  We develop an alternative approach to the homological spectrum through the
lens of definable subcategories. This culminates in a proof that the
homological spectrum is homeomorphic to a quotient of the Ziegler spectrum.
Along the way, we characterise injective objects in homological residue fields
in terms of the definable subcategory corresponding to a given homological
prime. We use these results to give a purity perspective on the relationship
between the homological and Balmer spectrum.
