Cohomology with supports; idempotent pairs

TL;DR

This chapter reviews foundational concepts of cohomology with supports, emphasizing duality, local-global interactions, and the role of idempotent pairs in the context of topological and scheme-theoretic frameworks.

Contribution

It introduces the concept of idempotent pairs as a key tool in duality theory, extending standard results to unbounded complexes and general support systems.

Findings

Refined results on cohomology with supports under non-noetherian conditions

Introduction of tensor-coreflectiveness and idempotent pairs in cohomology

Analysis of cohomology in formal schemes and topologically noetherian rings

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.