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Abstract A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen–Macaulay subscheme of 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 , 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 and a vertex decomposition of the Stanley–Reisner complex of its polarization , we give conditions that allow for the lifting of an associated basic double G‐link of to a basic double G‐link of itself. We use the relationship we develop in the process to show that the Stanley–Reisner complexes of polarizations of stable Cohen– Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner 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. 

There is an abundance of deep literature on the use of free resolutions to study modules and vector bundle resolutions to study coherent sheaves. When studying a module over the Cox ring of a smooth projective toric variety X, each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which are amenable to algebraic and combinatorial study and also capture desirable geometric information. In this extended abstract, we continue this program in the combinatorially-rich Stanley–Reisner setting. In particular, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings. After augmenting the simplicial complexes associated to these Stanley–Reisner rings with a coloring that reflects the product structure on X, our primary tool is Reisner’s criterion, whose conclusion we interpret in the virtual setting. We also provide two constructions of short virtual resolutions for use beyond the Stanley–Reisner case.