Spectrum of some triangulated categories

A local-global principle for the telescope conjecture

“…The question is especially important when X has a coarse moduli scheme X. If moreover X is smooth, a recent paper of Dubey and Mallick [14] together with a positive answer to the question would produce a classification of all ⊗-closed smashing localizations of D qc (X): they would be in bijection with specialization closed subsets of X. In particular, the theorems and statements of this section all lead to classification theorems.…”

“…Proof. The coarse moduli space of X is X, so by [14] there is an isomorphism Spc D perf (X) Spc D perf (X), where Spc denotes the spectrum of Balmer [4]. This means that there is a bijection between the thick ⊗-ideals in these two ⊗-triangulated categories.…”

Perfect complexes on algebraic stacks

“…Theorem A, as well as the theory developed to establish it, have been used to classify the thick tensor ideals in [Hal16] (generalizing work of [Kri09, DM12]), to resolve the telescope conjecture for algebraic stacks [HR17] (extending [Ant14]), and for results on dg-enhancements [CS16, BLS16].…”

Filtrations Via Tensor Actions

“…In [1] Paul Balmer has introduced tensor triangular geometry providing a unified framework in which to view the various classification theorems for thick tensor ideals of tensor triangulated categories in fields varying from algebraic geometry to KK-theory. Since then the study of the spectrum of a tensor triangulated category has taken on a life of its own; new examples have been computed [9], [10], the abstract theory has been further developed (of course with applications) [3], [2], and the framework has been extended to apply to a wider variety of examples [5], [18]. The current work is concerned with extending some of the abstract theory developed by Balmer in [2] to the context of tensor actions.…”