Cotilting sheaves on Noetherian schemes
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Mathematische Zeitschrift
Cotilting sheaves on Noetherian schemes 1 · Jan Štˇ ovíˇcek2 ˇ Pavel Coupek
Received: 16 September 2017 / Accepted: 8 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
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 pureinjective 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 of quasi-coherent sheaves 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. Keywords Grothendieck category · Cotilting objects · Pure-injective objects · Noetherian scheme · Classification
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Cotilting objects in Grothendieck categories . . . . . . . . . . . 3 Pure-injectivity of cotilting objects . . . . . . . . . . . . . . . . 4 Derived equivalences . . . . . . . . . . . . . . . . . . . . . . . . 5 Torsion pairs in categories of sheaves . . . . . . . . . . . . . . . 6 Classification of cotilting sheaves . . . . . . . . . . . . . . . . . Appendix A. Ext-functors and products in abelian categories . . . . Appendix B. Quasi-coherent sheaves on locally Noetherian schemes B.1 Injective sheaves on locally Noetherian schemes . . . . . . .
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ˇ Pavel Coupek was partially supported by the institutional grant SVV 260456 of the Charles University and partially supported by the Ross Fellowship of Purdue University. Jan Šˇtovíˇcek was supported by Neuron Fund for Support of Science.
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Jan Šˇtovíˇcek [email protected]
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Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA
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Department of Algebra, Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague 8, Czech Republic
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ˇ P. Coupek, J. Štˇovíˇcek B.2 Supports and associated points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 The closed monoidal structure on sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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