On the stability and regularity of the multiplier ideals of monomial ideals

Let a ? ?[x ₁, . . . , x _d ] be a monomial ideal and J (a) its multiplier ideal which is also a monomial ideal. It is proved that if a is strongly stable or squarefree strongly stable then so is J (a). Denote the maximal degree of minimal generators of a by d(a). When a is strongly stable or squarefree strongly stable, it is shown that the Castelnuovo-Mumford regularity of J (a) is less than or equal to d(a). As a corollary, one gets a vanishing result on the ideal sheaf](widetilde {mathcal{J}left( a right)}) on ? ^d–1 associated to J (a) that H ⁱ(? ^d–1;(widetilde {mathcal{J}left( a right)})(s–i)) = 0, for all i > 0 and s ≥ d(a).