Cotilting Sheaves on Noetherian Schemes
Abstract
We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme. Finally, we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks.
Cite
@article{arxiv.1707.01677,
title = {Cotilting Sheaves on Noetherian Schemes},
author = {Pavel Čoupek and Jan Šťovíček},
journal= {arXiv preprint arXiv:1707.01677},
year = {2021}
}
Comments
39 pages; version 2: improvements in Section 3 (Theorem 3.10 characterizes cotilting torsion-free classes in arbitrary Grothendieck categories) and Section 6 (we show that any Noetherian scheme has a torsion-free generator), Remark 2.2 and some misprints corrected, references updated