What makes a complex a virtual resolution?

Abstract

Virtual resolutions are homological representations of finitely generated Pic ( X ) \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.

Virtual resolutions of points in P1×P1

“…In their original work [2], Berkesch, Erman, and Smith gave multiple strategies for constructing virtual resolutions. Loper [15] identified two algebraic conditions that characterize when a chain complex is virtual. Kennedy [13] gave an equivalent algebraic definition for a complex to be a virtual resolution.…”

“…By focusing on this family, we can take advantage of many known algebraic and geometric properties about them, e.g., as developed in [6,8,9,11,12]. At the same time, we complement Gao, Li, Loper, and Matto's paper [15] which studies points in P 1 × P 1 that are virtual complete intersections. The contribution of this manuscript is two-fold, as we now describe.…”