Theorem 2.5 in Conca and Herzog, Castelnuovo-Mumford regularity of products of ideals http://arxiv.org/abs/math/0210065, says that if $R$ is a polynomial ring over a field $k$, $I$ a homogeneous ideal of $R$ such that $\dim R /I \le 1$ and $M$ is a finitely generated, graded $R$-module, then $reg(IM) \le reg(M) + reg(I)$, where $reg(\cdot)$ denotes Castelnuovo-Mumford regularity.

Definition: An element $x \in R$ is called almost regular on $M$, if it is not a zero divisor of $M / H_m^0(M)$.

The proof begins by the statement "let $x \in R_1$ be almost regular on $M, M/IM, R/I$". The proof does not seem to treat the case where no such $x$ exists, i.e. no element of $R_1$ is almost regular on $M/IM$. I suspect this is true because the statement of the theorem holds true in that case.

Question: Suppose that no element of $R_1$ is almost regular on $M /IM$. Then why is it true that $reg(IM) \le reg(M) + reg(I)$?


1 Answer 1


The point is that if the base field is infinite (as one can assume after a field extension) a general enough linear form is almost regular. This follows from the so called prime avoidance.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .