Patch-density in tensor-triangular geometry
Paul Balmer, Martin Gallauer
TL;DR
The paper develops a patch-density framework in tensor-triangular geometry, showing that a family of tt-functors $\{F_i\}$ jointly detects $\otimes$-nilpotence if and only if the union of their spectral images $\bigcup_i\mathrm{Im}(\mathrm{Spc}(F_i))$ is patch-dense in $\mathrm{Spc}(\mathscr{K})$. It also proves a reconstruction principle: a patch-dense subset with restricted supports determines $\mathrm{Spc}(\mathscr{K})$ via its constructible closure, enabling recovery of tt-geometric information from partial data. The work provides natural patch-dense sources—such as finite/visible loci, retractable limits, profinite equivariance, and support-variety frameworks—that yield practical criteria for nilpotence detection and spectrum-descriptions in diverse settings, including stable homotopy theory, derived categories of profinite groups, and representation theory. Overall, it connects dense spectral subsets to robust detection results and demonstrates how patch-density drives both conceptual understanding and concrete applications in tt-geometry.
Abstract
The spectrum of a tensor-triangulated category carries a compact Hausdorff topology, called the constructible topology, also known as the patch topology. We prove that patch-dense subsets detect tt-ideals and we prove that any infinite family of tt-functors that detects nilpotence provides such a patch-dense subset. We review several applications and examples in algebra, in topology and in the representation theory of profinite groups.