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Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the Schwartz sense. The de Rham differential is defined on such forms and defines a complex $$0\to \underline{\Omega}^{0,-\infty}_X\overset{d}{\to}\underline{\Omega}^{1,-\infty}_X\overset{d}{\to}\underline{\Omega}^{2,-\infty}_X\overset{d}{\to}\dots$$. It is known to be a resolution of the constant sheaf $\underline{\mathbb{R}}$ (the Poincare lemma).

Let $A\subset X$ be a closed subset. Consider the space of global sections of the above distributional de Rham complex with support contained in $A$. Does the $i$th cohomology of this complex coincide with the $i$th right derived functor of the functor $\Gamma_A$ of global sections supported in $A$?

Remark. If instead of distributional de Rham complex one takes smooth, the answer to the above question would be negative in general.

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    $\begingroup$ This is the dual complex of the graded Lie algebra $\Gamma_c(\bigwedge TM)$ of smooth multivector fields. You could also call them co-currents. The book of DeRham and large parts of geometric measure theory are devoted to them. If $A$ is a submanifold, then this duality breaks down since the distr. forms with support in $A$ also feel transversal directions. Maybe this helps. $\endgroup$ – Peter Michor Nov 6 '17 at 8:06

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