I am learning about an algorithm to construct a projective resolution of the coordinate ring of a variety $X$ as a $k[x_1,...,x_n]$-module using Gröbner bases and syzygies (more specifically, it gives the minimal free resolution). It is possible to compute some topological information from this projective resolution; for instance, one can use it to very easily compute the Hilbert polynomial of $k[X]$, which tells you the dimension, degree, and genus of $X$.
I am curious as to whether there is any finer topological information about $X$ that can be computed using this projective resolution. For instance, are there any interesting right exact functors from $R$-Mod whose left derived functors give topological information? The only right exact functor I can think of is $\otimes$, but the only information I can think of that can be learned from Tor is whether or not $X$ is irreducible. Are there any different functors that provide more interesting information in this setting?
To be more specific, I am really interested in partial calculations of the singular cohomology. I would like some finer notion of the presence of "holes" in $X$ than is given by the genus. However, methods of computing any sort of topological information are welcome.