Say, you have an ideal $I$ of a polynomial ring $R = K\lbrack X_1,\ldots,X_n \rbrack$ over an algebraically closed field $K$ (you can assume $K = \mathbb{C}$). What does a minimal free resolution of $R/I$ as an $R$-module tell me about the variety defined by $I$? Anything at all?

I know that this question is very similar but the answers were not satisfying for me as they essentially reduce to graded settings. I really do not want to assume that $I$ is homogenous. In this case a minimal free resolution is not unique and so the "Betti numbers" (i.e., the ranks of the terms in a minimal free resolution) are not unique. Are they still interesting?

To get something unique I could still do the following two things:

Localize $R/I$ in some point. Then minimal free resolutions and the Betti numbers are unique. But what do they tell me about $R/I$? How do they relate to the non-localized ones?

I could homogenize $I$ (i.e., consider the projective closure of the variety). Again minimal free resolutions and the Betti numbers are unique. But what do they tell me about $R/I$? How do they relate to the non-projectivized ones?

I know there is a bunch of literature on free resolutions and syzygies (book by Eisenbud for example) but usually everything is restricted to the graded setting.