Zero-generic initial ideals

Given a homogeneous ideal I of a polynomial ring and a monomial order , we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to and denote it with . When char , a zero-generic initial ideal is the usual generic initial ideal. We show that is endowed with many interesting properties and, quite surprisingly, it also satisfies Green's Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0.