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.
Highlights
In a famous paper, Buchsbaum and Eisenbud present two criteria that completely determine whether or not a chain complex is exact over a Noetherian ring [BE73]
This is done without examining the homology of the complex. These criteria are useful in investigating a module by examining the minimal free resolution. These criteria can be used to study the geometry of projective space
When considering homogeneous B-saturated primes of the Cox ring S of X, we need only consider the homogeneous primes of codimension at most the dimension of X that do not contain any prime components of the irrelevant ideal B
Summary
Buchsbaum and Eisenbud present two criteria that completely determine whether or not a chain complex is exact over a Noetherian ring [BE73]. Any complex of graded free S-modules that is exact will be a virtual resolution. This includes notation, relevant definitions, and some preliminary facts about B-saturated homogeneous prime ideals. These primes are important, because the homogeneous localization of a module at a B-saturated homogeneous prime corresponds to taking the stalk of a sheaf over the toric variety X. It ends with a useful result concerning unbounded virtual resolutions.
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