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.

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

References in corpus (3)

• The tilting-cotilting correspondence
• Tilting classes over commutative rings
• On flat generators and Matlis duality for quasicoherent sheaves

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• High Frobenius pushforwards generate the bounded derived category
• Tilting complexes and codimension functions over commutative noetherian rings
• Purity in categories of sheaves
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• The Yoneda Ext and arbitrary coproducts in abelian categories