The local-to-global principle via topological properties of the tensor triangular support (Q6611785)

In this paper, the author establishes generalizations of the local-to-global principle in tensor triangular geometry, working with a rigidly-compactly generated tensor triangulated category \(\mathcal{T}\) that has a monoidal model (hence has well-behaved homotopy colimits). The Balmer spectrum \(\mathrm{Spc}(\mathcal{T}^c)\) of its compact objects is equipped with the inverse topology and the author works with Sanders' tensor triangular support \(\mathrm{Supp} (-)\) [\textit{W. T. Sanders}, ``Support and vanishing for non-Noetherian rings and tensor triangulated categories'', Preprint, \url{arXiv:1710.10199}]. He assumes that \(\mathcal{T}\) has the detection property (i.e., if \(t\) is an object of \(\mathcal{T}\) such that \(\mathrm{Supp} (t) = \emptyset\), then \(t=0\)).\N\NThe main result is a local version of a result of [\textit{C. Zou}, ``Support theories for non-Noetherian tensor triangulated categories'', Preprint, \url{arXiv:2312.08596}]. The author proves that, if \(\mathrm{Supp} (t) \subseteq \mathrm{Spc} (\mathcal{T}^c)\) is weakly-scattered with respect to the inverse topology, then the local-to-global principle holds for \(t\). Explicitly\N\[\N\mathrm{Loc}^{\otimes}\langle t \rangle = \mathrm{Loc}^{\otimes}\langle t \otimes g (W_i) \mid i \in \mathcal{I} \rangle\N\]\Nfor each cover \(\{ W_i \mid i \in \mathcal{I} \}\) of \(\mathrm{Spc}(\mathcal{T}^c)\) by weakly visible subsets, where \(g(W_i)\) is the idempotent associated to \(W_i\).\N\NThis yields a positive response to a question of [\textit{T. Barthel} et al., ``Cosupport in tensor triangular geometry'', Preprint, \url{arXiv:2303.13480}]. Namely, if \(\mathrm{Spc} (\mathcal{T}^c)\) is weakly noetherian and \(\mathrm{Supp} (t)\) is a noetherian subspace, then the local-to-global principle holds for \(t\).\N\NIt also provides a local version of a result of [\textit{G. Stevenson}, J. Algebra 473, 406--429 (2017; Zbl 1428.18023)]: if \(\mathrm{Spc} (\mathcal{T}^c)\) has the constructible topology and \(\mathrm{Supp} (t) \subseteq \mathrm{Spc} (\mathcal{T}^c)\) is a scattered subset, then the local-to-global principle holds for \(t\).\N\NAs an application, the author considers the derived category \(\mathrm{D} (R)\) of an absolutely flat ring \(R\) that is not semi-artinian. He shows that the support of an injective superdecomposable module is contained in the maximal perfect subset of \(\mathrm{Spc}(\mathrm{D} (R)^c) \) and gives an example where the support coincides with this.