Algebraic Topology · Problem 17

Cohomology — Allen Hatcher

Mathematics — Topology · direct-limits-homology

Problem

Show that a direct limit of exact sequences is exact. More generally, show that homology commutes with direct limits: If

\{C_\alpha, f_{\alpha\beta}\}

is a directed system of chain complexes, with the maps

f_{\alpha\beta}: C_\alpha \to C_\beta

chain maps, then

H_n(\varinjlim C_\alpha) = \varinjlim H_n(C_\alpha)