Homological support of big objects in tensor-triangulated categories (Q2209257)

Let \(\mathcal{T}\) be a `big' tensor-triangulated category, meaning a rigidly compactly generated one, as in [\textit{P. Balmer} and \textit{G. Favi}, Proc. Lond. Math. Soc. (3) 102, No. 6, 1161--1185 (2011; Zbl 1220.18009)] and \(\mathcal{T}^c\) its subcategory of compact objects. In recent years, new tools have emerged, among which the homological residue fields of [\textit{P. Balmer}, Tunis. J. Math. 2, No. 2, 359--378 (2020; Zbl 1427.18010); \textit{P. Balmer} et al., Sel. Math., New Ser. 25, No. 1, Paper No. 13, 36 p. (2019; Zbl 1409.18011)]. These consist of homological tensor-functors \[ \bar{h}_\mathcal{B}: \mathcal{T}\longrightarrow \bar{\mathcal{A}}_\mathcal{B}\] to various tensor-abelian categories \(\mathcal{A}_\mathcal{B}\). The parameter \(\mathcal{B}\) lives in the homological spectrum \(\mbox{Spc}^h(\mathcal{T}^c)\) defined in [Balmer, loc. cit.]. Using homological residue fields, the author assigns to every object \(X\) of \(\mathcal{T}\) a support \(\mbox{Supp}(X)\) as a subset of \(\mbox{Spc}^{h}(\mathcal{T}^c)\) and proves a tensor-product formula.