Some of the hypothesis on gradedness might be redundant but I explicit them in order to let the context in which I am working clear enough in case that could help.
Let $M$ be an $\mathbb{N}$-graded free $R$-module ($R$ a graded commutative $k$-algebra with $1$ and of infinite cardinality) and suppose that $G:=\{m_{1},\dots, m_{r}\}\in M_{e}$ for some integer $e$ generates $M_{\geq e}$ as an $R$-module. If $G$ is not a basis, then we have some linear dependence relations of the form $$\sum_{i=1}^{r}a_{i}m_{i}=0$$ with some $0\neq a_{i}\in R.$ My question is if there is a meaningful way to coumpute (maybe up to multiplication by a scalar in $R$ or degree) how many of these dependence relations are there.
I would be interested in being able to say something like there is a finite number of them (up to some equivalence) for some concept of finiteness that we could introduce. In general, I am interested in finding up to some interesting equivalence relation how many of these linear dependence relations are there among the elements of the generating set $G.$