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I'm currently trying to learn about syzygies. Most material is written in the context of graded rings and/or graded modules but I'm interested in a specific question about local rings. Hence I need to translate results from one context to another.

To ask a precise question: suppose you have a finitely generated module $M$ over a (noetherian) local ring $(A,\mathfrak{m},k)$. In this case one can also speak about the associated graded objects: $N=gr_{\mathfrak{m}}M$ over $G=gr_{\mathfrak{m}}A$. Suppose you know the (minimal) free resolutions $\{F_s=A^{\oplus\beta_s}\}$ of $M$ and $\{H_i=\oplus_j G(-j)^{\beta_{i,j}}\}$ of $N$. Here the $\beta_{i,j}$ are graded Betti numbers which represent the (minimal) number of generators for $H_i$ in degree $j$ and (I would like to say (see (2), below)) the $\beta_{s}$ are the total(?) Betti numbers.

1) How do these two free resolutions relate?

2) Is it true - at least - something like the formulae $\beta_s=\sum_j\beta_{s,j}$?

I would also appreciate if someone could indicate a (well-known?) reference doing this kind of translations. Thank you.

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1) Your question is somewhat broad, but in some sense the general answer is "they don't", or at least that the relationship is not well understood in general, and I think constructing a minimal graded free resolution of $\operatorname{gr}_{\mathfrak{m}}(M)$ is quite a hard problem. Given a minimal free resolution $F_{\bullet}$ of $M$ over a local ring, one can construct an associated graded complex $\operatorname{gr}_{\mathfrak{m}} (F_{\bullet})$ of free modules, but it is rarely acyclic. In fact, $\operatorname{gr}_{\mathfrak{m}}(F_{\bullet})$ is a minimal graded free resolution of $\operatorname{gr}_{\mathfrak{m}}(M)$ if and only if $M$ is Koszul (this is sometimes taken as the definition of Koszul). See this paper for a reference on Koszul modules.

2) No. For example, let $S=k[\![x,y,z]\!]$ with $\mathfrak{n}=(x,y,z)$ and consider the $S$-module $R=S/I$, where $I$ is the ideal $I=(xz-y^3,yz-x^4,z^2-x^3y^2)$; one can prove $R \cong k[\![t^4,t^5,t^{11}]\!]$. This is a classical example of a domain of dimension $1$ whose associated graded ring is not Cohen-Macaulay, and it can be found, for instance, as Example 1.3 in these notes. Concretely, one can show that $\operatorname{gr}_{\mathfrak{n}}(R) \cong k[x,y,z]/(xz,yz,z^2,y^4)$. We observe that $z$ is a nonzero socle element of $\operatorname{gr}_{\mathfrak{n}}(R)$, so $\operatorname{gr}_{\mathfrak{n}}(R)$ has depth $0$. In particular, the Auslander-Buchsbaum formula tells us that the minimal graded $\operatorname{gr}_{\mathfrak{n}}(S)$-free resolution of $\operatorname{gr}_{\mathfrak{n}}(R)$ does not even have the same length as the minimal $S$-free resolution of $M$. That is, these modules have different projective dimensions.

The takeaway from these things is that the associated graded operation does not behave well with taking homology or with homological properties.

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