What makes a complex a virtual resolution?

Loper, Michael

doi:10.1090/btran/91

What makes a complex a virtual resolution?
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by Michael C. Loper HTML | PDF

Trans. Amer. Math. Soc. Ser. B 8 (2021), 885-898

Abstract:

Virtual resolutions are homological representations of finitely generated $\text {Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

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