BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Claudia Polini (University of Notre Dame) DTSTART:20201106T130000Z DTEND:20201106T140000Z DTSTAMP:20260423T035055Z UID:VCAS/6 DESCRIPTION:Title: The core of monomial ideals\nby Claudia Polini (University of Notre Dame) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\n Let $I$ be a monomial ideal. Even though there may not exist any\nproper r eduction of $I$ which is monomial (or even homogeneous)\, the\nintersectio n of all reductions\, the core\, is again a monomial ideal.\nThe integral closure and the adjoint of a monomial ideal are again\nmonomial ideals and can be described in terms of the Newton\npolyhedron of $I$. Such a descri ption cannot exist for the core\,\nsince the Newton polyhedron only recove rs the integral closure of\nthe ideal\, whereas the core may change when p assing from $I$ to\nits integral closure. When attempting to derive any ki nd of combinatorial\ndescription for the core of a monomial ideal from the known colon\nformulas\, one faces the problem that the colon formula invo lves\nnon-monomial ideals\, unless $I$ has a reduction $J$ generated by a\ nmonomial regular sequence. Instead\, in joint work with Ulrich and\nVitul li we exploit the existence of such non-monomial reductions to\ndevise an interpretation of the core in terms of monomial\noperations. This algorit hm provides a new interpretation of the\ncore as the largest monomial idea l contained in a general locally\nminimal reduction of $I$. In recent join t work with Fouli Montano\, \nand Ulrich we extend this formula to a large class of monomial ideals \nand we study the core of lex-segment monomial ideals generated in one-degree.\n LOCATION:https://researchseminars.org/talk/VCAS/6/ END:VEVENT END:VCALENDAR