On a class of monomial ideals


  • Keivan Borna Kharazmi University
  • Raheleh Jafari IPM, School of Mathematics, Tehran, Iran.


Associated primes, Castelnuovou-Mumford regularity, powers of ideals, primary ideals, local cohomology


Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC if there exists a prime ideal $\fp\in\Ass_S\,S/I$ for which $\h(\fp)$ equals to the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I=\cap^k_{i=1}Q_i$ be the irreducible decomposition of $I$ and let $m(I)=\max\{|Q_i|-\h(Q_i) ,\; 1\leq i\leq k \}$ where $|Q_i|$ denotes the total degree of $Q_i$. Then it can be seen that when $I$ is primary, $\reg(S/I)=m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\reg(S/I) = m(I)$ provided that $|\Ass_S\,S/I|\leq2$. We also prove that $m(I^n)\leq n\max\{|Q_i| ,\; 1\leq i\leq~k\}-\h(I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\reg(S/(I+w^dS)^n)\leq\reg(S/I)+nd-1$ for all $n\geq 1$ and $d$ is large enough. Finally we implement an algorithm for computation of $m(I)$. 10.1017/S0004972712001037