Cotilting Sheaves on Noetherian Schemes

TL;DR

This paper develops a comprehensive theory of cotilting sheaves on Noetherian schemes, linking their properties to torsion pairs, invariance under line bundle twists, and explicit classifications for curves.

Contribution

It introduces a new framework for cotilting objects in Grothendieck categories, characterizes their behavior on Noetherian schemes, and provides explicit descriptions for curves.

Findings

Cotilting objects are always pure-injective.

Cotilting classes are invariant under line bundle twists iff they are closed under injective envelopes.

Explicit classification of cotilting sheaves on curves.

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.