Further remarks on systems of interlocking exact sequences

In a system of interlocking sequences, the assumption that threeout of the four sequences are exact does not guarantee theexactness of the fourth. In 1967, Hilton proved that,with the additional condition that it is differential at thecrossing points, the fourth sequence is also exact. In this paper,we trace such a diagram and analyze the relation between thekernels and the images, in the case that the fourth sequence isnot necessarily exact. Regarding the exactness of the fourthsequence, we remark that the exactness of the other threesequences does guarantee the exactness of the fourth atnoncrossing points. As to a crossing point p, we needthe extra criterion that the fourth sequence is differential. One notices that the condition, for thefourth sequence, that kernel ⊇ image at p turns out to be equivalent to the “opposite” condition kernel ⊆ image. Next, for the kernel and the image at p of the fourth sequence,even though they may not coincide, they are not fardifferent—they always have the same cardinality as sets, andbecome isomorphic after taking quotients by a subgroup which iscommon to both. We demonstrate these phenomena with an example