Here are two beliefs. I think everybody will agree that one of them, at least, is false. I adhere to the second one.
Belief 1. There is no simple generalization of the Hodge Theorem to noncompact manifolds.
Belief 2. The most naive statement which would, if true, generalize the Hodge Theorem to noncompact manifolds is this.
The inclusion of the complex of coclosed harmonic forms into the de Rham complex of a riemannian manifold is a quasi-isomorphism.
This statement happens to be true.
Here is a reference:
http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Hodgegaillard/
The simplest example is that of the real line with its standard metric. In degree zero the complex of coclosed harmonic forms is $\mathbb C\oplus\mathbb Cx$, and in degree one it is $\mathbb Cdx$, which gives the right cohomology.
Here is the (trivial) algebra background.
Let $A$ be a module over some unnamed ring, and let $d,\delta$ be two endomorphisms of $A$ satisfying $d^2=0=\delta^2$. Put $\Delta:=d\delta+\delta d$. Assume $A=\Delta A+A_{d,\delta}$ where $A_{d,\delta}$ stands for $\ker d\cap\ker\delta$. Write $A_{\delta,\Delta}$ for $\ker\Delta\cap\ker\delta$.
We claim that the natural map $$f:H(A_{\delta,\Delta},d)\to $H(A_{\delta,\Delta},d)\to H(A,d)$$ between homology modules is bijective.
Injectivity. Assume $\delta da=0$ form some $a$ in $A$. We must find an $x$ in $A_{\delta,\Delta}$ such that $dx=da$. We have $a=\Delta b+c$ for some $b\in A$ and some $c\in A_{d,\delta}$. One easily checks that $x:=\delta db+c$ does the trick.
Surjectivity. Let $a$ be in $\ker d$. We must find $x\in A$, $y\in A_{d,\delta}$ such that $a=dx+y$. We have $a=\Delta b+c$ for some $b\in A$ and some $c\in A_{d,\delta}$. One easily checks that $x:=\delta b$, $y:=\delta db+c$ works.
