The core of monomial ideals

Abstract: Let $I$ be a monomial ideal. Even though there may not exist any proper reduction of $I$ which is monomial (or even homogeneous), the intersection of all reductions, the core, is again a monomial ideal. The integral closure and the adjoint of a monomial ideal are again monomial ideals and can be described in terms of the Newton polyhedron of $I$. Such a description cannot exist for the core, since the Newton polyhedron only recovers the integral closure of the ideal, whereas the core may change when passing from $I$ to its integral closure. When attempting to derive any kind of combinatorial description for the core of a monomial ideal from the known colon formulas, one faces the problem that the colon formula involves non-monomial ideals, unless $I$ has a reduction $J$ generated by a monomial regular sequence. Instead, in joint work with Ulrich and Vitulli we exploit the existence of such non-monomial reductions to devise an interpretation of the core in terms of monomial operations. This algorithm provides a new interpretation of the core as the largest monomial ideal contained in a general locally minimal reduction of $I$. In recent joint work with Fouli Montano, and Ulrich we extend this formula to a large class of monomial ideals and we study the core of lex-segment monomial ideals generated in one-degree.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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