Let us consider de Rham chain complex $\Omega^{*}_c(\mathbb{R}^n)$ with compact support. There is a well-defined inner product on such a complex - the Hodge inner product, i.e. \begin{equation} \int_{\mathbb{R}^n}*\alpha\wedge\beta, \end{equation} where $\alpha,\beta \in \Omega^{*}_c(\mathbb{R}^n)$. We know that de Rham cohomologies of such a complex is a subspace of $\Omega^{*}_c(\mathbb{R}^n)$ and given by \begin{equation} H^{*}_c(\mathbb{R}^n)=\begin{cases} \mathbb{R}, & \text{in dimension n}.\\ 0, & \text{otherwise}. \end{cases} \end{equation} Naively, it seems that I can restrict the above inner product to a subspace - that is, to a cohomologies and get well-defined pairing between two elements of $H^{*}_c(\mathbb{R}^n)$ or at least on some chosen representatives. But there are just top-forms in $H^{*}_c(\mathbb{R}^n)$ and all pairings are zero. So, the inner product is degenerate.
I can't figure out where my logic breaks. Why does this happen?
Because as I know there exists an inner product between representatives on de Rham cohomologies of non-compact support forms, see e.g. Inner product of De Rham cohomology classes.