Next: 3.2 Regular Extensions Up: 3. The Extension of Previous: 3. The Extension of

## 3.1 Criteria for Regularity

In the case of a graded ring and a graded ideal we obtain a first necessary und sufficient condition for the regularity of over using on the computation of Hilbert series.

Consider the exact sequence

where the left most mapping is the multiplication by the class of in . We denote this mapping by , too. Assume that is regular over , i.e., is not a zero divisor in . Then the mapping is a monomorphism and the Hilbert series of is the difference of those of and , where denotes the module with the degree shifted by . If is not regular the Hilbert series of is different from this difference. Hence, the knowledge of the standard base of and provide us with a criterion for the regularity of .

Let denote the Hilbert serie of the -module .

Lemma 3.2   The element is -regular if and only if

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

Lemma 3.3   The element is -regular if and only if .

PROOF: We have . If is not a zero divisor over then any element of lies in .
Consider the module of all syzygies of . Then the -th component of represents the relative syzygies of over . The ideal generated by the -th component of is just .

Definition 3.4   We define

and call the -th extension ideal of with respect to the sequence .

Next: 3.2 Regular Extensions Up: 3. The Extension of Previous: 3. The Extension of
| ZCA Home | Reports |