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Better spelling "DeRham", not derham... I can't figure out how to change this... moderators? The cohomology of the complex of differential forms on a smooth manifold with differential given by exterior derivative.
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Leafwise de Rham cohomology (A true definition of differential forms along leaves)
The thing you are confused about is that leafwise differential forms are not invariant under foliation charts. Basic forms are invariant under change of foliation charts and foliated maps (in your not …