  This chapter sets out preliminaries for the duality theory in later chapters.
An underlying idea is that local cohomology functors are higher derived
functors of colocalizations (a.k.a.~coreflections).
  Predominantly well-known facts about cohomology with supports--often under
"finitary" conditions that obtain, e.g., under noetherian hypotheses--and its
local and global interactions with quasi-coherence and with colimits, are
reviewed from both the topological and scheme-theoretic perspectives. Some
refinements of standard results are needed to accommodate certain features
involving unbounded complexes and general systems of supports.
  An important attribute of such cohomology is "tensor-coreflectiveness," in
its avatar--ultimately in the context of closed categories--as "idempotent
pair," a notion which plays an important role in the sequel.
  Some basic facts about linearly topologized noetherian rings and their maps,
related to cohomology with supports, and subsumed under properties of
idempotent pairs, are brought forth; and similarly for the less-familiar
context of formal schemes.
