Cohomology with supports; idempotent pairs

Abstract

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.