  For each object in a tensor triangulated category, we construct a natural
continuous map from the object's support---a closed subset of the category's
triangular spectrum---to the Zariski spectrum of a certain commutative ring of
endomorphisms. When applied to the unit object this recovers a construction of
P. Balmer. These maps provide an iterative approach for understanding the
spectrum of a tensor triangulated category by starting with the comparison map
for the unit object and iteratively analyzing the fibers of this map via
"higher" comparison maps. We illustrate this approach for the stable homotopy
category of finite spectra. In fact, the same underlying construction produces
a whole collection of new comparison maps, including maps associated to (and
defined on) each closed subset of the triangular spectrum. These latter maps
provide an alternative strategy for analyzing the spectrum by iteratively
building a filtration of closed subsets by pulling back filtrations of affine
schemes.
