Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves

When studying a graded module $M$ over the Cox ring of smooth projective toric variety $X$, there are two standard types resolutions commonly used to glean information: free and vector bundle its sheafification. Each approach comes with own challenges. There is geometric information that fail encode, while can resist study using algebraic combinatorial techniques. Recently, Berkesch, Erman, Smith introduced virtual resolutions, which capture desirable also amenable study. The theory includes notion virtually Cohen--Macaulay property, though tools for assessing modules have only recently started be developed. In this paper, we continue research program in related ways. first that, when $X$ product spaces, produce large new class Stanley--Reisner rings, show via explicit constructions appropriate reflecting underlying structure. second an arbitrary develop homological property. Some these give exclusionary criteria, others constructive methods producing suitably short resolutions. We use establish relationships among arithmetically, geometrically, properties.