# Free resolutions of affine (non-projective!) varieties

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:

1. 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?

2. 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.

• what do these Betti numbers tell you in the graded setting? – pro Jun 5 '15 at 8:36
• Quote from Eisenbud's book: "Hilbert originally studied free resolutions because their discrete invariants, the graded Betti numbers, determine the Hilbert function. But the graded Betti numbers contain more information than the Hilbert function. [...]" – user74608 Jun 5 '15 at 8:42
• thanks. (additional comment: in general having a free resolution is useful to compute Tor and thus intersection numbers with other subvarieties) – pro Jun 5 '15 at 9:34
• The length of the resolution is an important invariant. If the length of the resolution equals the height of the ideal, then the quotient ring is Cohen-Macaulay. In particular, for Cohen-Macaulay quotient rings of codimension 2, this is the basis for the Hilbert-Schaps(-Burch) theorem. – Jason Starr Jun 5 '15 at 12:12