Topology Seminar, 
2-3 PM Tuesday Jan 21, 2025,
Nelson Room, 1146 Faculty Administration Bldg.
Speaker:  Anish Chedalavada, Johns Hopkins University

Title: A Derived Refinement of a Classical Theorem in TT-Geometry

Abstract: The Balmer spectrum of a tensor-triangulated category is the space of its prime thick tensor ideals, much like the spectrum of a ring; its open sets roughly classify Verdier localizations of the category, analogous to the way that the spectrum of a ring classifies localizations of a ring. Using some higher category theory, it is possible to equip this space with a structure sheaf and perform gluing/descent arguments as one would do in algebraic geometry. As an application, we show that the Balmer spectrum of (the homotopy category of) a symmetric monoidal, rigid, idempotent complete stable infinity-category (henceforth 2CAlg) is naturally locally spectrally ringed, and that there is a  natural map of locally spectrally ringed spaces $$y: Spc(Perf_X) to X$$ where X is a (potentially nonconnective) qcqs spectral scheme. We also show that $$Map_{2CAlg}(Perf_X, K) = Map_{LRS}(Spc(K), X)$$ after passing to spectra and postcomposing with the map y. This will demonstrate the existence of spectral refinements of some well-known support varieties in tt-geometry; as one potential use-case, this provides descent spectral sequences based on the cohomology of the underlying variety. If time permits, we will try to explain the argument and indicate some open avenues that remain. Parts of the above are joint work with Ko Aoki, Tobias Barthel, Tomer Schlank, and Greg Stevenson.