Let $K$ be the field of Hahn series in an indeterminate $t$ with exponents in $\mathbb{R}$, coefficients in $\mathbb{F}_2$ and valuation $v$. For each $q\in\mathbb{R}$, we set $$I_q:=\{a\in K:v(a)\geq q\},$$ $$I_{>q}:=\{a\in K:v(a)>q\}$$ and $A:=I_0$. I am new to projective resolutions and want to find them for some simple cases of $A$-modules. It is clear that $$0\to A\to A\to 0$$ is a proj. res. for $A$ and that $$0\to I_q\to A\to A/I_q\to 0$$ is a proj. res. for $A/I_q$ for each $q>0$. I am now trying to find a proj. res. for $A/I_{>q}$, ($q\geq 0$), but the non-principal ideals make things tricky.
For example, $$0\to I_{>q}\to A\to A/I_{>q}\to 0$$ is not a projective resolution since $I_{>q}$ is not projective. What is a way around this?
Since $A$ is local, an $A$-module is projective iff free i.e. a direct sum of copies of $A$. To find a proj. res. for $K$, I presumably need some epimorphism $\alpha:\bigoplus_{n=1}^\infty A\to K$ and need to be able to find $\text{ker}(\alpha)$, but with all such morphisms I find, I cannot find the kernel. Similarly for $I_{>0}$.
Is there a good way of finding such resolutions?