A classical theorem of Hopkins, Neeman and
Thomason can be stated in the following conceptual way. For R a
commutative ring, the compactly generated localising subcategories of
the derived category D(R) form a coherent frame, Hochster dual to the
Zariski frame (the frame of radical ideals in R). I'll explain the
statement, contrasting its original formulation with the above
formulation, which belongs to the setting of point-free topology, and
sketch the proof, which exploits cellularisation techniques. Next I'll
explain how also the Zariski frame itself can be realised inside
D(R). Finally I'll comment on the global case, Thomason's theorem for
coherent schemes (i.e. quasi-separated and quasi-compact), whose proof
in this approach is related to recent developments in constructive
algebraic geometry. This is joint work with Wolfgang Pitsch (TAMS
2017).