Torsion pairs in silting theory

Abstract

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.

Changes in v2: new Proposition 4.5, weaker assumptions in Lemma 4.8 and some minor changes throughout

References in corpus (2)

• The Ziegler spectrum for derived-discrete algebras
• tCG Torsion Pairs

Cited by in corpus (17)

• Silting objects
• Definability and approximations in triangulated categories
• $t$-Structures on stable derivators and Grothendieck hearts
• $t$-Structures with Grothendieck hearts via functor categories
• Telescope conjecture for homotopically smashing t-structures over commutative noetherian rings
• From weight structures to (orthogonal) $t$-structures and back
• Hearts for commutative noetherian rings: torsion pairs and derived equivalences
• A functorial approach to rank functions on triangulated categories
• Tilting complexes and codimension functions over commutative noetherian rings
• Mutation and torsion pairs
• Singular equivalences to locally coherent hearts of commutative noetherian rings
• Finitely cosilting modules
• Gluing compactly generated t-structures over stalks of affine schemes
• Right triangulated categories: As extriangulated categories, aisles and co-aisles
• Mutation and the Gabriel spectrum
• Definable functors between triangulated categories
• A note on gluing cosilting objects