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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.

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  • $\begingroup$ Sometimes if you are lucky you can get topological information from the resolution. In this article, the author uses the resolution to compute the Hodge numbers of a Calabi-Yau variety. $\endgroup$ – Fredrik Meyer Oct 3 '16 at 12:19
  • $\begingroup$ @FredrikMeyer Thank you for the link. This paper is quite far beyond my level of comprehension. Could you perhaps point me towards the section that addresses my question? $\endgroup$ – Ethan MacBrough Oct 3 '16 at 19:38
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    $\begingroup$ I noticed now that he skips the calculation in the article. In the thesis (click here), he does the whole computation. However, he uses a resolution of both $I_X$ and $I_X^2$ to do this. The calculation is on page 16. $\endgroup$ – Fredrik Meyer Oct 4 '16 at 9:30

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