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For any essentially small triangulated category, the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way, we establish Mayer–Vietoris sequences for pairs of commuting subcategories. [ABSTRACT FROM AUTHOR]

We discuss the notion of strongly unbounded type for abelian length categories; this is closely related to the Second Brauer–Thrall Conjecture for Artin algebras. A new ingredient is the space of characters in the sense of Crawley-Boevey. [ABSTRACT FROM AUTHOR]

Inclusion-preserving maps from modules over an Artin algebra to complete partially ordered sets are studied. This yields a filtration of the Ziegler spectrum that is indexed by all Gabriel–Roiter measures. Another application is a compactness result for the set of subcategories of finitely presented modules that are closed under submodules. [ABSTRACT FROM AUTHOR]