Cosupport for tensor triangulated categories (Q6642556)

By constructing local homology and cohomology functors on a triangulated category \(\mathcal{T}\), \textit{D. Benson} et al. [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 575--621 (2008; Zbl 1171.18007); J. Reine Angew. Math. 673, 161--207 (2012; Zbl 1271.18012)] proposed a new method for defining the notions of support and cosupport for objects in \(\mathcal{T}\). Suitably specialized approach recovers, for example, the support theory of \textit{H.-B. Foxby} [J. Pure Appl. Algebra 15, 149--172 (1979; Zbl 0411.13006)] and \textit{A. Neeman} [Topology 31, No. 3, 519--532 (1992; Zbl 0793.18008)] for commutative noetherian rings, the theory of \textit{L. L. Avramov} [Invent. Math. 96, No. 1, 71--101 (1989; Zbl 0677.13004)] \textit{L. L. Avramov} and \textit{R.-O. Buchweitz} [Invent. Math. 142, No. 2, 285--318 (2000; Zbl 0999.13008)] and for complete intersection local rings, and varieties for representations of finite groups according to \textit{D. J. Benson} et al. [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597--615 (1996; Zbl 0888.20003)]. \textit{T. Barthel} et al. [``Cosupport in tensor triangular geometry'', Preprint, \url{arXiv:2303.13480}] developed a theory of cosupport in tensor triangular geometry and provided a conceptual foundation for cosupport as categorically dual to support. In general, the theory of cosupport is not completely satisfactory as its construction is not well-understood compared to support.\N\NBy creating a dual functor of localization, the so-called colocalization functor, in a compactly generated tensor triangulated category \(\mathcal{T}\), the authors develop a theory of cosupport that is dual to that of support in \(\mathcal{T}\). They further show that there are many properties, similar or dual, between cosupport and support in \(\mathcal{T}\). For instance, the cosupport of an object can be detected by the cosupport of its cohomology; or the cosupport of a non-zero object is non-empty. Also some characterizations, properties, and relations between cosupport and support are given.