, I have a few questions related to this:
- Is the assumption $\dim_\Bbbk A_n < \infty$ for all $n \in \mathbb{N}$ strictly necessary? If $\dim_\Bbbk A_n = \infty$ for some $n$, then the above construction doesn't seem to allow me to consider a formal infinite sum of basis elements of $A_n$. Is there a way around this?
- Are there any general advantages of looking at $\widehat{A}$ rather than $A$? For example, I believe that when $A = \Bbbk[x,y,z]/(xy-z^3)$, we get $\widehat{A} = \Bbbk[[x,y,z]]/(xy-z^3)$, where the finitely generated module category of the former is not Krull-Schmidt but that of the latter is.
- Can this process be extended to an arbitrary ring $R$? At the very least, a closed form for the product of two infinite series doesn't always seem to be possible.
Yes, this is always possible. No assumption on the dimensions are necessary, but you crucially need a $\mathbb{Z}_{\ge 0}$-grading and not a $\mathbb{Z}$-grading for multiplication to be well-defined. As an abelian group this ring is just $\prod_{n \in \mathbb{Z}_{\ge 0}} A_n$ and multiplication can be given by a componentwise formula (essentially the Cauchy product).
Versions of this ring come up in topology and geometry, where certain infinite sums of characteristic classes are used such as the Todd class or Chern character.
I don't know what you mean by extending the process to an arbitrary ring.