Let $R=\mathbb{C}[X_1,\ldots,X_r]$ be a polynomial ring and consider a finitely generated and free differential graded $R$-module $M$ with differential $d$. Lets say that the degree of the variables $X_i$ is $2$. We know that the cohomology $H^*(M,d)$ is a finitely generated graded $R$-module.
My question is the following: Looking at $(M,d)$, is there a way to see that we can choose generators of $H^*(M,d)$ below a certain degree $N$ depending on $d$?
Let me explain where I'm coming from: If we are given a matrix $D\in R^{n\times n}$ (where $n=rk(M)$ and $D$ corresponds to $d$), then $Ker(D)$ is a finitely generated submodule of $R^n$. I would like find $N$, depending on the degrees of the entries of $D$, such that I can always choose a set of generators of $Ker(D)$ where the components of my generators are polynomials in degree $<N$.
I feel like if the degrees of entries of $D$ are small, so should the degrees of generators of $Ker(D)$.
