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##

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

.

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** Up:** 3. The Extension of
** Previous:** 3. The Extension of
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