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Castelnuovo-Mumford regularity; powers of ideal; defect sequence
When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $\mathfrak {m}$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., ${\rm reg}(I^m)=dm+e$ for $m\gg 0$ and some integers $d$, $ e$. This motivates writing ${\rm reg}(I^m)=dm+e_m$ for every $m\geq 0$. The sequence $e_m$, called the \emph {defect sequence} of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after proving various results about the regularity of monomial ideals and their powers, we give several bounds and restrictions on $e_m$ and its first differences when $I$ is a primary monomial ideal. Our theorems extend the previous results about $\mathfrak {m}$-primary ideals in the monomial case. We also use our results to obtatin information about the regularity of powers of a monomial ideal using its primary decomposition. Finally, we study another interesting phenomenon related to the defect sequence, namely that of regularity jump, where we give an infinite family of ideals with regularity jumps at the second power.
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