Symmetric algebra of an ideal and syzygies

Question

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow I\longrightarrow 0$ be a finite presentation of $I.$ What is the relation between symmetric algebra of $I,$ i.e. $Sym(I)$ and the syzygies matrix?

Answer 1

Quite in general, let $M$ be an $R$-module represented as the cockernel of a linear map $f: G\to F$ of free R-moduels $F$ and $G$ of rank $n$ and $m$. Then $\text{Sym}(M)$ is isomorphic (as an R-algebra) to $\text{Sym}(F)/J$ where J is the ideal of $\text{Sym}(F)$ generated by the image of $f$ (the syzygies of $M$). Choosing basis $\text{Sym}(F)$ is isomorphic to $R[x_1,\dots,x_n]$ and $J$ is generated by $m$ elements of degree 1 that correspond to the syzygies of $M$.

These are well known properties of the symmetric algebra, see for example Bourbaki Algebra I: Chapters 1-3.