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
through a brief introduction to point-free topology, and
explain how it relates to the original formulation. I'll
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.