... the ability to co-operate is more important.

I think, the correct sentence is "the ability of co-operation is more important"
I don't understand, why the structure "noun + to + v + be" is correct.

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    Ability of denotes the agent: the ability of the acrobat, the ability of the watchmaker. Ability to denotes the activity: the ability to whistle, the ability to perform a somersault. – Hugh Nov 23 '15 at 12:44

"Cooperation" is a noun, while "to cooperate" is a verb. At least in American English usage, I cannot think of an example in which "the ability of cooperation" would be correct. Try it with a different word. You wouldn't write, "the ability of application." One would say (or write), "the ability to apply the core principles of the organization," for instance. That said, "the ability to cooperate is more important" is a weak statement in itself. More important than what? Than acting alone? Again, you wouldn't say, "the ability of action is more important."

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