Exactness of cech homology theory

ABSTRACT It is known that the Cech homology sequence of a compact pair (X, A) over a compact coefficient group is exact. It is proved that the Cech homology sequence of a compact pair (X, A) over an algebraically compact coefficient group is exact, too. To show this it is proved that if the Cech homology sequence of a pair (X, A) (not necessarily a compact pair) over a coefficient group G is exact and H is a direct summand of the group G, then the Cech homology sequence of the pair (X, A) over the group H is exact, too. In showing this we work in the category InvM(Comp) of inverse systems of chain complexes over a fixed directed set M which is shown to form a category.