The core of ideals in arbitrary characteristic

The core of zero-dimensional monomial ideals

“…Now instead we assume that the ideal be 0-dimensional in order to assure that the core is a finite intersection of reductions. We then use the results of [23,13,6] to obtain explicit formulas for the core in global rings. Proposition 2.1.…”

Blowups and Fibers of Morphisms

“…It follows that satisfies as well because it suffices to consider prime ideals containing and because the ideal has height . One uses the argument that (2) implies (3) to see that the property of the extended Rees ring implies the bigraded isomorphism By [44, 2.2] and [20, 1.2] the core can be recovered from the canonical module of the extended Rees ring. Therefore, , which is the second equality in (6).…”

“…The fact that the core can detect geometric properties was already apparent in the work of Hyry and Smith [29, 30] and in [20], where the Cayley–Bacharach property of zero-dimensional schemes is characterized in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. The equality (where is the height of the ideal ) has also been investigated by Hyry and Smith [29, 30] in their work on the conjecture of Kawamata.…”

Core of ideals in integral domains

“…In 2007, Polini, Ulrich, and Vitulli [48] gave some remarkable results on the computation of the core of zero-dimensional monomial ideals. In 2008, Fouli, Polini and Ulrich [20] studied the core in arbitrary characteristic and, in 2010, the same authors [21] investigated the annihilators of graded components of the canonical module and the core of standard graded algebras. In this latter paper, for example, the authors characterized Cayley-Bacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring.…”