3.1 Criteria for Regularity

In the case of a graded ring $R$ and a graded ideal $I$ we obtain a first necessary und sufficient condition for the regularity of $f_i$ over $I_{i-1}$ using on the computation of Hilbert series.

Consider the exact sequence

$$R/I_{i-1}\rightarrow R/I_{i-1}\rightarrow R/I_i\rightarrow 0,$$

where the left most mapping is the multiplication by the class of $f_i$ in $R/I_{i-1}$. We denote this mapping by $f_i$, too. Assume that $f_i$ is regular over $I_{i-1}$, i.e., $f_i$ is not a zero divisor in $R/I_{i-1}$. Then the mapping $f_i$ is a monomorphism and the Hilbert series of $R/I_i$ is the difference of those of $R/I_{i-1}$ and $(R/I_{i-1})[deg(f_i)]$, where $(R/I_{i-1})[deg(f_i)]$ denotes the module $R/I_{i-1}$ with the degree shifted by $deg(f_i)$. If $f_i$ is not regular the Hilbert series of $R/I_i$ is different from this difference. Hence, the knowledge of the standard base of $I_{i-1}$ and $I_i$ provide us with a criterion for the regularity of $f_i$.

Let $h_M(t)$ denote the Hilbert serie of the $R$-module $M$.

Lemma 3.2   The element $f_i\in R$ is $I_{i-1}$-regular if and only if

$$h_{R/I_i}(t)=h_{R/I_{i-1}}(t)-t^{-deg(f_i)}h_{R/I_{i-1}}(t).$$

There is a second, simple, but, nevertheless very usefull criterion (not assuming gradedness).

Lemma 3.3   The element $f_i\in R$ is $I_{i-1}$-regular if and only if $(I_{i-1}:f_i)=I_{i-1}$.

PROOF: We have $I_{i-1}\subset (I_{i-1}:f_i)$. If $f_i$ is not a zero divisor over $I_{i-1}$ then any element of $(I_{i-1}:f_i)$ lies in $I_{i-1}$.$\Box$
Consider the module $M_i=Syz(I_i)\subset R^i$ of all syzygies of $I_i$. Then the $i$-th component of $M_i$ represents the relative syzygies of $f_i$ over $R/I_{i-1}$. The ideal generated by the $i$-th component of $M_i$ is just $(I_{i-1}:f_i)$.

Definition 3.4   We define

$$J_i:=J(f_i,I_{i-1}):=(I_{i-1}:f_i)$$

and call $J_i$ the $i$-th extension ideal of $I$ with respect to the sequence $\{f_1,\ldots,f_n\}$.