Big categories, big spectra
Abstract
We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories is spatial in general, but supposing it is as in the examples we understand, we show it is equipped with a morphism to the Balmer spectrum which detects the failure of the telescope conjecture and we develop the corresponding support theory. The new invariant, the big spectrum, results from taking the entire collection of localizing ideals seriously and considering prime localizing ideals. Although there are, in principle, a proper class of localizing ideals, we are able to prove the existence of at least one big prime lying over every Balmer prime. We conclude with a pair of examples illustrating our constructions.
Cite
@article{arxiv.2109.11934,
title = {Big categories, big spectra},
author = {Scott Balchin and Greg Stevenson},
journal= {arXiv preprint arXiv:2109.11934},
year = {2023}
}
Comments
Updated to address a serious error in the argument that the frame of smashing ideals was spatial in the previous version. Added an appendix explaining why this argument, as given, cannot work. The status of spatiality is that it is again an open problem