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For each object in a tensor triangulated category, we construct a naturalcontinuous map from the object's support---a closed subset of the category'striangular spectrum---to the Zariski spectrum of a certain commutative ring ofendomorphisms. When applied to the unit object this recovers a construction ofP. Balmer. These maps provide an iterative approach for understanding thespectrum of a tensor triangulated category by starting with the comparison mapfor the unit object and iteratively analyzing the fibers of this map via"higher" comparison maps. We illustrate this approach for the stable homotopycategory of finite spectra. In fact, the same underlying construction producesa whole collection of new comparison maps, including maps associated to (anddefined on) each closed subset of the triangular spectrum. These latter mapsprovide an alternative strategy for analyzing the spectrum by iterativelybuilding a filtration of closed subsets by pulling back filtrations of affineschemes.

Higher comparison maps for the spectrum of a tensor triangulated category