I have a commutative unital $\mathbb{F}_2$-algebra $Q$ of countably infinite dimension, and I want to see if it has a $\mathbb{Z}$-grading where $Q_i=0$ for $i<0$ and $Q_0=\mathbb{F}_2$ and $Q_i$ is finite for $i>0$. I also impose the condition $q^2\in\{0,1\}$ for all $q\in Q$.
I know that the exterior algebra $E$ on countably infinitely many generators $\{x_i\}_{i\geq 0}$ is a commutative unital $\mathbb{F}_2$-algebra and has such a $\mathbb{Z}$-grading (also $x^2\in\{0,1\}$ for all $x\in E$), see Proposition 1.5 of this paper. This made me wonder if it is possible in a more general setting.
It is clear that we can choose a countably infinite basis for $Q$, but not clear how this can be used to form a grading. How might I go about this?
If such a grading existed, then in particular there is a surjection $Q\to Q_0$, the quotient by the ideal $Q_{>0}:=\bigoplus_{i>0}Q_i$. A $\mathbb F_2$-algebra need not admit such a surjection to a finite-dimensional $\mathbb F_2$-algebra.
For example, if $Q=\overline{\mathbb F_2}$ is the algebraic closure of $\mathbb F_2$, then $Q$ is a field, so the only surjection it has is to itself: $Q=Q$.