Polarization and Gorenstein liaison

A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen-Macaulay subscheme of Pⁿ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of Pⁿ⁺¹, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley-Reisner complexes. Given a monomial ideal I and a vertex decomposition of the Stanley-Reisner complex of its polarization P(I), we give conditions that allow for the lifting of an associated basic double G-link of P(I) to a basic double G-link of I itself. We use the relationship we develop in the process to show that the Stanley-Reisner complexes of polarizations of artinian monomial ideals and of stable Cohen-Macaulay monomial ideals are vertex decomposable, recovering and strengthening the recent result of Fl{}ystad and Mafi that these complexes are shellable. We then introduce and study polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.