3. The Extension of free Resolutions

Let $R$ be a polynomial ring over some base field $k$ and $I=(f_1,\ldots,f_n)$ an ideal in $R$. Denote by $I_i=(f_1,\ldots,f_i)$, $i=0,1,\ldots,n$ the (sub-)ideal generated by the first $i$ elements of $\{f_1,\ldots,f_n\}$. We obtain a chain of ideals

$$0=I_0\subset I_1\subset\ldots\subset I_{n-1}\subset I_n=I,$$

where each ideal differs from the preceeding by just one generator. We may further assume that $f_i\in R\setminus I_{i-1}$.

Definition 3.1   An element $f_i\in R$ is called $I_{i-1}$-regular if $f_i$ does not represent a zero divisor of $R/I_{i-1}$. A sequence of elements $\{f_1,\ldots,f_n\}$ is called regular if each $f_i$ is $I_{i-1}$-regular.