Point-free topology and Hochster duality in derived categories
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 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.