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The Poincare lemma on current states that:

If $U$ is a star-shaped open set in $\mathbb R^n$ and $T$ is a $k$-current on $U$ such that $dT=0$, then there is a $k-1$-current $S$ on $U$ such that $dS = T$.

I wish to find a reference for this result. I know that a proof is given in Demailly's "Complex analytic and algebraic geometry", but it seems to me that this is an online note and is not published in book form. I wish to have a "more reliable" reference, such as a published book or some peer-reviewed journals. Thanks in advanced.

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There's the book of de Rham himself, "Differentiable Manifolds, Forms, Currents, Harmonic Forms", which is quite old, but reliable.

In particular, at the beginning on Chapter IV ("Homologies"), he proves that every closed current is "homologous" to a smooth form. The notation is old, but the concepts are essentially the same.

There must be other, more modern sources, but I am not aware of any.

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